Integrand size = 31, antiderivative size = 310 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\frac {B \text {arctanh}(\sin (c+d x))}{b^4 d}-\frac {\left (3 a^2 A b^5+2 A b^7+2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B-8 a b^6 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^4 (a+b)^{7/2} d}+\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {a^2 \left (5 A b^3+3 a^3 B-8 a b^2 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {a \left (a^2 A b^3-16 A b^5+9 a^5 B-28 a^3 b^2 B+34 a b^4 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]
B*arctanh(sin(d*x+c))/b^4/d-(3*A*a^2*b^5+2*A*b^7+2*B*a^7-7*B*a^5*b^2+8*B*a ^3*b^4-8*B*a*b^6)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b )^(7/2)/b^4/(a+b)^(7/2)/d+1/3*a*(A*b-B*a)*sec(d*x+c)^2*tan(d*x+c)/b/(a^2-b ^2)/d/(a+b*sec(d*x+c))^3+1/6*a^2*(5*A*b^3+3*B*a^3-8*B*a*b^2)*tan(d*x+c)/b^ 3/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2-1/6*a*(A*a^2*b^3-16*A*b^5+9*B*a^5-28*B* a^3*b^2+34*B*a*b^4)*tan(d*x+c)/b^3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))
Time = 1.92 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.19 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\frac {\cos (c+d x) (A+B \sec (c+d x)) \left (\frac {24 \left (3 a^2 A b^5+2 A b^7+2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B-8 a b^6 B\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-24 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {2 a b \left (8 a^4 A b^3+a^2 A b^5+36 A b^7-6 a^7 B-5 a^5 b^2 B+38 a^3 b^4 B-72 a b^6 B-6 a b \left (-a^2 A b^3-9 A b^5+5 a^5 B-15 a^3 b^2 B+20 a b^4 B\right ) \cos (c+d x)+a^2 \left (4 a^2 A b^3+11 A b^5-6 a^5 B+17 a^3 b^2 B-26 a b^4 B\right ) \cos (2 (c+d x))\right ) \sin (c+d x)}{\left (-a^2+b^2\right )^3 (b+a \cos (c+d x))^3}\right )}{24 b^4 d (B+A \cos (c+d x))} \]
(Cos[c + d*x]*(A + B*Sec[c + d*x])*((24*(3*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^3*b^4*B - 8*a*b^6*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2] )/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) - 24*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 24*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (2*a*b*(8*a^ 4*A*b^3 + a^2*A*b^5 + 36*A*b^7 - 6*a^7*B - 5*a^5*b^2*B + 38*a^3*b^4*B - 72 *a*b^6*B - 6*a*b*(-(a^2*A*b^3) - 9*A*b^5 + 5*a^5*B - 15*a^3*b^2*B + 20*a*b ^4*B)*Cos[c + d*x] + a^2*(4*a^2*A*b^3 + 11*A*b^5 - 6*a^5*B + 17*a^3*b^2*B - 26*a*b^4*B)*Cos[2*(c + d*x)])*Sin[c + d*x])/((-a^2 + b^2)^3*(b + a*Cos[c + d*x])^3)))/(24*b^4*d*(B + A*Cos[c + d*x]))
Time = 2.15 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.18, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.516, Rules used = {3042, 4517, 3042, 4578, 25, 3042, 4568, 27, 3042, 4486, 3042, 4257, 4318, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 4517 |
| \(\displaystyle \frac {\int \frac {\sec ^2(c+d x) \left (3 \left (a^2-b^2\right ) B \sec ^2(c+d x)-3 b (A b-a B) \sec (c+d x)+2 a (A b-a B)\right )}{(a+b \sec (c+d x))^3}dx}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (3 \left (a^2-b^2\right ) B \csc \left (c+d x+\frac {\pi }{2}\right )^2-3 b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a (A b-a B)\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4578 |
