Integrand size = 28, antiderivative size = 238 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{5/2} d}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}+\frac {7}{10 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac {89}{20 a^2 d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}-\frac {361 \sqrt {a+i a \tan (c+d x)}}{60 a^3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {707 i \sqrt {a+i a \tan (c+d x)}}{60 a^3 d \sqrt {\tan (c+d x)}} \] 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)^(3/2)/(a+I*a*tan(d*x+c))^(5/2)+7/10/a/d/tan (d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(3/2)+89/20/a^2/d/tan(d*x+c)^(3/2)/(a+I*a *tan(d*x+c))^(1/2)-361/60*(a+I*a*tan(d*x+c))^(1/2)/a^3/d/tan(d*x+c)^(3/2)+ 707/60*I*(a+I*a*tan(d*x+c))^(1/2)/a^3/d/tan(d*x+c)^(1/2)
Time = 1.38 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\frac {15 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) (i a \tan (c+d x))^{3/2}+\frac {2 a \sqrt {a+i a \tan (c+d x)} \left (-40 i-200 \tan (c+d x)-1305 i \tan ^2(c+d x)+1760 \tan ^3(c+d x)+707 i \tan ^4(c+d x)\right )}{(-i+\tan (c+d x))^3}}{120 a^4 d \tan ^{\frac {3}{2}}(c+d x)} \] Input:
Integrate[1/(Tan[c + d*x]^(5/2)*(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]]]*(I*a*Tan[c + d*x])^(3/2) + (2*a*Sqrt[a + I*a*Tan[c + d*x]]*(-40*I - 200*Tan[c + d*x] - (1305*I)*Tan[c + d*x]^2 + 1760*Tan[c + d*x]^3 + (707*I )*Tan[c + d*x]^4))/(-I + Tan[c + d*x])^3)/(120*a^4*d*Tan[c + d*x]^(3/2))
Time = 1.51 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.08, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3042, 4042, 27, 3042, 4079, 27, 3042, 4079, 27, 3042, 4081, 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.
| \(\displaystyle \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^{5/2} (a+i a \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4042 |
| \(\displaystyle \frac {\int \frac {13 a-8 i a \tan (c+d x)}{2 \tan ^{\frac {5}{2}}(c+d x) (i \tan (c+d x) a+a)^{3/2}}dx}{5 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {13 a-8 i a \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (i \tan (c+d x) a+a)^{3/2}}dx}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {13 a-8 i a \tan (c+d x)}{\tan (c+d x)^{5/2} (i \tan (c+d x) a+a)^{3/2}}dx}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\int \frac {3 \left (47 a^2-42 i a^2 \tan (c+d x)\right )}{2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {i \tan (c+d x) a+a}}dx}{3 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {47 a^2-42 i a^2 \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {i \tan (c+d x) a+a}}dx}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {47 a^2-42 i a^2 \tan (c+d x)}{\tan (c+d x)^{5/2} \sqrt {i \tan (c+d x) a+a}}dx}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (361 a^3-356 i a^3 \tan (c+d x)\right )}{2 \tan ^{\frac {5}{2}}(c+d x)}dx}{a^2}+\frac {89 a^2}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (361 a^3-356 i a^3 \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)}dx}{2 a^2}+\frac {89 a^2}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (361 a^3-356 i a^3 \tan (c+d x)\right )}{\tan (c+d x)^{5/2}}dx}{2 a^2}+\frac {89 a^2}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \int -\frac {\sqrt {i \tan (c+d x) a+a} \left (722 \tan (c+d x) a^4+707 i a^4\right )}{2 \tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {722 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {89 a^2}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (722 \tan (c+d x) a^4+707 i a^4\right )}{\tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {722 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {89 a^2}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (722 \tan (c+d x) a^4+707 i a^4\right )}{\tan (c+d x)^{3/2}}dx}{3 a}-\frac {722 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {89 a^2}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {2 \int \frac {15 a^5 \sqrt {i \tan (c+d x) a+a}}{2 \sqrt {\tan (c+d x)}}dx}{a}-\frac {1414 i a^4 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {722 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {89 a^2}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {15 a^4 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {1414 i a^4 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {722 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {89 a^2}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {15 a^4 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {1414 i a^4 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {722 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {89 a^2}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \frac {\frac {\frac {-\frac {-\frac {30 i a^6 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {1414 i a^4 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {722 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {89 a^2}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {89 a^2}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}+\frac {-\frac {722 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\frac {(15-15 i) a^{9/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {1414 i a^4 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}}{2 a^2}}{2 a^2}+\frac {7 a}{d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}}{10 a^2}+\frac {1}{5 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}\) |
Input:
Int[1/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]
Output:
1/(5*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)) + ((7*a)/(d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + ((89*a^2)/(d*Tan[c + d*x]^(3/ 2)*Sqrt[a + I*a*Tan[c + d*x]]) + ((-722*a^3*Sqrt[a + I*a*Tan[c + d*x]])/(3 *d*Tan[c + d*x]^(3/2)) - (((15 - 15*I)*a^(9/2)*ArcTanh[((1 + I)*Sqrt[a]*Sq rt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - ((1414*I)*a^4*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]))/(3*a))/(2*a^2))/(2*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. 660 vs. \(2 (189 ) = 378\).
