Integrand size = 28, antiderivative size = 400 \[ \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\frac {8 a^2 \text {arctanh}(\sin (c+d x))}{b^6 d}+\frac {\text {arctanh}(\sin (c+d x))}{2 b^4 d}+\frac {2 \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{b^6 d}+\frac {4 a^3 \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 \sqrt {a^2+b^2} d}+\frac {3 a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 b^4 \sqrt {a^2+b^2} d}+\frac {6 a \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {4 a \sec (c+d x)}{b^5 d}-\frac {a^2+b^2}{3 b^3 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {3 a (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac {4 a^2}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac {2 \left (a^2+b^2\right )}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}+\frac {\sec (c+d x) \tan (c+d x)}{2 b^4 d} \] Output:
8*a^2*arctanh(sin(d*x+c))/b^6/d+1/2*arctanh(sin(d*x+c))/b^4/d+2*(a^2+b^2)* arctanh(sin(d*x+c))/b^6/d+4*a^3*arctanh((b*cos(d*x+c)-a*sin(d*x+c))/(a^2+b ^2)^(1/2))/b^6/(a^2+b^2)^(1/2)/d+3/2*a*arctanh((b*cos(d*x+c)-a*sin(d*x+c)) /(a^2+b^2)^(1/2))/b^4/(a^2+b^2)^(1/2)/d+6*a*(a^2+b^2)^(1/2)*arctanh((b*cos (d*x+c)-a*sin(d*x+c))/(a^2+b^2)^(1/2))/b^6/d-4*a*sec(d*x+c)/b^5/d-1/3*(a^2 +b^2)/b^3/d/(a*cos(d*x+c)+b*sin(d*x+c))^3+3/2*a*(b*cos(d*x+c)-a*sin(d*x+c) )/b^4/d/(a*cos(d*x+c)+b*sin(d*x+c))^2-4*a^2/b^5/d/(a*cos(d*x+c)+b*sin(d*x+ c))-2*(a^2+b^2)/b^5/d/(a*cos(d*x+c)+b*sin(d*x+c))+1/2*sec(d*x+c)*tan(d*x+c )/b^4/d
Time = 2.89 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.34 \[ \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=-\frac {\sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (4 b^3 \left (a^2+b^2\right )+18 b^2 \left (a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b \sin (c+d x))+6 b \left (12 a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2+48 a b (a \cos (c+d x)+b \sin (c+d x))^3+\frac {60 a \left (4 a^2+3 b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right ) (a \cos (c+d x)+b \sin (c+d x))^3}{\sqrt {a^2+b^2}}+30 \left (4 a^2+b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^3-30 \left (4 a^2+b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^3-\frac {3 b^2 (a \cos (c+d x)+b \sin (c+d x))^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {48 a b \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^3}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {3 b^2 (a \cos (c+d x)+b \sin (c+d x))^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {48 a b \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^3}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{12 b^6 d (a+b \tan (c+d x))^4} \] Input:
Integrate[Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]
Output:
-1/12*(Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])*(4*b^3*(a^2 + b^2) + 18*b^2*(a^2 + b^2)*Sin[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x]) + 6*b *(12*a^2 + b^2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + 48*a*b*(a*Cos[c + d* x] + b*Sin[c + d*x])^3 + (60*a*(4*a^2 + 3*b^2)*ArcTanh[(-b + a*Tan[(c + d* x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/Sqrt[a^2 + b^ 2] + 30*(4*a^2 + b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3 - 30*(4*a^2 + b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3 - (3*b^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (48*a*b*Sin[( c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(Cos[(c + d*x)/2] - Sin[( c + d*x)/2]) + (3*b^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(Cos[(c + d*x)/ 2] + Sin[(c + d*x)/2])^2 - (48*a*b*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Si n[c + d*x])^3)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(b^6*d*(a + b*Tan[c + d*x])^4)
Time = 4.38 (sec) , antiderivative size = 782, normalized size of antiderivative = 1.96, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {3042, 3585, 3042, 3573, 3042, 3555, 3042, 3553, 219, 3573, 3042, 3553, 219, 3585, 3042, 3555, 3042, 3553, 219, 3573, 3042, 3553, 219, 3583, 3042, 3553, 219, 4255, 3042, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^4}dx\) |
| \(\Big \downarrow \) 3585 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4}dx}{b^2}+\frac {\int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3573 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {a \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {a \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3555 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {a \left (\frac {\int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}+\frac {\int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {a \left (\frac {\int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}+\frac {\int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {a \left (-\frac {\int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{2 d \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}+\frac {\int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3573 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \sec (c+d x)dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {a \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3585 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {2 a \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \sec ^3(c+d x)dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 3555 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (\frac {\int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (\frac {\int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (-\frac {\int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{2 d \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 a \left (-\frac {2 a \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 3573 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}-\frac {2 a \left (-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \sec (c+d x)dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \sec (c+d x)dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}-\frac {2 a \left (-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (-\frac {2 a \left (\frac {a \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (\frac {a \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (-\frac {2 a \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 3583 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (-\frac {2 a \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \sec (c+d x)dx}{b^2}+\frac {\sec (c+d x)}{b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \sec (c+d x)dx}{b^2}+\frac {\sec (c+d x)}{b d}\right )}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (-\frac {2 a \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\sec (c+d x)}{b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\sec (c+d x)}{b d}\right )}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}+\frac {\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {2 a \left (\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\frac {\sec (c+d x)}{b d}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\left (a^2+b^2\right ) \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}-\frac {2 a \left (\frac {\sec (c+d x)}{b d}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\left (a^2+b^2\right ) \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}\right )}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (-\frac {2 a \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}+\frac {\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {2 a \left (\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \left (-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}\right )}{b^2}+\frac {\frac {\sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \sec (c+d x)dx}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}+\frac {\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {2 a \left (\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}}{b^2}\right )}{b^2}+\frac {\frac {\frac {\sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\left (a^2+b^2\right ) \left (\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \left (-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}\right )}{b^2}}{b^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}+\frac {\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}+\frac {\text {arctanh}(\sin (c+d x))}{b^2 d}}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {2 a \left (\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}+\frac {\text {arctanh}(\sin (c+d x))}{b^2 d}\right )}{b^2}+\frac {-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a \text {arctanh}(\sin (c+d x))}{b^2 d}+\frac {\sec (c+d x)}{b d}}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}+\frac {\text {arctanh}(\sin (c+d x))}{b^2 d}\right )}{b^2}-\frac {2 a \left (-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a \text {arctanh}(\sin (c+d x))}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}+\frac {\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}}{b^2}}{b^2}\) |
Input:
Int[Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]
Output:
((a^2 + b^2)*(-1/3*1/(b*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (a*(-1/2* ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]]/((a^2 + b^2)^(3 /2)*d) - (b*Cos[c + d*x] - a*Sin[c + d*x])/(2*(a^2 + b^2)*d*(a*Cos[c + d*x ] + b*Sin[c + d*x])^2)))/b^2 + (ArcTanh[Sin[c + d*x]]/(b^2*d) + (a*ArcTanh [(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]* d) - 1/(b*d*(a*Cos[c + d*x] + b*Sin[c + d*x])))/b^2))/b^2 - (2*a*((-((a*Ar cTanh[Sin[c + d*x]])/(b^2*d)) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^2*d) + Sec[c + d*x]/(b*d))/b^2 + ((a ^2 + b^2)*(-1/2*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]] /((a^2 + b^2)^(3/2)*d) - (b*Cos[c + d*x] - a*Sin[c + d*x])/(2*(a^2 + b^2)* d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)))/b^2 - (2*a*(ArcTanh[Sin[c + d*x]] /(b^2*d) + (a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/ (b^2*Sqrt[a^2 + b^2]*d) - 1/(b*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))))/b^2) )/b^2 + ((-2*a*(-((a*ArcTanh[Sin[c + d*x]])/(b^2*d)) - (Sqrt[a^2 + b^2]*Ar cTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^2*d) + Sec[c + d*x]/(b*d)))/b^2 + ((a^2 + b^2)*(ArcTanh[Sin[c + d*x]]/(b^2*d) + (a*ArcT anh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^ 2]*d) - 1/(b*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))))/b^2 + (ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d))/b^2)/b^2
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-d^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x _Symbol] :> Simp[(b*Cos[c + d*x] - a*Sin[c + d*x])*((a*Cos[c + d*x] + b*Sin [c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[(n + 2)/((n + 1)*(a^ 2 + b^2)) Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && NeQ[n, -2]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_)/co s[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(a*Cos[c + d*x] + b*Sin[c + d*x])^ (n + 1)/(b*d*(n + 1)), x] + (Simp[1/b^2 Int[(a*Cos[c + d*x] + b*Sin[c + d *x])^(n + 2)/Cos[c + d*x], x], x] - Simp[a/b^2 Int[(a*Cos[c + d*x] + b*Si n[c + d*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[cos[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-Cos[c + d*x]^(m + 1)/(b*d*(m + 1) ), x] + (-Simp[a/b^2 Int[Cos[c + d*x]^(m + 1), x], x] + Simp[(a^2 + b^2)/ b^2 Int[Cos[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) / ; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
