Integrand size = 28, antiderivative size = 383 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=-\frac {4 a^3 \text {arctanh}(\sin (c+d x))}{b^6 d}-\frac {3 a \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {6 a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{b^6 d}-\frac {8 a^2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 b^4 d}-\frac {2 \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}+\frac {4 a^2 \sec (c+d x)}{b^5 d}+\frac {2 \left (a^2+b^2\right ) \sec (c+d x)}{b^5 d}+\frac {\sec ^3(c+d x)}{3 b^3 d}-\frac {\left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac {4 a \left (a^2+b^2\right )}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac {3 a \sec (c+d x) \tan (c+d x)}{2 b^4 d} \] Output:
-4*a^3*arctanh(sin(d*x+c))/b^6/d-3/2*a*arctanh(sin(d*x+c))/b^4/d-6*a*(a^2+ b^2)*arctanh(sin(d*x+c))/b^6/d-8*a^2*(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-1/2*(a^2+b^2)^(1/2)*arctanh((b*cos(d *x+c)-a*sin(d*x+c))/(a^2+b^2)^(1/2))/b^4/d-2*(a^2+b^2)^(3/2)*arctanh((b*co s(d*x+c)-a*sin(d*x+c))/(a^2+b^2)^(1/2))/b^6/d+4*a^2*sec(d*x+c)/b^5/d+2*(a^ 2+b^2)*sec(d*x+c)/b^5/d+1/3*sec(d*x+c)^3/b^3/d-1/2*(a^2+b^2)*(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*(a^2+b^2)/b^5/d/(a* cos(d*x+c)+b*sin(d*x+c))-3/2*a*sec(d*x+c)*tan(d*x+c)/b^4/d
Result contains complex when optimal does not.
Time = 2.25 (sec) , antiderivative size = 688, normalized size of antiderivative = 1.80 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\frac {\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (\frac {6 b^2 \left (a^2+b^2\right )^2 \sin (c+d x)}{a}+\frac {6 (a-i b) (a+i b) b \left (8 a^2-b^2\right ) (a \cos (c+d x)+b \sin (c+d x))}{a}+2 b \left (36 a^2+13 b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2+60 \sqrt {a^2+b^2} \left (4 a^2+b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right ) (a \cos (c+d x)+b \sin (c+d x))^2+30 a \left (4 a^2+3 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2-30 a \left (4 a^2+3 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2+\frac {b^2 (-9 a+b) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 b^3 \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2 b \left (36 a^2+13 b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {2 b^3 \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {b^2 (9 a+b) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {2 b \left (36 a^2+13 b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{12 b^6 d (a+b \tan (c+d x))^3} \] Input:
Integrate[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]
Output:
(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*((6*b^2*(a^2 + b^2)^2*Si n[c + d*x])/a + (6*(a - I*b)*(a + I*b)*b*(8*a^2 - b^2)*(a*Cos[c + d*x] + b *Sin[c + d*x]))/a + 2*b*(36*a^2 + 13*b^2)*(a*Cos[c + d*x] + b*Sin[c + d*x] )^2 + 60*Sqrt[a^2 + b^2]*(4*a^2 + b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/S qrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + 30*a*(4*a^2 + 3*b^2) *Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x] )^2 - 30*a*(4*a^2 + 3*b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a*Cos [c + d*x] + b*Sin[c + d*x])^2 + (b^2*(-9*a + b)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*b^3*Sin[(c + d*x)/ 2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/ 2])^3 + (2*b*(36*a^2 + 13*b^2)*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (2*b^3*Sin[(c + d*x)/2] *(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2] )^3 + (b^2*(9*a + b)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2 ] + Sin[(c + d*x)/2])^2 - (2*b*(36*a^2 + 13*b^2)*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(12*b ^6*d*(a + b*Tan[c + d*x])^3)
Time = 4.20 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.82, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {3042, 3585, 3042, 3583, 3042, 3583, 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.