| \(\displaystyle \frac {\frac {\int -\frac {\sec (c+d x) \left (-6 b \left (a^2-b^2\right )^2 B \sec ^2(c+d x)+\left (3 B a^5-10 b^2 B a^3+A b^3 a^2+12 b^4 B a-6 A b^5\right ) \sec (c+d x)+2 a b \left (3 B a^3-8 b^2 B a+5 A b^3\right )\right )}{(a+b \sec (c+d x))^2}dx}{2 b^2 \left (a^2-b^2\right )}+\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int \frac {\sec (c+d x) \left (-6 b \left (a^2-b^2\right )^2 B \sec ^2(c+d x)+\left (3 B a^5-10 b^2 B a^3+A b^3 a^2+12 b^4 B a-6 A b^5\right ) \sec (c+d x)+2 a b \left (3 B a^3-8 b^2 B a+5 A b^3\right )\right )}{(a+b \sec (c+d x))^2}dx}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (-6 b \left (a^2-b^2\right )^2 B \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (3 B a^5-10 b^2 B a^3+A b^3 a^2+12 b^4 B a-6 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a b \left (3 B a^3-8 b^2 B a+5 A b^3\right )\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4568 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int -\frac {3 \sec (c+d x) \left (b \left (2 A b^6-6 a B b^5+3 a^2 A b^4+2 a^3 B b^3-a^5 B b\right )-2 b \left (a^2-b^2\right )^3 B \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {3 \int \frac {\sec (c+d x) \left (b \left (2 A b^6-6 a B b^5+3 a^2 A b^4+2 a^3 B b^3-a^5 B b\right )-2 b \left (a^2-b^2\right )^3 B \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {3 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (b \left (2 A b^6-6 a B b^5+3 a^2 A b^4+2 a^3 B b^3-a^5 B b\right )-2 b \left (a^2-b^2\right )^3 B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4486 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {3 \left (\left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx-2 B \left (a^2-b^2\right )^3 \int \sec (c+d x)dx\right )}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {3 \left (\left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx-2 B \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx\right )}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {3 \left (\left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 B \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}\right )}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {3 \left (\frac {\left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{b}-\frac {2 B \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}\right )}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {3 \left (\frac {\left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{b}-\frac {2 B \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}\right )}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {3 \left (\frac {2 \left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}-\frac {2 B \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}\right )}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {3 \left (\frac {2 \left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}-\frac {2 B \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}\right )}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\) |
(a*(A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[ c + d*x])^3) + ((a^2*(5*A*b^3 + 3*a^3*B - 8*a*b^2*B)*Tan[c + d*x])/(2*b^2* (a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*((-2*(a^2 - b^2)^3*B*ArcTanh[S in[c + d*x]])/d + (2*(3*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^ 3*b^4*B - 8*a*b^6*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/ (Sqrt[a - b]*Sqrt[a + b]*d)))/(b*(a^2 - b^2)) + (a*(a^2*A*b^3 - 16*A*b^5 + 9*a^5*B - 28*a^3*b^2*B + 34*a*b^4*B)*Tan[c + d*x])/((a^2 - b^2)*d*(a + b* Sec[c + d*x])))/(2*b^2*(a^2 - b^2)))/(3*b*(a^2 - b^2))
3.4.37.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[( e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[B/b Int[Csc[e + f*x], x], x] + Simp[(A*b - a*B)/b Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x ] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*d^2*( A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 2)/(b*f*(m + 1)*(a^2 - b^2))), x] - Simp[d/(b*(m + 1)*(a^2 - b^2)) Int[( a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*Simp[a*d*(A*b - a*B)*( n - 2) + b*d*(A*b - a*B)*(m + 1)*Csc[e + f*x] - (a*A*b*d*(m + n) - d*B*(a^2 *(n - 1) + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f , A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[ n, 1]