Int[cos[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a^2 + b^2)/b^2 Int[Cos[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^n, x], x] + (Simp[1/b^2 I nt[Cos[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] - Simp[ 2*(a/b^2) Int[Cos[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] & & LtQ[m, -1]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 4.89 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-b -8 a}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-20 a^{2}-5 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{6}}+\frac {\frac {2 \left (\frac {b^{2} \left (9 a^{4}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 a}+\frac {b \left (12 a^{6}-39 a^{4} b^{2}-4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a^{2}}-\frac {b^{2} \left (108 a^{6}-57 a^{4} b^{2}-4 a^{2} b^{4}-4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a^{3}}-\frac {b \left (12 a^{6}-50 a^{4} b^{2}-9 a^{2} b^{4}-2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a^{2}}+\frac {b^{2} \left (63 a^{4}+10 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {b \left (36 a^{4}+5 a^{2} b^{2}+2 b^{4}\right )}{6}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{3}}-\frac {5 a \left (4 a^{2}+3 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}}{b^{6}}-\frac {1}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-b +8 a}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (20 a^{2}+5 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{6}}}{d}\) | \(452\) |
| default | \(\frac {\frac {1}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-b -8 a}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-20 a^{2}-5 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{6}}+\frac {\frac {2 \left (\frac {b^{2} \left (9 a^{4}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 a}+\frac {b \left (12 a^{6}-39 a^{4} b^{2}-4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a^{2}}-\frac {b^{2} \left (108 a^{6}-57 a^{4} b^{2}-4 a^{2} b^{4}-4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a^{3}}-\frac {b \left (12 a^{6}-50 a^{4} b^{2}-9 a^{2} b^{4}-2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a^{2}}+\frac {b^{2} \left (63 a^{4}+10 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {b \left (36 a^{4}+5 a^{2} b^{2}+2 b^{4}\right )}{6}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{3}}-\frac {5 a \left (4 a^{2}+3 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}}{b^{6}}-\frac {1}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-b +8 a}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (20 a^{2}+5 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{6}}}{d}\) | \(452\) |
| risch | \(-\frac {360 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}+240 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}+60 a^{4} {\mathrm e}^{i \left (d x +c \right )}-105 a^{2} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+20 a^{2} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-300 i a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}-60 i a \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-150 i a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}+250 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+20 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-105 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}+60 a^{4} {\mathrm e}^{9 i \left (d x +c \right )}-15 b^{4} {\mathrm e}^{9 i \left (d x +c \right )}+240 a^{4} {\mathrm e}^{7 i \left (d x +c \right )}+20 b^{4} {\mathrm e}^{7 i \left (d x +c \right )}+60 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+150 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+22 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+20 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-15 b^{4} {\mathrm e}^{i \left (d x +c \right )}+300 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{3} b^{5} d}+\frac {10 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{b^{6} d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 b^{4} d}+\frac {10 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}\, d \,b^{6}}+\frac {15 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}\, d \,b^{4}}-\frac {10 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}\, d \,b^{6}}-\frac {15 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}\, d \,b^{4}}-\frac {10 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{b^{6} d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 b^{4} d}\) | \(661\) |
Input:
int(sec(d*x+c)^3/(a*cos(d*x+c)+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(1/2/b^4/(tan(1/2*d*x+1/2*c)-1)^2-1/2*(-b-8*a)/b^5/(tan(1/2*d*x+1/2*c) -1)+1/2/b^6*(-20*a^2-5*b^2)*ln(tan(1/2*d*x+1/2*c)-1)+2/b^6*((1/2*b^2*(9*a^ 4+2*b^4)/a*tan(1/2*d*x+1/2*c)^5+1/2*b*(12*a^6-39*a^4*b^2-4*b^6)/a^2*tan(1/ 2*d*x+1/2*c)^4-1/3/a^3*b^2*(108*a^6-57*a^4*b^2-4*a^2*b^4-4*b^6)*tan(1/2*d* x+1/2*c)^3-1/a^2*b*(12*a^6-50*a^4*b^2-9*a^2*b^4-2*b^6)*tan(1/2*d*x+1/2*c)^ 2+1/2/a*b^2*(63*a^4+10*a^2*b^2+2*b^4)*tan(1/2*d*x+1/2*c)+1/6*b*(36*a^4+5*a ^2*b^2+2*b^4))/(tan(1/2*d*x+1/2*c)^2*a-2*b*tan(1/2*d*x+1/2*c)-a)^3-5/2*a*( 4*a^2+3*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*d*x+1/2*c)-2*b)/(a^2 +b^2)^(1/2)))-1/2/b^4/(tan(1/2*d*x+1/2*c)+1)^2-1/2*(-b+8*a)/b^5/(tan(1/2*d *x+1/2*c)+1)+1/2/b^6*(20*a^2+5*b^2)*ln(tan(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 820 vs. \(2 (378) = 756\).