| \(\displaystyle \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^4 (a \cos (c+d x)+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3585 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {2 a \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^4 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}\) |
| \(\Big \downarrow \) 3583 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \sec ^3(c+d x)dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3583 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \sec (c+d x)dx}{b^2}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (-\frac {\left (a^2+b^2\right ) \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}+\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\) |
| \(\Big \downarrow \) 3585 |
| \(\displaystyle \frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {2 a \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \sec ^3(c+d x)dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 3555 |
| \(\displaystyle \frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (\frac {\int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (\frac {\int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (-\frac {\int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{2 d \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {2 a \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 3573 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}-\frac {2 a \left (-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \sec (c+d x)dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \sec (c+d x)dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}-\frac {2 a \left (-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {2 a \left (\frac {a \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (\frac {a \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {2 a \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}\) |
| \(\Big \downarrow \) 3583 |
| \(\displaystyle \frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {2 a \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \sec (c+d x)dx}{b^2}+\frac {\sec (c+d x)}{b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \sec (c+d x)dx}{b^2}+\frac {\sec (c+d x)}{b d}\right )}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {2 a \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\sec (c+d x)}{b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\sec (c+d x)}{b d}\right )}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {2 a \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {-\frac {\left (a^2+b^2\right ) \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\sec (c+d x)}{b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \left (-\frac {\left (a^2+b^2\right ) \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\sec (c+d x)}{b d}\right )}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {2 a \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {2 a \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}+\frac {\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {2 a \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}+\frac {-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}}{b^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}\right )}{b^2}-\frac {2 a \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}+\frac {\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}}{b^2}\right )}{b^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {2 a \left (\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}+\frac {\text {arctanh}(\sin (c+d x))}{b^2 d}\right )}{b^2}+\frac {-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a \text {arctanh}(\sin (c+d x))}{b^2 d}+\frac {\sec (c+d x)}{b d}}{b^2}\right )}{b^2}+\frac {\frac {\left (a^2+b^2\right ) \left (-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a \text {arctanh}(\sin (c+d x))}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}}{b^2}-\frac {2 a \left (\frac {\left (a^2+b^2\right ) \left (\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}+\frac {\text {arctanh}(\sin (c+d x))}{b^2 d}\right )}{b^2}-\frac {2 a \left (-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a \text {arctanh}(\sin (c+d x))}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}+\frac {\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}}{b^2}\right )}{b^2}\) |
Input:
Int[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]
Output:
((a^2 + b^2)*((-((a*ArcTanh[Sin[c + d*x]])/(b^2*d)) - (Sqrt[a^2 + b^2]*Arc Tanh[(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*((-2*a*(-((a*ArcTanh[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)*(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 + (ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d ))/b^2))/b^2 + (Sec[c + d*x]^3/(3*b*d) + ((a^2 + b^2)*(-((a*ArcTanh[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*(ArcTanh[Si n[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 = 2.79 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.16
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {\frac {b^{2} \left (7 a^{4}+5 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}+\frac {b \left (8 a^{6}-9 a^{4} b^{2}-15 a^{2} b^{4}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}}-\frac {b^{2} \left (25 a^{4}+23 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-4 a^{4} b -\frac {7 a^{2} b^{3}}{2}+\frac {b^{5}}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}-\frac {5 \left (4 a^{4}+5 a^{2} b^{2}+b^{4}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{6}}-\frac {1}{3 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {b +3 a}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {12 a^{2}+3 a b +5 b^{2}}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {5 a \left (4 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{6}}+\frac {1}{3 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {b -3 a}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-12 a^{2}+3 a b -5 b^{2}}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {5 a \left (4 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{6}}}{d}\) | \(444\) |
