Integrand size = 26, antiderivative size = 230 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx=\frac {\left (\frac {7}{16}-\frac {5 i}{16}\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}-\frac {\left (\frac {7}{16}-\frac {5 i}{16}\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {\left (\frac {7}{16}+\frac {5 i}{16}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {\cot ^{\frac {3}{2}}(c+d x)}{3 a d (i a+a \cot (c+d x))^2}+\frac {5 \sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )} \] Output:
(-7/32+5/32*I)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/a^3/d+(-7/32+5/ 32*I)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/a^3/d+(7/32+5/32*I)*arcta nh(2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^(1/2)/a^3/d+1/6*cot(d*x+c)^( 5/2)/d/(I*a+a*cot(d*x+c))^3+1/3*cot(d*x+c)^(3/2)/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 = 0.41 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx=\frac {\cot ^{\frac {3}{2}}(c+d x) \left (-\frac {4 \cot ^3(c+d x)}{(i+\cot (c+d x))^3}+6 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-i \tan (c+d x)\right )+\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},i \tan (c+d x)\right )+\frac {3 i}{-i+\tan (c+d x)}\right )}{24 a^3 d} \] Input:
Integrate[Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^3,x]
Output:
(Cot[c + d*x]^(3/2)*((-4*Cot[c + d*x]^3)/(I + Cot[c + d*x])^3 + 6*Hypergeo metric2F1[-3/2, 1, -1/2, (-I)*Tan[c + d*x]] + Hypergeometric2F1[-3/2, 1, - 1/2, I*Tan[c + d*x]] + (3*I)/(-I + Tan[c + d*x])))/(24*a^3*d)
Time = 1.06 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.15, 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, 4041, 27, 3042, 4078, 27, 3042, 4078, 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.
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)}}{(a+i a \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {\cot ^{\frac {7}{2}}(c+d x)}{(a \cot (c+d x)+i a)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}{\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^3}dx\) |
| \(\Big \downarrow \) 4041 |
| \(\displaystyle \frac {\int -\frac {\cot ^{\frac {3}{2}}(c+d x) (5 i a-11 a \cot (c+d x))}{2 (\cot (c+d x) a+i a)^2}dx}{6 a^2}+\frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\int \frac {\cot ^{\frac {3}{2}}(c+d x) (5 i a-11 a \cot (c+d x))}{(\cot (c+d x) a+i a)^2}dx}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (11 \tan \left (c+d x+\frac {\pi }{2}\right ) a+5 i a\right )}{\left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{12 a^2}\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {\int \frac {12 \sqrt {\cot (c+d x)} \left (2 i a^2-3 a^2 \cot (c+d x)\right )}{\cot (c+d x) a+i a}dx}{4 a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \int \frac {\sqrt {\cot (c+d x)} \left (2 i a^2-3 a^2 \cot (c+d x)\right )}{\cot (c+d x) a+i a}dx}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (3 \tan \left (c+d x+\frac {\pi }{2}\right ) a^2+2 i a^2\right )}{i a-a \tan \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (\frac {\int \frac {5 i a^3-7 a^3 \cot (c+d x)}{2 \sqrt {\cot (c+d x)}}dx}{2 a^2}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (\frac {\int \frac {5 i a^3-7 a^3 \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{4 a^2}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (\frac {\int \frac {7 \tan \left (c+d x+\frac {\pi }{2}\right ) a^3+5 i a^3}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a^2}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (\frac {\int -\frac {a^3 (5 i-7 \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{2 a^2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {\int \frac {a^3 (5 i-7 \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{2 a^2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \int \frac {5 i-7 \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\left (\frac {7}{2}+\frac {5 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {7}{2}-\frac {5 i}{2}\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\left (\frac {7}{2}+\frac {5 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {7}{2}-\frac {5 i}{2}\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\left (\frac {7}{2}+\frac {5 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {7}{2}-\frac {5 i}{2}\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\left (\frac {7}{2}+\frac {5 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {7}{2}-\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\left (\frac {7}{2}+\frac {5 i}{2}\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\left (\frac {7}{2}-\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\left (\frac {7}{2}+\frac {5 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\left (\frac {7}{2}-\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\left (\frac {7}{2}+\frac {5 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\left (\frac {7}{2}-\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{2 d}-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {5 a^2 \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}-\frac {a \left (\left (\frac {7}{2}+\frac {5 i}{2}\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\left (\frac {7}{2}-\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{2 d}\right )}{a^2}-\frac {4 a \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