Time = 0.17 (sec) , antiderivative size = 820, normalized size of antiderivative = 2.05 \[ \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)^3/(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="fricas" )
Output:
1/12*(6*a^2*b^5 + 6*b^7 - 30*(4*a^6*b - 3*a^4*b^3 - 8*a^2*b^5 - b^7)*cos(d *x + c)^4 - 20*(11*a^4*b^3 + 13*a^2*b^5 + 2*b^7)*cos(d*x + c)^2 + 15*((4*a ^6 - 9*a^4*b^2 - 9*a^2*b^4)*cos(d*x + c)^5 + 3*(4*a^4*b^2 + 3*a^2*b^4)*cos (d*x + c)^3 + ((12*a^5*b + 5*a^3*b^3 - 3*a*b^5)*cos(d*x + c)^4 + (4*a^3*b^ 3 + 3*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 + b^2)*log((2*a*b*cos( d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 - 2*a^2 - b^2 - 2*sqrt( a^2 + b^2)*(b*cos(d*x + c) - a*sin(d*x + c)))/(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)) + 15*((4*a^7 - 7*a^5*b^2 - 14*a^ 3*b^4 - 3*a*b^6)*cos(d*x + c)^5 + 3*(4*a^5*b^2 + 5*a^3*b^4 + a*b^6)*cos(d* x + c)^3 + ((12*a^6*b + 11*a^4*b^3 - 2*a^2*b^5 - b^7)*cos(d*x + c)^4 + (4* a^4*b^3 + 5*a^2*b^5 + b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(sin(d*x + c) + 1) - 15*((4*a^7 - 7*a^5*b^2 - 14*a^3*b^4 - 3*a*b^6)*cos(d*x + c)^5 + 3*( 4*a^5*b^2 + 5*a^3*b^4 + a*b^6)*cos(d*x + c)^3 + ((12*a^6*b + 11*a^4*b^3 - 2*a^2*b^5 - b^7)*cos(d*x + c)^4 + (4*a^4*b^3 + 5*a^2*b^5 + b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-sin(d*x + c) + 1) - 30*(10*(a^5*b^2 + a^3*b^4)*co s(d*x + c)^3 + (a^3*b^4 + a*b^6)*cos(d*x + c))*sin(d*x + c))/((a^5*b^6 - 2 *a^3*b^8 - 3*a*b^10)*d*cos(d*x + c)^5 + 3*(a^3*b^8 + a*b^10)*d*cos(d*x + c )^3 + ((3*a^4*b^7 + 2*a^2*b^9 - b^11)*d*cos(d*x + c)^4 + (a^2*b^9 + b^11)* d*cos(d*x + c)^2)*sin(d*x + c))
\[ \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{4}}\, dx \] Input:
integrate(sec(d*x+c)**3/(a*cos(d*x+c)+b*sin(d*x+c))**4,x)
Output:
Integral(sec(c + d*x)**3/(a*cos(c + d*x) + b*sin(c + d*x))**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 936 vs. \(2 (378) = 756\).