| default | \(\frac {-\frac {2 \left (\frac {\frac {b^{2} \left (7 a^{4}+5 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}+\frac {b \left (8 a^{6}-9 a^{4} b^{2}-15 a^{2} b^{4}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}}-\frac {b^{2} \left (25 a^{4}+23 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-4 a^{4} b -\frac {7 a^{2} b^{3}}{2}+\frac {b^{5}}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}-\frac {5 \left (4 a^{4}+5 a^{2} b^{2}+b^{4}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{6}}-\frac {1}{3 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {b +3 a}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {12 a^{2}+3 a b +5 b^{2}}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {5 a \left (4 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{6}}+\frac {1}{3 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {b -3 a}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-12 a^{2}+3 a b -5 b^{2}}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {5 a \left (4 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{6}}}{d}\) | \(444\) |
| risch | \(\frac {-90 i a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}+60 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-180 i a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}+100 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+140 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+15 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}+15 a^{2} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+140 a^{2} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+250 a^{2} b^{2} {\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 )}-20 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-15 b^{4} {\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 )}+360 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}+22 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+180 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}-60 i a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-100 i a \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+90 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \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 )^{2} b^{5} d}+\frac {10 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right ) a^{2}}{d \,b^{6}}+\frac {5 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{2 d \,b^{4}}-\frac {10 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right ) a^{2}}{d \,b^{6}}-\frac {5 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{2 d \,b^{4}}+\frac {10 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{b^{6} d}+\frac {15 a \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 )}+i\right )}{b^{6} d}-\frac {15 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 b^{4} d}\) | \(693\) |
Input:
int(sec(d*x+c)^4/(a*cos(d*x+c)+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-2/b^6*((1/2*b^2*(7*a^4+5*a^2*b^2-2*b^4)/a*tan(1/2*d*x+1/2*c)^3+1/2*b *(8*a^6-9*a^4*b^2-15*a^2*b^4+2*b^6)/a^2*tan(1/2*d*x+1/2*c)^2-1/2*b^2*(25*a ^4+23*a^2*b^2-2*b^4)/a*tan(1/2*d*x+1/2*c)-4*a^4*b-7/2*a^2*b^3+1/2*b^5)/(ta n(1/2*d*x+1/2*c)^2*a-2*b*tan(1/2*d*x+1/2*c)-a)^2-5/2*(4*a^4+5*a^2*b^2+b^4) /(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)) )-1/3/b^3/(tan(1/2*d*x+1/2*c)-1)^3-1/2*(b+3*a)/b^4/(tan(1/2*d*x+1/2*c)-1)^ 2-1/2*(12*a^2+3*a*b+5*b^2)/b^5/(tan(1/2*d*x+1/2*c)-1)+5/2*a*(4*a^2+3*b^2)/ b^6*ln(tan(1/2*d*x+1/2*c)-1)+1/3/b^3/(tan(1/2*d*x+1/2*c)+1)^3-1/2*(b-3*a)/ b^4/(tan(1/2*d*x+1/2*c)+1)^2-1/2*(-12*a^2+3*a*b-5*b^2)/b^5/(tan(1/2*d*x+1/ 2*c)+1)-5/2*a*(4*a^2+3*b^2)/b^6*ln(tan(1/2*d*x+1/2*c)+1))
Time = 0.16 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.47 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\frac {4 \, b^{5} + 30 \, {\left (4 \, a^{4} b + a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{4} + 20 \, {\left (2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (4 \, a^{4} - 3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (4 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (4 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 15 \, {\left ({\left (4 \, a^{5} - a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (4 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left ({\left (4 \, a^{5} - a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (4 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 10 \, {\left (a b^{4} \cos \left (d x + c\right ) - 6 \, {\left (3 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{12 \, {\left (2 \, a b^{7} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + b^{8} d \cos \left (d x + c\right )^{3} + {\left (a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right )^{5}\right )}} \] Input:
integrate(sec(d*x+c)^4/(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
1/12*(4*b^5 + 30*(4*a^4*b + a^2*b^3 - b^5)*cos(d*x + c)^4 + 20*(2*a^2*b^3 + b^5)*cos(d*x + c)^2 + 15*((4*a^4 - 3*a^2*b^2 - b^4)*cos(d*x + c)^5 + 2*( 4*a^3*b + a*b^3)*cos(d*x + c)^4*sin(d*x + c) + (4*a^2*b^2 + b^4)*cos(d*x + c)^3)*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^5 - a^3*b^2 - 3*a*b^4)*cos(d*x + c)^5 + 2*(4*a^4*b + 3*a ^2*b^3)*cos(d*x + c)^4*sin(d*x + c) + (4*a^3*b^2 + 3*a*b^4)*cos(d*x + c)^3 )*log(sin(d*x + c) + 1) + 15*((4*a^5 - a^3*b^2 - 3*a*b^4)*cos(d*x + c)^5 + 2*(4*a^4*b + 3*a^2*b^3)*cos(d*x + c)^4*sin(d*x + c) + (4*a^3*b^2 + 3*a*b^ 4)*cos(d*x + c)^3)*log(-sin(d*x + c) + 1) - 10*(a*b^4*cos(d*x + c) - 6*(3* a^3*b^2 + 2*a*b^4)*cos(d*x + c)^3)*sin(d*x + c))/(2*a*b^7*d*cos(d*x + c)^4 *sin(d*x + c) + b^8*d*cos(d*x + c)^3 + (a^2*b^6 - b^8)*d*cos(d*x + c)^5)
\[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(sec(d*x+c)**4/(a*cos(d*x+c)+b*sin(d*x+c))**3,x)
Output:
Integral(sec(c + d*x)**4/(a*cos(c + d*x) + b*sin(c + d*x))**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 902 vs. \(2 (361) = 722\).
