Integrand size = 28, antiderivative size = 199 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{5/2} d}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}+\frac {17}{30 a d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {151}{60 a^2 d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {317 \sqrt {a+i a \tan (c+d x)}}{60 a^3 d \sqrt {\tan (c+d x)}} \] Output:
(1/8+1/8*I)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2 ))/a^(5/2)/d+1/5/d/tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(5/2)+17/30/a/d/tan (d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(3/2)+151/60/a^2/d/tan(d*x+c)^(1/2)/(a+I* a*tan(d*x+c))^(1/2)-317/60*(a+I*a*tan(d*x+c))^(1/2)/a^3/d/tan(d*x+c)^(1/2)
Time = 1.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\frac {15 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)}+\frac {2 \sqrt {a+i a \tan (c+d x)} \left (-120 i+615 \tan (c+d x)+800 i \tan ^2(c+d x)-317 \tan ^3(c+d x)\right )}{(-i+\tan (c+d x))^3}}{120 a^3 d \sqrt {\tan (c+d x)}} \] Input:
Integrate[1/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]
Output:
(15*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[I*a*Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[I*a*Tan[c + d*x]] + (2*Sqrt[a + I*a*Tan[c + d*x]]*(-120*I + 61 5*Tan[c + d*x] + (800*I)*Tan[c + d*x]^2 - 317*Tan[c + d*x]^3))/(-I + Tan[c + d*x])^3)/(120*a^3*d*Sqrt[Tan[c + d*x]])
Time = 1.16 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.07, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {3042, 4042, 27, 3042, 4079, 27, 3042, 4079, 27, 3042, 4081, 27, 3042, 4027, 218}
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.
| \(\displaystyle \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^{3/2} (a+i a \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4042 |
| \(\displaystyle \frac {\int \frac {11 a-6 i a \tan (c+d x)}{2 \tan ^{\frac {3}{2}}(c+d x) (i \tan (c+d x) a+a)^{3/2}}dx}{5 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {11 a-6 i a \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (i \tan (c+d x) a+a)^{3/2}}dx}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {11 a-6 i a \tan (c+d x)}{\tan (c+d x)^{3/2} (i \tan (c+d x) a+a)^{3/2}}dx}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\int \frac {83 a^2-68 i a^2 \tan (c+d x)}{2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {i \tan (c+d x) a+a}}dx}{3 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {83 a^2-68 i a^2 \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {i \tan (c+d x) a+a}}dx}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {83 a^2-68 i a^2 \tan (c+d x)}{\tan (c+d x)^{3/2} \sqrt {i \tan (c+d x) a+a}}dx}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (317 a^3-302 i a^3 \tan (c+d x)\right )}{2 \tan ^{\frac {3}{2}}(c+d x)}dx}{a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (317 a^3-302 i a^3 \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)}dx}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (317 a^3-302 i a^3 \tan (c+d x)\right )}{\tan (c+d x)^{3/2}}dx}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \int \frac {15 i a^4 \sqrt {i \tan (c+d x) a+a}}{2 \sqrt {\tan (c+d x)}}dx}{a}-\frac {634 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {15 i a^3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {634 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {15 i a^3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {634 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \frac {\frac {\frac {\frac {30 a^5 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {634 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{2 a^2}+\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {151 a^2}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\frac {(15+15 i) a^{7/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {634 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{2 a^2}}{6 a^2}+\frac {17 a}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}\) |
Input:
Int[1/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]
Output:
1/(5*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)) + ((17*a)/(3*d*Sqr t[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + ((151*a^2)/(d*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (((15 + 15*I)*a^(7/2)*ArcTanh[((1 + I )*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (634*a^3*Sq rt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]))/(2*a^2))/(6*a^2))/(10*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/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*a*(b/f) Subst[Int[1/(a*c - b*d - 2* a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && N eQ[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[((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*d - B*c)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(a*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c* m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Fr eeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (158 ) = 316\).
