Integrand size = 26, antiderivative size = 275 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=-\frac {\left (\frac {5}{16}-\frac {7 i}{16}\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {\left (\frac {5}{16}-\frac {7 i}{16}\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}-\frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{3 a d (i a+a \cot (c+d x))^2}+\frac {5 \sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {\left (\frac {5}{32}+\frac {7 i}{32}\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^3 d}-\frac {\left (\frac {5}{32}+\frac {7 i}{32}\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^3 d} \]
(5/32-7/32*I)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+(5/32-7/32 *I)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+(5/64+7/64*I)*ln(1+co t(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)-(5/64+7/64*I)*ln(1+cot(d* x+c)+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)-1/6*cot(d*x+c)^(1/2)/d/(I*a+a *cot(d*x+c))^3+1/3*I*cot(d*x+c)^(1/2)/a/d/(I*a+a*cot(d*x+c))^2+5/8*cot(d*x +c)^(1/2)/d/(I*a^3+a^3*cot(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\frac {i \sqrt {\cot (c+d x)} \left (32 i+(i+\cot (c+d x)) \left (64-i (i+\cot (c+d x)) \left (120+(i+\cot (c+d x)) \sqrt {\tan (c+d x)} \left (15 \sqrt {2} \left (2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )+56 i \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)\right )\right )\right )\right )}{192 a^3 d (i+\cot (c+d x))^3} \]
((I/192)*Sqrt[Cot[c + d*x]]*(32*I + (I + Cot[c + d*x])*(64 - I*(I + Cot[c + d*x])*(120 + (I + Cot[c + d*x])*Sqrt[Tan[c + d*x]]*(15*Sqrt[2]*(2*ArcTan [1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] ] + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*S qrt[Tan[c + d*x]] + Tan[c + d*x]]) + (56*I)*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2))))))/(a^3*d*(I + Cot[c + d*x])^3)
Time = 1.07 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.97, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.885, Rules used = {3042, 4156, 3042, 4042, 27, 3042, 4079, 27, 3042, 4079, 27, 3042, 4017, 25, 27, 1482, 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.
| \(\displaystyle \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cot (c+d x)^{7/2} (a+i a \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {1}{\sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^3}dx\) |
| \(\Big \downarrow \) 4042 |
| \(\displaystyle \frac {\int -\frac {11 i a-5 a \cot (c+d x)}{2 \sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)^2}dx}{6 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {11 i a-5 a \cot (c+d x)}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)^2}dx}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {5 \tan \left (c+d x+\frac {\pi }{2}\right ) a+11 i a}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle -\frac {\frac {\int \frac {12 \left (2 i \cot (c+d x) a^2+3 a^2\right )}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)}dx}{4 a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 \int \frac {2 i \cot (c+d x) a^2+3 a^2}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)}dx}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3 \int \frac {3 a^2-2 i a^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {\int -\frac {7 i a^3-5 a^3 \cot (c+d x)}{2 \sqrt {\cot (c+d x)}}dx}{2 a^2}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 \left (-\frac {\int \frac {7 i a^3-5 a^3 \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{4 a^2}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3 \left (-\frac {\int \frac {5 \tan \left (c+d x+\frac {\pi }{2}\right ) a^3+7 i a^3}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a^2}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle -\frac {\frac {3 \left (-\frac {\int -\frac {a^3 (7 i-5 \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{2 a^2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {a^3 (7 i-5 \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{2 a^2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {a \int \frac {7 i-5 \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {a \left (\left (\frac {5}{2}+\frac {7 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {5}{2}-\frac {7 i}{2}\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {a \left (\left (\frac {5}{2}+\frac {7 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {5}{2}-\frac {7 i}{2}\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {a \left (\left (\frac {5}{2}+\frac {7 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {5}{2}-\frac {7 i}{2}\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {a \left (\left (\frac {5}{2}+\frac {7 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {5}{2}-\frac {7 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {a \left (\left (\frac {5}{2}+\frac {7 i}{2}\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\left (\frac {5}{2}-\frac {7 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {a \left (\left (\frac {5}{2}+\frac {7 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\left (\frac {5}{2}-\frac {7 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {a \left (\left (\frac {5}{2}+\frac {7 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\left (\frac {5}{2}-\frac {7 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {a \left (\left (\frac {5}{2}+\frac {7 i}{2}\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\left (\frac {5}{2}-\frac {7 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 i a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}-\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
