Integrand size = 25, antiderivative size = 335 \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=-\frac {2 \cot ^3(c+d x) (3-\sec (c+d x))}{3 a d (e \cot (c+d x))^{7/2}}-\frac {\cot ^3(c+d x) \csc (c+d x) \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sqrt {\sin (2 c+2 d x)}}{3 a d (e \cot (c+d x))^{7/2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{7/2} \tan ^{\frac {7}{2}}(c+d x)}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{7/2} \tan ^{\frac {7}{2}}(c+d x)}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{7/2} \tan ^{\frac {7}{2}}(c+d x)}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{7/2} \tan ^{\frac {7}{2}}(c+d x)} \]
-2/3*cot(d*x+c)^3*(3-sec(d*x+c))/a/d/(e*cot(d*x+c))^(7/2)+1/3*cot(d*x+c)^3 *csc(d*x+c)*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d*x)*EllipticF(cos(c+ 1/4*Pi+d*x),2^(1/2))*sin(2*d*x+2*c)^(1/2)/a/d/(e*cot(d*x+c))^(7/2)+1/2*arc tan(-1+2^(1/2)*tan(d*x+c)^(1/2))/a/d/(e*cot(d*x+c))^(7/2)*2^(1/2)/tan(d*x+ c)^(7/2)+1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/a/d/(e*cot(d*x+c))^(7/2)*2 ^(1/2)/tan(d*x+c)^(7/2)-1/4*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/a/d/ (e*cot(d*x+c))^(7/2)*2^(1/2)/tan(d*x+c)^(7/2)+1/4*ln(1+2^(1/2)*tan(d*x+c)^ (1/2)+tan(d*x+c))/a/d/(e*cot(d*x+c))^(7/2)*2^(1/2)/tan(d*x+c)^(7/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 12.94 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=-\frac {4 \sqrt {e \cot (c+d x)} \csc (c+d x) \left (3-3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\tan ^2(c+d x)\right )+3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\tan ^2(c+d x)\right )+\cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right ) \left (1+\sqrt {\sec ^2(c+d x)}\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right )}{3 a d e^4} \]
(-4*Sqrt[e*Cot[c + d*x]]*Csc[c + d*x]*(3 - 3*Hypergeometric2F1[-1/2, 1/4, 5/4, -Tan[c + d*x]^2] + 3*Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2 ] + Cot[c + d*x]^2*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])*(1 + S qrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(3*a*d*e^4)
Time = 1.11 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.74, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.040, Rules used = {3042, 4388, 3042, 4376, 25, 3042, 4369, 27, 3042, 4372, 3042, 3094, 3042, 3053, 3042, 3120, 3957, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a) (e \cot (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a) (e \cot (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 4388 |
| \(\displaystyle \frac {\int \frac {\tan ^{\frac {7}{2}}(c+d x)}{\sec (c+d x) a+a}dx}{\tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{\tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {\int -\left ((a-a \sec (c+d x)) \tan ^{\frac {3}{2}}(c+d x)\right )dx}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int (a-a \sec (c+d x)) \tan ^{\frac {3}{2}}(c+d x)dx}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4369 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}-\frac {2}{3} \int \frac {3 a-a \sec (c+d x)}{2 \sqrt {\tan (c+d x)}}dx}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}-\frac {1}{3} \int \frac {3 a-a \sec (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}-\frac {1}{3} \int \frac {3 a-a \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\cot \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4372 |
| \(\displaystyle -\frac {\frac {1}{3} \left (a \int \frac {\sec (c+d x)}{\sqrt {\tan (c+d x)}}dx-3 a \int \frac {1}{\sqrt {\tan (c+d x)}}dx\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{3} \left (a \int \frac {\sec (c+d x)}{\sqrt {\tan (c+d x)}}dx-3 a \int \frac {1}{\sqrt {\tan (c+d x)}}dx\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3094 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}}dx}{\sqrt {\cos (c+d x)} \sqrt {\tan (c+d x)}}-3 a \int \frac {1}{\sqrt {\tan (c+d x)}}dx\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}}dx}{\sqrt {\cos (c+d x)} \sqrt {\tan (c+d x)}}-3 a \int \frac {1}{\sqrt {\tan (c+d x)}}dx\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}}dx}{\sqrt {\tan (c+d x)}}-3 a \int \frac {1}{\sqrt {\tan (c+d x)}}dx\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}}dx}{\sqrt {\tan (c+d x)}}-3 a \int \frac {1}{\sqrt {\tan (c+d x)}}dx\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}-3 a \int \frac {1}{\sqrt {\tan (c+d x)}}dx\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}-\frac {3 a \int \frac {1}{\sqrt {\tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}-\frac {6 a \int \frac {1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}-\frac {6 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}-\frac {6 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d}\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}-\frac {6 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}-\frac {6 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}-\frac {6 a \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}-\frac {6 a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}-\frac {6 a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}-\frac {6 a \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}\right )+\frac {2 \sqrt {\tan (c+d x)} (3 a-a \sec (c+d x))}{3 d}}{a^2 \tan ^{\frac {7}{2}}(c+d x) (e \cot (c+d x))^{7/2}}\) |
-((((-6*a*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2])/2 + (-1/2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Ta n[c + d*x]]/(2*Sqrt[2]))/2))/d + (a*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d *x]*Sqrt[Sin[2*c + 2*d*x]])/(d*Sqrt[Tan[c + d*x]]))/3 + (2*(3*a - a*Sec[c + d*x])*Sqrt[Tan[c + d*x]])/(3*d))/(a^2*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x ]^(7/2)))
3.3.48.3.1 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, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f }, x]
Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) Int[ 1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x], x] /; FreeQ[{b, e, f}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-e)*(e*Cot[c + d*x])^(m - 1)*((a*m + b*(m - 1)*Csc [c + d*x])/(d*m*(m - 1))), x] - Simp[e^2/m Int[(e*Cot[c + d*x])^(m - 2)*( a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m , 1]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(e*Cot[c + d*x])^m, x], x] + Simp[b Int[ (e*Cot[c + d*x])^m*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m}, x]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n) Int[(e*Cot[c + d*x])^(m + 2* n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a ^2 - b^2, 0] && ILtQ[n, 0]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*((a_) + (b_.)*sec[(c_.) + (d_.)*(x _)])^(n_.), x_Symbol] :> Simp[(e*Cot[c + d*x])^m*Tan[c + d*x]^m Int[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^m, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && !IntegerQ[m]
Result contains complex when optimal does not.
Time = 9.80 (sec) , antiderivative size = 976, normalized size of antiderivative = 2.91
-1/6/a/d*2^(1/2)*(3*I*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+ c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c )+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(d*x+c)^2-3*I*(cot(d*x+c)-csc(d*x+c)+ 1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*Ell ipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(d*x+c)^ 2+3*I*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d *x+c)-cot(d*x+c)+1)^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1 /2*I,1/2*2^(1/2))*cos(d*x+c)-3*I*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+ c)-csc(d*x+c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*EllipticPi((csc(d*x+c )-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(d*x+c)+8*(csc(d*x+c)-cot( d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1 /2)*EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))*cos(d*x+c)^2-3* (cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*x+c)- cot(d*x+c)+1)^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1 /2*2^(1/2))*cos(d*x+c)^2-3*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc (d*x+c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*EllipticPi((csc(d*x+c)-cot( d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(d*x+c)^2+8*(csc(d*x+c)-cot(d*x+ c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)* EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))*cos(d*x+c)-3*(cot(d *x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticPi((csc...
Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
\[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]