Input:
Int[Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^3,x]
Output:
Cot[c + d*x]^(5/2)/(6*d*(I*a + a*Cot[c + d*x])^3) - ((-4*a*Cot[c + d*x]^(3 /2))/(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*((-7/2 + (5*I)/2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[C ot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + (7/2 + (5*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*Sqrt[2]))) )/(2*d)))/a^2)/(12*a^2)
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[(-(b*c - a*d))*(a + b*Tan[e + f*x])^m* ((c + d*Tan[e + f*x])^(n - 1)/(2*a*f*m)), x] + Simp[1/(2*a^2*m) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1)) - d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && GtQ[n, 1] && (In tegerQ[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*b - a*B))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(2*a*f*m)), x] + Simp[1/(2*a^2*m) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f* x])^(n - 1)*Simp[A*(a*c*m + b*d*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a *A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, 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 = 0.62 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.56
| method | result | size |
| derivativedivides | \(\frac {-\frac {\arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{4 \left (\sqrt {2}-i \sqrt {2}\right )}-\frac {-9 \cot \left (d x +c \right )^{\frac {5}{2}}-\frac {38 i \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+5 \sqrt {\cot \left (d x +c \right )}}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}-\frac {3 \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{i \sqrt {2}+\sqrt {2}}\right )}{2 \left (i \sqrt {2}+\sqrt {2}\right )}}{d \,a^{3}}\) | \(128\) |
| default | \(\frac {-\frac {\arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{4 \left (\sqrt {2}-i \sqrt {2}\right )}-\frac {-9 \cot \left (d x +c \right )^{\frac {5}{2}}-\frac {38 i \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+5 \sqrt {\cot \left (d x +c \right )}}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}-\frac {3 \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{i \sqrt {2}+\sqrt {2}}\right )}{2 \left (i \sqrt {2}+\sqrt {2}\right )}}{d \,a^{3}}\) | \(128\) |
Input:
int(cot(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d/a^3*(-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*(-9*cot(d*x+c)^(5/2)-38/3*I*cot(d*x+c)^(3/2)+5*cot(d*x+c)^(1/2) )/(I+cot(d*x+c))^3-3/2/(I*2^(1/2)+2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(I*2^ (1/2)+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. \(2 (175) = 350\).
Time = 0.12 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.27 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")
Output:
-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) + 14*I*e^(4*I*d*x + 4*I*c) + 5 *I*e^(2*I*d*x + 2*I*c) + I))*e^(-6*I*d*x - 6*I*c)/(a^3*d)
\[ \int \frac {\sqrt {\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx=\frac {i \int \frac {\sqrt {\cot {\left (c + d x \right )}}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx}{a^{3}} \] Input:
integrate(cot(d*x+c)**(1/2)/(a+I*a*tan(d*x+c))**3,x)
Output:
I*Integral(sqrt(cot(c + d*x))/(tan(c + d*x)**3 - 3*I*tan(c + d*x)**2 - 3*t an(c + d*x) + I), x)/a**3
Exception generated. \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(cot(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx=-\frac {i \, {\left (\left (18 i + 18\right ) \, \sqrt {2} \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + \left (3 i - 3\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + \frac {2 \, {\left (15 \, \tan \left (d x + c\right )^{\frac {5}{2}} - 38 i \, \tan \left (d x + c\right )^{\frac {3}{2}} - 27 \, \sqrt {\tan \left (d x + c\right )}\right )}}{{\left (\tan \left (d x + c\right ) - i\right )}^{3}}\right )}}{48 \, a^{3} d} \] Input:
integrate(cot(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")
Output:
-1/48*I*((18*I + 18)*sqrt(2)*arctan((1/2*I + 1/2)*sqrt(2)*sqrt(tan(d*x + c ))) + (3*I - 3)*sqrt(2)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c))) + 2*(15*tan(d*x + c)^(5/2) - 38*I*tan(d*x + c)^(3/2) - 27*sqrt(tan(d*x + c )))/(tan(d*x + c) - I)^3)/(a^3*d)
Timed out. \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx=\int \frac {\sqrt {\mathrm {cot}\left (c+d\,x\right )}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \] Input:
int(cot(c + d*x)^(1/2)/(a + a*tan(c + d*x)*1i)^3,x)
Output:
int(cot(c + d*x)^(1/2)/(a + a*tan(c + d*x)*1i)^3, x)
\[ \int \frac {\sqrt {\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx=-\frac {\int \frac {\sqrt {\cot \left (d x +c \right )}}{\tan \left (d x +c \right )^{3} i +3 \tan \left (d x +c \right )^{2}-3 \tan \left (d x +c \right ) i -1}d x}{a^{3}} \] Input:
int(cot(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^3,x)
Output:
( - int(sqrt(cot(c + d*x))/(tan(c + d*x)**3*i + 3*tan(c + d*x)**2 - 3*tan( c + d*x)*i - 1),x))/a**3