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e _.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S ymbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cot[e + f*x]*((a + b*Csc[e + f*x] )^(m + 1)/(b*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^ 2, 0]
Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[ (e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x _Symbol] :> Simp[a*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x ])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - Simp[1/(b^2*(m + 1)*(a^2 - b^ 2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[b*(m + 1)*((-a)*(b *B - a*C) + A*b^2) + (b*B*(a^2 + b^2*(m + 1)) - a*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1 ]
Time = 1.98 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.55
| method | result | size |
| derivativedivides | \(\frac {\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}+\frac {\frac {2 \left (-\frac {\left (2 A \,a^{2} b^{3}+3 A a \,b^{4}+6 A \,b^{5}-2 B \,a^{5}+B \,a^{4} b +6 B \,a^{3} b^{2}-4 B \,a^{2} b^{3}-12 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (A \,a^{2} b^{3}+9 A \,b^{5}-3 B \,a^{5}+11 B \,a^{3} b^{2}-18 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (2 A \,a^{2} b^{3}-3 A a \,b^{4}+6 A \,b^{5}-2 B \,a^{5}-B \,a^{4} b +6 B \,a^{3} b^{2}+4 B \,a^{2} b^{3}-12 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (3 A \,a^{2} b^{5}+2 A \,b^{7}+2 B \,a^{7}-7 B \,a^{5} b^{2}+8 B \,a^{3} b^{4}-8 B a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{b^{4}}-\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}}{d}\) | \(482\) |
| default | \(\frac {\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}+\frac {\frac {2 \left (-\frac {\left (2 A \,a^{2} b^{3}+3 A a \,b^{4}+6 A \,b^{5}-2 B \,a^{5}+B \,a^{4} b +6 B \,a^{3} b^{2}-4 B \,a^{2} b^{3}-12 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (A \,a^{2} b^{3}+9 A \,b^{5}-3 B \,a^{5}+11 B \,a^{3} b^{2}-18 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (2 A \,a^{2} b^{3}-3 A a \,b^{4}+6 A \,b^{5}-2 B \,a^{5}-B \,a^{4} b +6 B \,a^{3} b^{2}+4 B \,a^{2} b^{3}-12 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (3 A \,a^{2} b^{5}+2 A \,b^{7}+2 B \,a^{7}-7 B \,a^{5} b^{2}+8 B \,a^{3} b^{4}-8 B a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{b^{4}}-\frac {B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}}{d}\) | \(482\) |
| risch | \(\text {Expression too large to display}\) | \(1688\) |
1/d*(B/b^4*ln(tan(1/2*d*x+1/2*c)+1)+2/b^4*((-1/2*(2*A*a^2*b^3+3*A*a*b^4+6* A*b^5-2*B*a^5+B*a^4*b+6*B*a^3*b^2-4*B*a^2*b^3-12*B*a*b^4)*a*b/(a-b)/(a^3+3 *a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5+2/3*(A*a^2*b^3+9*A*b^5-3*B*a^5+11 *B*a^3*b^2-18*B*a*b^4)*a*b/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2 *c)^3-1/2*(2*A*a^2*b^3-3*A*a*b^4+6*A*b^5-2*B*a^5-B*a^4*b+6*B*a^3*b^2+4*B*a ^2*b^3-12*B*a*b^4)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)) /(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3-1/2*(3*A*a^2*b^5+2* A*b^7+2*B*a^7-7*B*a^5*b^2+8*B*a^3*b^4-8*B*a*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4- b^6)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1 /2)))-B/b^4*ln(tan(1/2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 1110 vs. \(2 (295) = 590\).
