Integrand size = 26, antiderivative size = 209 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {25}{16}-\frac {21 i}{16}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} a^2 d}-\frac {25 \sqrt {\cot (c+d x)}}{8 a^2 d}+\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{8 a^2 d (i+\cot (c+d x))}+\frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2} \] Output:
(25/32+21/32*I)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/a^2/d+(25/32+2 1/32*I)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/a^2/d+(25/32-21/32*I)*a rctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^(1/2)/a^2/d-25/8*cot(d*x +c)^(1/2)/a^2/d+7/8*cot(d*x+c)^(3/2)/a^2/d/(I+cot(d*x+c))+1/4*cot(d*x+c)^( 5/2)/d/(I*a+a*cot(d*x+c))^2
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.48 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.69 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\cot ^{\frac {9}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}-\frac {-\frac {23 \cot ^{\frac {5}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-i \tan (c+d x)\right )}{10 d}-\frac {\cot ^{\frac {5}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},i \tan (c+d x)\right )}{5 d}+\frac {3 i \cot ^{\frac {5}{2}}(c+d x)}{2 d (i-\tan (c+d x))}}{4 a^2} \] Input:
Integrate[Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^2,x]
Output:
-1/4*Cot[c + d*x]^(9/2)/(d*(I*a + a*Cot[c + d*x])^2) - ((-23*Cot[c + d*x]^ (5/2)*Hypergeometric2F1[-5/2, 1, -3/2, (-I)*Tan[c + d*x]])/(10*d) - (Cot[c + d*x]^(5/2)*Hypergeometric2F1[-5/2, 1, -3/2, I*Tan[c + d*x]])/(5*d) + (( (3*I)/2)*Cot[c + d*x]^(5/2))/(d*(I - Tan[c + d*x])))/(4*a^2)
Time = 1.00 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.16, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.808, Rules used = {3042, 4156, 3042, 4041, 27, 3042, 4078, 3042, 4011, 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.
| \(\displaystyle \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot (c+d x)^{3/2}}{(a+i a \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {\cot ^{\frac {7}{2}}(c+d x)}{(a \cot (c+d x)+i a)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}{\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^2}dx\) |
| \(\Big \downarrow \) 4041 |
| \(\displaystyle \frac {\int -\frac {\cot ^{\frac {3}{2}}(c+d x) (5 i a-9 a \cot (c+d x))}{2 (\cot (c+d x) a+i a)}dx}{4 a^2}+\frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\int \frac {\cot ^{\frac {3}{2}}(c+d x) (5 i a-9 a \cot (c+d x))}{\cot (c+d x) a+i a}dx}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (9 \tan \left (c+d x+\frac {\pi }{2}\right ) a+5 i a\right )}{i a-a \tan \left (c+d x+\frac {\pi }{2}\right )}dx}{8 a^2}\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\int \sqrt {\cot (c+d x)} \left (21 i a^2-25 a^2 \cot (c+d x)\right )dx}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (25 \tan \left (c+d x+\frac {\pi }{2}\right ) a^2+21 i a^2\right )dx}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}+\int \frac {21 i \cot (c+d x) a^2+25 a^2}{\sqrt {\cot (c+d x)}}dx}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}+\int \frac {25 a^2-21 i a^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}+\frac {2 \int -\frac {a^2 (21 i \cot (c+d x)+25)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {2 \int \frac {a^2 (21 i \cot (c+d x)+25)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \int \frac {21 i \cot (c+d x)+25}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \left (\left (\frac {25}{2}-\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {25}{2}+\frac {21 i}{2}\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \left (\left (\frac {25}{2}-\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \left (\left (\frac {25}{2}-\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \left (\left (\frac {25}{2}-\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \left (\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \left (\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \left (\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\frac {50 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 \cot ^{\frac {3}{2}}(c+d x)}{d (\cot (c+d x)+i)}}{8 a^2}\) |
Input:
Int[Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^2,x]
Output:
Cot[c + d*x]^(5/2)/(4*d*(I*a + a*Cot[c + d*x])^2) - ((-7*Cot[c + d*x]^(3/2 ))/(d*(I + Cot[c + d*x])) + ((50*a^2*Sqrt[Cot[c + d*x]])/d - (2*a^2*((25/2 + (21*I)/2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + (25/2 - (21*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]))))/d)/(2*a^2))/(8*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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
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 = 0.53 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.62
| method | result | size |
| derivativedivides | \(\frac {-2 \sqrt {\cot \left (d x +c \right )}-\frac {i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}+\frac {-\frac {11 i \cot \left (d x +c \right )^{\frac {3}{2}}}{2}+\frac {9 \sqrt {\cot \left (d x +c \right )}}{2}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {23 i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{i \sqrt {2}+\sqrt {2}}\right )}{4 \left (i \sqrt {2}+\sqrt {2}\right )}}{a^{2} d}\) | \(130\) |
| default | \(\frac {-2 \sqrt {\cot \left (d x +c \right )}-\frac {i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}+\frac {-\frac {11 i \cot \left (d x +c \right )^{\frac {3}{2}}}{2}+\frac {9 \sqrt {\cot \left (d x +c \right )}}{2}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {23 i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{i \sqrt {2}+\sqrt {2}}\right )}{4 \left (i \sqrt {2}+\sqrt {2}\right )}}{a^{2} d}\) | \(130\) |
Input:
int(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/a^2/d*(-2*cot(d*x+c)^(1/2)-1/2*I/(2^(1/2)-I*2^(1/2))*arctan(2*cot(d*x+c) ^(1/2)/(2^(1/2)-I*2^(1/2)))+1/4*(-11/2*I*cot(d*x+c)^(3/2)+9/2*cot(d*x+c)^( 1/2))/(I+cot(d*x+c))^2+23/4*I/(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. 509 vs. \(2 (156) = 312\).