Time = 0.14 (sec) , antiderivative size = 936, normalized size of antiderivative = 2.34 \[ \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)^3/(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="maxima" )
Output:
-1/6*(2*(60*a^7 + 5*a^5*b^2 + 2*a^3*b^4 + 6*(55*a^6*b + 5*a^4*b^3 + a^2*b^ 5)*sin(d*x + c)/(cos(d*x + c) + 1) - 2*(120*a^7 - 280*a^5*b^2 - 25*a^3*b^4 - 6*a*b^6)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2*(510*a^6*b - 105*a^4*b ^3 + 2*a^2*b^5 - 4*b^7)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2*(180*a^7 - 635*a^5*b^2 - 65*a^3*b^4 - 18*a*b^6)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 2*(540*a^6*b - 195*a^4*b^3 - 2*a^2*b^5 - 8*b^7)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 6*(40*a^7 - 140*a^5*b^2 - 5*a^3*b^4 - 6*a*b^6)*sin(d*x + c)^ 6/(cos(d*x + c) + 1)^6 - 2*(210*a^6*b - 75*a^4*b^3 + 2*a^2*b^5 - 4*b^7)*si n(d*x + c)^7/(cos(d*x + c) + 1)^7 + 3*(20*a^7 - 45*a^5*b^2 - 4*a*b^6)*sin( d*x + c)^8/(cos(d*x + c) + 1)^8 + 6*(5*a^6*b + a^2*b^5)*sin(d*x + c)^9/(co s(d*x + c) + 1)^9)/(a^6*b^5 + 6*a^5*b^6*sin(d*x + c)/(cos(d*x + c) + 1) + 6*a^5*b^6*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - a^6*b^5*sin(d*x + c)^10/(c os(d*x + c) + 1)^10 - (5*a^6*b^5 - 12*a^4*b^7)*sin(d*x + c)^2/(cos(d*x + c ) + 1)^2 - 8*(3*a^5*b^6 - a^3*b^8)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2 *(5*a^6*b^5 - 18*a^4*b^7)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*(9*a^5*b ^6 - 4*a^3*b^8)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 2*(5*a^6*b^5 - 18*a^ 4*b^7)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 8*(3*a^5*b^6 - a^3*b^8)*sin(d *x + c)^7/(cos(d*x + c) + 1)^7 + (5*a^6*b^5 - 12*a^4*b^7)*sin(d*x + c)^8/( cos(d*x + c) + 1)^8) - 15*(4*a^2 + 3*b^2)*a*log((b - a*sin(d*x + c)/(cos(d *x + c) + 1) + sqrt(a^2 + b^2))/(b - a*sin(d*x + c)/(cos(d*x + c) + 1) ...
Time = 0.22 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.37 \[ \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)^3/(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="giac")
Output:
1/6*(15*(4*a^2 + b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^6 - 15*(4*a^2 + b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^6 + 15*(4*a^3 + 3*a*b^2)*log(ab s(2*a*tan(1/2*d*x + 1/2*c) - 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^6) + 6*(b*tan(1/2* d*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1/2*c)^2 + b*tan(1/2*d*x + 1/2*c) - 8*a )/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*b^5) + 2*(27*a^6*b*tan(1/2*d*x + 1/2*c)^ 5 + 6*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 36*a^7*tan(1/2*d*x + 1/2*c)^4 - 117 *a^5*b^2*tan(1/2*d*x + 1/2*c)^4 - 12*a*b^6*tan(1/2*d*x + 1/2*c)^4 - 216*a^ 6*b*tan(1/2*d*x + 1/2*c)^3 + 114*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 + 8*a^2*b^ 5*tan(1/2*d*x + 1/2*c)^3 + 8*b^7*tan(1/2*d*x + 1/2*c)^3 - 72*a^7*tan(1/2*d *x + 1/2*c)^2 + 300*a^5*b^2*tan(1/2*d*x + 1/2*c)^2 + 54*a^3*b^4*tan(1/2*d* x + 1/2*c)^2 + 12*a*b^6*tan(1/2*d*x + 1/2*c)^2 + 189*a^6*b*tan(1/2*d*x + 1 /2*c) + 30*a^4*b^3*tan(1/2*d*x + 1/2*c) + 6*a^2*b^5*tan(1/2*d*x + 1/2*c) + 36*a^7 + 5*a^5*b^2 + 2*a^3*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 - 2*b*tan(1/2* d*x + 1/2*c) - a)^3*a^3*b^5))/d