-1/6*Sqrt[Cot[c + d*x]]/(d*(I*a + a*Cot[c + d*x])^3) - (((-4*I)*a*Sqrt[Cot [c + d*x]])/(d*(I*a + a*Cot[c + d*x])^2) + (3*((-5*a^2*Sqrt[Cot[c + d*x]]) /(2*d*(I*a + a*Cot[c + d*x])) + (a*((-5/2 + (7*I)/2)*(-(ArcTan[1 - Sqrt[2] *Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqr t[2]) + (5/2 + (7*I)/2)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqr t[2]))))/(2*d)))/a^2)/(12*a^2)
3.8.50.3.1 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[((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[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) In t[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Time = 2.19 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.47
| method | result | size |
| derivativedivides | \(\frac {-\frac {\arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4 \left (\sqrt {2}-i \sqrt {2}\right )}+\frac {5 \left (\cot ^{\frac {5}{2}}\left (d x +c \right )\right )+\frac {38 i \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-9 \left (\sqrt {\cot }\left (d x +c \right )\right )}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {3 \arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{2 \left (\sqrt {2}+i \sqrt {2}\right )}}{a^{3} d}\) | \(128\) |
| default | \(\frac {-\frac {\arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4 \left (\sqrt {2}-i \sqrt {2}\right )}+\frac {5 \left (\cot ^{\frac {5}{2}}\left (d x +c \right )\right )+\frac {38 i \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-9 \left (\sqrt {\cot }\left (d x +c \right )\right )}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {3 \arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{2 \left (\sqrt {2}+i \sqrt {2}\right )}}{a^{3} d}\) | \(128\) |
1/a^3/d*(-1/4/(2^(1/2)-I*2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(2^(1/2)-I*2^( 1/2)))+1/8*(5*cot(d*x+c)^(5/2)+38/3*I*cot(d*x+c)^(3/2)-9*cot(d*x+c)^(1/2)) /(I+cot(d*x+c))^3+3/2/(2^(1/2)+I*2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(2^(1/ 2)+I*2^(1/2))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (206) = 412\).
Time = 0.27 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.90 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=-\frac {{\left (12 \, a^{3} d \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-2 \, {\left (8 \, {\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - 12 \, a^{3} d \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-2 \, {\left (8 \, {\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - 12 \, a^{3} d \sqrt {\frac {9 i}{16 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left (4 \, {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {9 i}{16 \, a^{6} d^{2}}} + 3\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3} d}\right ) + 12 \, a^{3} d \sqrt {\frac {9 i}{16 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {{\left (4 \, {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {9 i}{16 \, a^{6} d^{2}}} - 3\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3} d}\right ) - \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} {\left (-20 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 26 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 7 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{48 \, a^{3} d} \]
-1/48*(12*a^3*d*sqrt(-1/64*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(-2*(8*(I*a ^3*d*e^(2*I*d*x + 2*I*c) - I*a^3*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2 *I*d*x + 2*I*c) - 1))*sqrt(-1/64*I/(a^6*d^2)) - I*e^(2*I*d*x + 2*I*c))*e^( -2*I*d*x - 2*I*c)) - 12*a^3*d*sqrt(-1/64*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)* log(-2*(8*(-I*a^3*d*e^(2*I*d*x + 2*I*c) + I*a^3*d)*sqrt((I*e^(2*I*d*x + 2* I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-1/64*I/(a^6*d^2)) - I*e^(2*I*d* x + 2*I*c))*e^(-2*I*d*x - 2*I*c)) - 12*a^3*d*sqrt(9/16*I/(a^6*d^2))*e^(6*I *d*x + 6*I*c)*log(1/4*(4*(a^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*sqrt((I*e^(2* I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(9/16*I/(a^6*d^2)) + 3) *e^(-2*I*d*x - 2*I*c)/(a^3*d)) + 12*a^3*d*sqrt(9/16*I/(a^6*d^2))*e^(6*I*d* x + 6*I*c)*log(-1/4*(4*(a^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*sqrt((I*e^(2*I* d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(9/16*I/(a^6*d^2)) - 3)*e ^(-2*I*d*x - 2*I*c)/(a^3*d)) - sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d* x + 2*I*c) - 1))*(-20*I*e^(6*I*d*x + 6*I*c) + 26*I*e^(4*I*d*x + 4*I*c) - 7 *I*e^(2*I*d*x + 2*I*c) + I))*e^(-6*I*d*x - 6*I*c)/(a^3*d)
Timed out. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\int { \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]