Time = 0.10 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.44 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {{\left (4 \, a^{2} d \sqrt {\frac {i}{16 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (2 \, {\left (4 \, {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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}{16 \, a^{4} 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 ) - 4 \, a^{2} d \sqrt {\frac {i}{16 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-2 \, {\left (4 \, {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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}{16 \, a^{4} 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 ) + 4 \, a^{2} d \sqrt {-\frac {529 i}{64 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac {{\left (8 \, {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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 {529 i}{64 \, a^{4} d^{2}}} + 23 i\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - 4 \, a^{2} d \sqrt {-\frac {529 i}{64 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {{\left (8 \, {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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 {529 i}{64 \, a^{4} d^{2}}} - 23 i\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} 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 (42 \, e^{\left (4 i \, d x + 4 i \, c\right )} - 9 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{16 \, a^{2} d} \] Input:
integrate(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
1/16*(4*a^2*d*sqrt(1/16*I/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*log(2*(4*(a^2*d*e ^(2*I*d*x + 2*I*c) - a^2*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(1/16*I/(a^4*d^2)) + I*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)) - 4*a^2*d*sqrt(1/16*I/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*log(-2*(4*( a^2*d*e^(2*I*d*x + 2*I*c) - a^2*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2* I*d*x + 2*I*c) - 1))*sqrt(1/16*I/(a^4*d^2)) - I*e^(2*I*d*x + 2*I*c))*e^(-2 *I*d*x - 2*I*c)) + 4*a^2*d*sqrt(-529/64*I/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*l og(1/8*(8*(a^2*d*e^(2*I*d*x + 2*I*c) - a^2*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-529/64*I/(a^4*d^2)) + 23*I)*e^(-2*I* d*x - 2*I*c)/(a^2*d)) - 4*a^2*d*sqrt(-529/64*I/(a^4*d^2))*e^(4*I*d*x + 4*I *c)*log(-1/8*(8*(a^2*d*e^(2*I*d*x + 2*I*c) - a^2*d)*sqrt((I*e^(2*I*d*x + 2 *I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-529/64*I/(a^4*d^2)) - 23*I)*e^ (-2*I*d*x - 2*I*c)/(a^2*d)) - sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*(42*e^(4*I*d*x + 4*I*c) - 9*e^(2*I*d*x + 2*I*c) - 1))*e^(- 4*I*d*x - 4*I*c)/(a^2*d)
\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=- \frac {\int \frac {\cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \] Input:
integrate(cot(d*x+c)**(3/2)/(a+I*a*tan(d*x+c))**2,x)
Output:
-Integral(cot(c + d*x)**(3/2)/(tan(c + d*x)**2 - 2*I*tan(c + d*x) - 1), x) /a**2
Exception generated. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.43 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {-\left (23 i + 23\right ) \, \sqrt {2} \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + \left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) - \frac {32}{\sqrt {\tan \left (d x + c\right )}} - \frac {2 \, {\left (9 \, \tan \left (d x + c\right )^{\frac {3}{2}} - 11 i \, \sqrt {\tan \left (d x + c\right )}\right )}}{{\left (\tan \left (d x + c\right ) - i\right )}^{2}}}{16 \, a^{2} d} \] Input:
integrate(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
1/16*(-(23*I + 23)*sqrt(2)*arctan((1/2*I + 1/2)*sqrt(2)*sqrt(tan(d*x + c)) ) + (2*I - 2)*sqrt(2)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c))) - 32/sqrt(tan(d*x + c)) - 2*(9*tan(d*x + c)^(3/2) - 11*I*sqrt(tan(d*x + c))) /(tan(d*x + c) - I)^2)/(a^2*d)
Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \] Input:
int(cot(c + d*x)^(3/2)/(a + a*tan(c + d*x)*1i)^2,x)
Output:
int(cot(c + d*x)^(3/2)/(a + a*tan(c + d*x)*1i)^2, x)
\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\int \frac {\sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )}{\tan \left (d x +c \right )^{2}-2 \tan \left (d x +c \right ) i -1}d x}{a^{2}} \] Input:
int(cot(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^2,x)
Output:
( - int((sqrt(cot(c + d*x))*cot(c + d*x))/(tan(c + d*x)**2 - 2*tan(c + d*x )*i - 1),x))/a**2