Time = 20.14 (sec) , antiderivative size = 1961, normalized size of antiderivative = 4.90 \[ \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\text {Too large to display} \] Input:
int(1/(cos(c + d*x)^3*(a*cos(c + d*x) + b*sin(c + d*x))^4),x)
Output:
(atanh((4000*a^3*tan(c/2 + (d*x)/2))/(1000*a*b^2 + 4000*a^3) + (1000*a*tan (c/2 + (d*x)/2))/(1000*a + (4000*a^3)/b^2))*(20*a^2 + 5*b^2))/(b^6*d) - (( 60*a^4 + 2*b^4 + 5*a^2*b^2)/(3*b^5) + (2*tan(c/2 + (d*x)/2)^9*(5*a^4 + b^4 ))/(a*b^4) + (2*tan(c/2 + (d*x)/2)^6*(6*b^6 - 40*a^6 + 5*a^2*b^4 + 140*a^4 *b^2))/(a^2*b^5) - (2*tan(c/2 + (d*x)/2)^7*(210*a^6 - 4*b^6 + 2*a^2*b^4 - 75*a^4*b^2))/(3*a^3*b^4) + (2*tan(c/2 + (d*x)/2)^2*(6*b^6 - 120*a^6 + 25*a ^2*b^4 + 280*a^4*b^2))/(3*a^2*b^5) - (2*tan(c/2 + (d*x)/2)^3*(510*a^6 - 4* b^6 + 2*a^2*b^4 - 105*a^4*b^2))/(3*a^3*b^4) - (2*tan(c/2 + (d*x)/2)^4*(18* b^6 - 180*a^6 + 65*a^2*b^4 + 635*a^4*b^2))/(3*a^2*b^5) - (tan(c/2 + (d*x)/ 2)^8*(4*b^6 - 20*a^6 + 45*a^4*b^2))/(a^2*b^5) + (2*tan(c/2 + (d*x)/2)*(55* a^4 + b^4 + 5*a^2*b^2))/(a*b^4) + (2*tan(c/2 + (d*x)/2)^5*(9*a^2 - 4*b^2)* (60*a^4 + 2*b^4 + 5*a^2*b^2))/(3*a^3*b^4))/(d*(tan(c/2 + (d*x)/2)^2*(12*a* b^2 - 5*a^3) - a^3*tan(c/2 + (d*x)/2)^10 - tan(c/2 + (d*x)/2)^8*(12*a*b^2 - 5*a^3) - tan(c/2 + (d*x)/2)^4*(36*a*b^2 - 10*a^3) + tan(c/2 + (d*x)/2)^6 *(36*a*b^2 - 10*a^3) - tan(c/2 + (d*x)/2)^3*(24*a^2*b - 8*b^3) - tan(c/2 + (d*x)/2)^7*(24*a^2*b - 8*b^3) + tan(c/2 + (d*x)/2)^5*(36*a^2*b - 16*b^3) + a^3 + 6*a^2*b*tan(c/2 + (d*x)/2) + 6*a^2*b*tan(c/2 + (d*x)/2)^9)) - (a*a tan(((a*(4*a^2 + 3*b^2)*(a^2 + b^2)^(1/2)*((8*(25*a^2*b^9 + 200*a^4*b^7 + 400*a^6*b^5))/b^14 + (8*tan(c/2 + (d*x)/2)*(50*a*b^11 + 650*a^3*b^9 + 1600 *a^5*b^7 + 800*a^7*b^5))/b^15 - (5*a*(4*a^2 + 3*b^2)*(a^2 + b^2)^(1/2)*...
Time = 0.24 (sec) , antiderivative size = 3036, normalized size of antiderivative = 7.59 \[ \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int(sec(d*x+c)^3/(a*cos(d*x+c)+b*sin(d*x+c))^4,x)
Output:
(120*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2) )*cos(c + d*x)*sin(c + d*x)**4*a**7*i - 270*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**4*a**5 *b**2*i - 270*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a** 2 + b**2))*cos(c + d*x)*sin(c + d*x)**4*a**3*b**4*i - 240*sqrt(a**2 + b**2 )*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**7*i + 180*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i) /sqrt(a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**5*b**2*i + 270*sqrt(a* *2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d* x)*sin(c + d*x)**2*a**3*b**4*i + 120*sqrt(a**2 + b**2)*atan((tan((c + d*x) /2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*a**7*i + 90*sqrt(a**2 + b** 2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*a**5* b**2*i + 360*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(c + d*x)**5*a**6*b*i + 150*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(c + d*x)**5*a**4*b**3*i - 90*sqr t(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(c + d*x)**5*a**2*b**5*i - 720*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(c + d*x)**3*a**6*b*i - 420*sqrt(a**2 + b**2)* atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(c + d*x)**3*a**4* b**3*i + 90*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a*...