Time = 28.00 (sec) , antiderivative size = 2278, normalized size of antiderivative = 7.35 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
[-1/12*(3*(2*B*a^7*b^3 - 7*B*a^5*b^5 + 8*B*a^3*b^7 + 3*A*a^2*b^8 - 8*B*a*b ^9 + 2*A*b^10 + (2*B*a^10 - 7*B*a^8*b^2 + 8*B*a^6*b^4 + 3*A*a^5*b^5 - 8*B* a^4*b^6 + 2*A*a^3*b^7)*cos(d*x + c)^3 + 3*(2*B*a^9*b - 7*B*a^7*b^3 + 8*B*a ^5*b^5 + 3*A*a^4*b^6 - 8*B*a^3*b^7 + 2*A*a^2*b^8)*cos(d*x + c)^2 + 3*(2*B* a^8*b^2 - 7*B*a^6*b^4 + 8*B*a^4*b^6 + 3*A*a^3*b^7 - 8*B*a^2*b^8 + 2*A*a*b^ 9)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*c os(d*x + c)^2 + 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^ 2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - 6*(B*a^8*b^3 - 4*B*a^6*b^5 + 6*B*a^4*b^7 - 4*B*a^2*b^9 + B*b^11 + (B*a^11 - 4*B*a^9*b^2 + 6*B*a^7*b^4 - 4*B*a^5*b^6 + B*a^3*b^8)*cos(d*x + c)^3 + 3*(B*a^10*b - 4* B*a^8*b^3 + 6*B*a^6*b^5 - 4*B*a^4*b^7 + B*a^2*b^9)*cos(d*x + c)^2 + 3*(B*a ^9*b^2 - 4*B*a^7*b^4 + 6*B*a^5*b^6 - 4*B*a^3*b^8 + B*a*b^10)*cos(d*x + c)) *log(sin(d*x + c) + 1) + 6*(B*a^8*b^3 - 4*B*a^6*b^5 + 6*B*a^4*b^7 - 4*B*a^ 2*b^9 + B*b^11 + (B*a^11 - 4*B*a^9*b^2 + 6*B*a^7*b^4 - 4*B*a^5*b^6 + B*a^3 *b^8)*cos(d*x + c)^3 + 3*(B*a^10*b - 4*B*a^8*b^3 + 6*B*a^6*b^5 - 4*B*a^4*b ^7 + B*a^2*b^9)*cos(d*x + c)^2 + 3*(B*a^9*b^2 - 4*B*a^7*b^4 + 6*B*a^5*b^6 - 4*B*a^3*b^8 + B*a*b^10)*cos(d*x + c))*log(-sin(d*x + c) + 1) + 2*(11*B*a ^8*b^3 - 2*A*a^7*b^4 - 43*B*a^6*b^5 + 7*A*a^5*b^6 + 68*B*a^4*b^7 - 23*A*a^ 3*b^8 - 36*B*a^2*b^9 + 18*A*a*b^10 + (6*B*a^10*b - 23*B*a^8*b^3 - 4*A*a^7* b^4 + 43*B*a^6*b^5 - 7*A*a^5*b^6 - 26*B*a^4*b^7 + 11*A*a^3*b^8)*cos(d*x...
\[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 844 vs. \(2 (295) = 590\).
Time = 0.40 (sec) , antiderivative size = 844, normalized size of antiderivative = 2.72 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
-1/3*(3*(2*B*a^7 - 7*B*a^5*b^2 + 8*B*a^3*b^4 + 3*A*a^2*b^5 - 8*B*a*b^6 + 2 *A*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan (1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*sqrt(-a^2 + b^2)) - 3*B*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^4 + 3*B*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^4 - (6*B*a^ 8*tan(1/2*d*x + 1/2*c)^5 - 15*B*a^7*b*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^6*b^2 *tan(1/2*d*x + 1/2*c)^5 - 6*A*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 45*B*a^5*b^ 3*tan(1/2*d*x + 1/2*c)^5 + 3*A*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^4*b^ 4*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 60*B*a^3*b ^5*tan(1/2*d*x + 1/2*c)^5 + 27*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 + 36*B*a^2 *b^6*tan(1/2*d*x + 1/2*c)^5 - 18*A*a*b^7*tan(1/2*d*x + 1/2*c)^5 - 12*B*a^8 *tan(1/2*d*x + 1/2*c)^3 + 56*B*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 + 4*A*a^5*b^ 3*tan(1/2*d*x + 1/2*c)^3 - 116*B*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 + 32*A*a^3 *b^5*tan(1/2*d*x + 1/2*c)^3 + 72*B*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 - 36*A*a *b^7*tan(1/2*d*x + 1/2*c)^3 + 6*B*a^8*tan(1/2*d*x + 1/2*c) + 15*B*a^7*b*ta n(1/2*d*x + 1/2*c) - 6*B*a^6*b^2*tan(1/2*d*x + 1/2*c) - 6*A*a^5*b^3*tan(1/ 2*d*x + 1/2*c) - 45*B*a^5*b^3*tan(1/2*d*x + 1/2*c) - 3*A*a^4*b^4*tan(1/2*d *x + 1/2*c) - 6*B*a^4*b^4*tan(1/2*d*x + 1/2*c) - 6*A*a^3*b^5*tan(1/2*d*x + 1/2*c) + 60*B*a^3*b^5*tan(1/2*d*x + 1/2*c) - 27*A*a^2*b^6*tan(1/2*d*x + 1 /2*c) + 36*B*a^2*b^6*tan(1/2*d*x + 1/2*c) - 18*A*a*b^7*tan(1/2*d*x + 1/...