Time = 1.61 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.12
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (1268 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )^{4}+60 i \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{4}-15 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{5}-5660 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\, \tan \left (d x +c \right )^{2}-60 i \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{2}-4468 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )^{3}+90 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{3}+2940 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )-15 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )+480 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{240 d \,a^{3} \sqrt {\tan \left (d x +c \right )}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (-\tan \left (d x +c \right )+i\right )^{4} \sqrt {-i a}}\) | \(620\) |
| default | \(-\frac {\sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (1268 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )^{4}+60 i \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{4}-15 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{5}-5660 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\, \tan \left (d x +c \right )^{2}-60 i \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{2}-4468 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )^{3}+90 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{3}+2940 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )-15 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )+480 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{240 d \,a^{3} \sqrt {\tan \left (d x +c \right )}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (-\tan \left (d x +c \right )+i\right )^{4} \sqrt {-i a}}\) | \(620\) |
Input:
int(1/tan(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/240/d*(a*(1+I*tan(d*x+c)))^(1/2)/a^3*(1268*(-I*a)^(1/2)*(a*tan(d*x+c)*( 1+I*tan(d*x+c)))^(1/2)*tan(d*x+c)^4+60*I*2^(1/2)*ln(-(-2*2^(1/2)*(-I*a)^(1 /2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+I*a-3*a*tan(d*x+c))/(tan(d*x+c)+ I))*a*tan(d*x+c)^4-15*2^(1/2)*ln(-(-2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*( 1+I*tan(d*x+c)))^(1/2)+I*a-3*a*tan(d*x+c))/(tan(d*x+c)+I))*a*tan(d*x+c)^5- 5660*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)*tan(d*x+c)^2-60*I* 2^(1/2)*ln(-(-2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2) +I*a-3*a*tan(d*x+c))/(tan(d*x+c)+I))*a*tan(d*x+c)^2-4468*I*(a*tan(d*x+c)*( 1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)*tan(d*x+c)^3+90*2^(1/2)*ln(-(-2*2^(1/2 )*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+I*a-3*a*tan(d*x+c))/( tan(d*x+c)+I))*a*tan(d*x+c)^3+2940*I*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2) *(-I*a)^(1/2)*tan(d*x+c)-15*2^(1/2)*ln(-(-2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d* x+c)*(1+I*tan(d*x+c)))^(1/2)+I*a-3*a*tan(d*x+c))/(tan(d*x+c)+I))*a*tan(d*x +c)+480*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2))/tan(d*x+c)^(1/ 2)/(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)/(-tan(d*x+c)+I)^4/(-I*a)^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (149) = 298\).
Time = 0.12 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.97 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-463 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 269 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 220 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 29 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} - 30 \, {\left (a^{3} d e^{\left (7 i \, d x + 7 i \, c\right )} - a^{3} d e^{\left (5 i \, d x + 5 i \, c\right )}\right )} \sqrt {\frac {i}{8 \, a^{5} d^{2}}} \log \left (i \, a^{3} d \sqrt {\frac {i}{8 \, a^{5} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right ) + 30 \, {\left (a^{3} d e^{\left (7 i \, d x + 7 i \, c\right )} - a^{3} d e^{\left (5 i \, d x + 5 i \, c\right )}\right )} \sqrt {\frac {i}{8 \, a^{5} d^{2}}} \log \left (-i \, a^{3} d \sqrt {\frac {i}{8 \, a^{5} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right )}{120 \, {\left (a^{3} d e^{\left (7 i \, d x + 7 i \, c\right )} - a^{3} d e^{\left (5 i \, d x + 5 i \, c\right )}\right )}} \] Input:
integrate(1/tan(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="fricas ")
Output:
1/120*(sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^(2*I*d*x + 2*I *c) + I)/(e^(2*I*d*x + 2*I*c) + 1))*(-463*I*e^(8*I*d*x + 8*I*c) - 269*I*e^ (6*I*d*x + 6*I*c) + 220*I*e^(4*I*d*x + 4*I*c) + 29*I*e^(2*I*d*x + 2*I*c) + 3*I) - 30*(a^3*d*e^(7*I*d*x + 7*I*c) - a^3*d*e^(5*I*d*x + 5*I*c))*sqrt(1/ 8*I/(a^5*d^2))*log(I*a^3*d*sqrt(1/8*I/(a^5*d^2))*e^(I*d*x + I*c) + 1/4*sqr t(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/( e^(2*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1)) + 30*(a^3*d*e^(7*I*d* x + 7*I*c) - a^3*d*e^(5*I*d*x + 5*I*c))*sqrt(1/8*I/(a^5*d^2))*log(-I*a^3*d *sqrt(1/8*I/(a^5*d^2))*e^(I*d*x + I*c) + 1/4*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))* (e^(2*I*d*x + 2*I*c) + 1)))/(a^3*d*e^(7*I*d*x + 7*I*c) - a^3*d*e^(5*I*d*x + 5*I*c))
Timed out. \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/tan(d*x+c)**(3/2)/(a+I*a*tan(d*x+c))**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/tan(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="maxima ")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/tan(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeDone
Timed out. \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {1}{{\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \] Input:
int(1/(tan(c + d*x)^(3/2)*(a + a*tan(c + d*x)*1i)^(5/2)),x)
Output:
int(1/(tan(c + d*x)^(3/2)*(a + a*tan(c + d*x)*1i)^(5/2)), x)
\[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {\int \frac {1}{\sqrt {\tan \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{3}-2 \sqrt {\tan \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{2} i -\sqrt {\tan \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )}d x}{\sqrt {a}\, a^{2}} \] Input:
int(1/tan(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x)
Output:
( - int(1/(sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)**3 - 2 *sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)**2*i - sqrt(tan( c + d*x))*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)),x))/(sqrt(a)*a**2)