Time = 0.13 (sec) , antiderivative size = 902, normalized size of antiderivative = 2.36 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)^4/(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
1/6*(2*(60*a^6 + 35*a^4*b^2 - 3*a^2*b^4 + (210*a^5*b + 125*a^3*b^3 - 6*a*b ^5)*sin(d*x + c)/(cos(d*x + c) + 1) - 2*(120*a^6 - 10*a^4*b^2 - 55*a^2*b^4 + 3*b^6)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2*(330*a^5*b + 205*a^3*b^3 - 12*a*b^5)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2*(180*a^6 - 95*a^4*b^2 - 120*a^2*b^4 + 9*b^6)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 12*(60*a^5*b + 35*a^3*b^3 - 3*a*b^5)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 6*(40*a^6 - 30*a^4*b^2 - 35*a^2*b^4 + 3*b^6)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 6* (50*a^5*b + 25*a^3*b^3 - 4*a*b^5)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 3* (20*a^6 - 15*a^4*b^2 - 15*a^2*b^4 + 2*b^6)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 3*(10*a^5*b + 5*a^3*b^3 - 2*a*b^5)*sin(d*x + c)^9/(cos(d*x + c) + 1 )^9)/(a^4*b^5 + 4*a^3*b^6*sin(d*x + c)/(cos(d*x + c) + 1) - 16*a^3*b^6*sin (d*x + c)^3/(cos(d*x + c) + 1)^3 + 24*a^3*b^6*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 16*a^3*b^6*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 4*a^3*b^6*sin(d *x + c)^9/(cos(d*x + c) + 1)^9 - a^4*b^5*sin(d*x + c)^10/(cos(d*x + c) + 1 )^10 - (5*a^4*b^5 - 4*a^2*b^7)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 2*(5* a^4*b^5 - 6*a^2*b^7)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 2*(5*a^4*b^5 - 6*a^2*b^7)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + (5*a^4*b^5 - 4*a^2*b^7)*s in(d*x + c)^8/(cos(d*x + c) + 1)^8) - 15*(4*a^3 + 3*a*b^2)*log(sin(d*x + c )/(cos(d*x + c) + 1) + 1)/b^6 + 15*(4*a^3 + 3*a*b^2)*log(sin(d*x + c)/(cos (d*x + c) + 1) - 1)/b^6 - 15*(4*a^4 + 5*a^2*b^2 + b^4)*log((b - a*sin(d...
Time = 0.23 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.33 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)^4/(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
-1/6*(15*(4*a^3 + 3*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^6 - 15*(4* a^3 + 3*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^6 + 15*(4*a^4 + 5*a^2* b^2 + b^4)*log(abs(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 ) + 2*(9*a*b*tan(1/2*d*x + 1/2*c)^5 + 36*a^2*tan(1/2*d*x + 1/2*c)^4 + 18*b ^2*tan(1/2*d*x + 1/2*c)^4 - 72*a^2*tan(1/2*d*x + 1/2*c)^2 - 24*b^2*tan(1/2 *d*x + 1/2*c)^2 - 9*a*b*tan(1/2*d*x + 1/2*c) + 36*a^2 + 14*b^2)/((tan(1/2* d*x + 1/2*c)^2 - 1)^3*b^5) + 6*(7*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 5*a^3*b^3 *tan(1/2*d*x + 1/2*c)^3 - 2*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 8*a^6*tan(1/2*d *x + 1/2*c)^2 - 9*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 - 15*a^2*b^4*tan(1/2*d*x + 1/2*c)^2 + 2*b^6*tan(1/2*d*x + 1/2*c)^2 - 25*a^5*b*tan(1/2*d*x + 1/2*c) - 23*a^3*b^3*tan(1/2*d*x + 1/2*c) + 2*a*b^5*tan(1/2*d*x + 1/2*c) - 8*a^6 - 7*a^4*b^2 + a^2*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 - 2*b*tan(1/2*d*x + 1/2*c ) - a)^2*a^2*b^5))/d