Time = 27.40 (sec) , antiderivative size = 9713, normalized size of antiderivative = 31.33 \[ \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
- ((tan(c/2 + (d*x)/2)*(2*B*a^6 + 3*A*a^2*b^4 - 2*A*a^3*b^3 + 12*B*a^2*b^4 - 4*B*a^3*b^3 - 6*B*a^4*b^2 - 6*A*a*b^5 + B*a^5*b))/((a + b)*(3*a*b^5 - b ^6 - 3*a^2*b^4 + a^3*b^3)) - (tan(c/2 + (d*x)/2)^5*(3*A*a^2*b^4 - 2*B*a^6 + 2*A*a^3*b^3 - 12*B*a^2*b^4 - 4*B*a^3*b^3 + 6*B*a^4*b^2 + 6*A*a*b^5 + B*a ^5*b))/((a*b^3 - b^4)*(a + b)^3) + (4*tan(c/2 + (d*x)/2)^3*(A*a^3*b^3 - 3* B*a^6 - 18*B*a^2*b^4 + 11*B*a^4*b^2 + 9*A*a*b^5))/(3*(a + b)^2*(b^5 - 2*a* b^4 + a^2*b^3)))/(d*(tan(c/2 + (d*x)/2)^2*(3*a*b^2 - 3*a^2*b - 3*a^3 + 3*b ^3) - tan(c/2 + (d*x)/2)^4*(3*a*b^2 + 3*a^2*b - 3*a^3 - 3*b^3) + 3*a*b^2 + 3*a^2*b + a^3 + b^3 - tan(c/2 + (d*x)/2)^6*(3*a*b^2 - 3*a^2*b + a^3 - b^3 ))) - (B*atan(((B*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 8*B^2*a^14 + 4*B^2* b^14 - 8*B^2*a*b^13 - 8*B^2*a^13*b + 12*A^2*a^2*b^12 + 9*A^2*a^4*b^10 + 44 *B^2*a^2*b^12 + 48*B^2*a^3*b^11 - 92*B^2*a^4*b^10 - 120*B^2*a^5*b^9 + 156* B^2*a^6*b^8 + 160*B^2*a^7*b^7 - 164*B^2*a^8*b^6 - 120*B^2*a^9*b^5 + 117*B^ 2*a^10*b^4 + 48*B^2*a^11*b^3 - 48*B^2*a^12*b^2 - 32*A*B*a*b^13 - 16*A*B*a^ 3*b^11 + 20*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 12*A*B*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10* a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) + (B*((8*(4*A*b^21 + 4*B*b^21 - 6*A*a^2*b^19 + 6*A*a^3*b^18 - 6*A*a^4*b^17 + 6*A*a^5*b^16 + 14*A*a^6*b^15 - 14*A*a^7*b^14 - 6*A*a^8*b^13 + 6*A*a^9*b^12 - 12*B*a^2*b^1 9 + 64*B*a^3*b^18 + 20*B*a^4*b^17 - 110*B*a^5*b^16 - 30*B*a^6*b^15 + 11...