Time = 20.21 (sec) , antiderivative size = 1203, normalized size of antiderivative = 3.14 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int(1/(cos(c + d*x)^4*(a*cos(c + d*x) + b*sin(c + d*x))^3),x)
Output:
((60*a^4 - 3*b^4 + 35*a^2*b^2)/(3*b^5) + (tan(c/2 + (d*x)/2)*(210*a^4 - 6* b^4 + 125*a^2*b^2))/(3*a*b^4) + (tan(c/2 + (d*x)/2)^8*(20*a^6 + 2*b^6 - 15 *a^2*b^4 - 15*a^4*b^2))/(a^2*b^5) - (2*tan(c/2 + (d*x)/2)^6*(40*a^6 + 3*b^ 6 - 35*a^2*b^4 - 30*a^4*b^2))/(a^2*b^5) - (2*tan(c/2 + (d*x)/2)^2*(120*a^6 + 3*b^6 - 55*a^2*b^4 - 10*a^4*b^2))/(3*a^2*b^5) + (2*tan(c/2 + (d*x)/2)^4 *(180*a^6 + 9*b^6 - 120*a^2*b^4 - 95*a^4*b^2))/(3*a^2*b^5) + (tan(c/2 + (d *x)/2)^9*(10*a^4 - 2*b^4 + 5*a^2*b^2))/(a*b^4) - (2*tan(c/2 + (d*x)/2)^7*( 50*a^4 - 4*b^4 + 25*a^2*b^2))/(a*b^4) + (4*tan(c/2 + (d*x)/2)^5*(60*a^4 - 3*b^4 + 35*a^2*b^2))/(a*b^4) - (2*tan(c/2 + (d*x)/2)^3*(330*a^4 - 12*b^4 + 205*a^2*b^2))/(3*a*b^4))/(d*(tan(c/2 + (d*x)/2)^8*(5*a^2 - 4*b^2) - tan(c /2 + (d*x)/2)^2*(5*a^2 - 4*b^2) + tan(c/2 + (d*x)/2)^4*(10*a^2 - 12*b^2) - tan(c/2 + (d*x)/2)^6*(10*a^2 - 12*b^2) - a^2*tan(c/2 + (d*x)/2)^10 + a^2 - 16*a*b*tan(c/2 + (d*x)/2)^3 + 24*a*b*tan(c/2 + (d*x)/2)^5 - 16*a*b*tan(c /2 + (d*x)/2)^7 + 4*a*b*tan(c/2 + (d*x)/2)^9 + 4*a*b*tan(c/2 + (d*x)/2))) - (atanh((3000*a^2*tan(c/2 + (d*x)/2))/(3000*a^2 + (7000*a^4)/b^2 + (4000* a^6)/b^4) + (7000*a^4*tan(c/2 + (d*x)/2))/(7000*a^4 + 3000*a^2*b^2 + (4000 *a^6)/b^2) + (4000*a^6*tan(c/2 + (d*x)/2))/(4000*a^6 + 3000*a^2*b^4 + 7000 *a^4*b^2))*(15*a*b^2 + 20*a^3))/(b^6*d) + (5*atanh((1000*a^2*(a^2 + b^2)^( 1/2))/(1000*a^2*b + (5000*a^4)/b + (4000*a^6)/b^3 + 10000*a^3*tan(c/2 + (d *x)/2) + 2000*a*b^2*tan(c/2 + (d*x)/2) + (8000*a^5*tan(c/2 + (d*x)/2))/...
Time = 0.31 (sec) , antiderivative size = 2108, normalized size of antiderivative = 5.50 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(sec(d*x+c)^4/(a*cos(d*x+c)+b*sin(d*x+c))^3,x)
Output:
( - 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)**4*a**4*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)**4*a **2*b**2*i + 60*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*b**4*i + 480*sqrt(a**2 + b**2)*a tan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*sin(c + d *x)**2*a**4*i - 120*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sq rt(a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**2*b**2*i - 60*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*s in(c + d*x)**2*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)*a**4*i - 60*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**2*b**2*i - 480*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**3*b*i - 120*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*b**3*i + 960*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 **3*b*i + 240*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*b**3*i - 480*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(c + d*x)*a**3*b*i - 120*sqrt(a* *2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(c +...