Time = 1.61 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.78
| method | result | size |
| derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (15 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 )^{6}-90 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}+2828 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 )^{5}+60 \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}+15 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}-12260 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}-60 \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}+9868 \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}+640 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 )-6020 \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}-160 \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} \tan \left (d x +c \right )^{\frac {3}{2}} \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}}\) | \(661\) |
| default | \(\frac {\sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (15 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 )^{6}-90 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}+2828 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 )^{5}+60 \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}+15 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}-12260 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}-60 \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}+9868 \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}+640 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 )-6020 \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}-160 \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} \tan \left (d x +c \right )^{\frac {3}{2}} \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}}\) | \(661\) |
Input:
int(1/tan(d*x+c)^(5/2)/(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)*(15*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)^6-90*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+2828*I*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*tan(d*x+c)^5+6 0*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+15*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)^2-12260*I*(-I*a)^(1/2)*(a*tan(d*x+c)*(1 +I*tan(d*x+c)))^(1/2)*tan(d*x+c)^3-60*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+9868*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*ta n(d*x+c)^4+640*I*(-I*a)^(1/2)*tan(d*x+c)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^( 1/2)-6020*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)*tan(d*x+c)^2- 160*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2))/a^3/tan(d*x+c)^(3/ 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. 445 vs. \(2 (178) = 356\).
Time = 0.11 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.87 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (983 \, e^{\left (10 i \, d x + 10 i \, c\right )} - 544 \, e^{\left (8 i \, d x + 8 i \, c\right )} - 1179 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 381 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 36 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3\right )} + 30 \, {\left (a^{3} d e^{\left (9 i \, d x + 9 i \, c\right )} - 2 \, a^{3} d e^{\left (7 i \, d x + 7 i \, c\right )} + a^{3} d e^{\left (5 i \, d x + 5 i \, c\right )}\right )} \sqrt {-\frac {i}{8 \, a^{5} d^{2}}} \log \left (a^{3} d \sqrt {-\frac {i}{8 \, a^{5} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right ) - 30 \, {\left (a^{3} d e^{\left (9 i \, d x + 9 i \, c\right )} - 2 \, a^{3} d e^{\left (7 i \, d x + 7 i \, c\right )} + a^{3} d e^{\left (5 i \, d x + 5 i \, c\right )}\right )} \sqrt {-\frac {i}{8 \, a^{5} d^{2}}} \log \left (-a^{3} d \sqrt {-\frac {i}{8 \, a^{5} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right )}{120 \, {\left (a^{3} d e^{\left (9 i \, d x + 9 i \, c\right )} - 2 \, a^{3} d e^{\left (7 i \, d x + 7 i \, c\right )} + a^{3} d e^{\left (5 i \, d x + 5 i \, c\right )}\right )}} \] Input:
integrate(1/tan(d*x+c)^(5/2)/(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))*(983*e^(10*I*d*x + 10*I*c) - 544*e^(8 *I*d*x + 8*I*c) - 1179*e^(6*I*d*x + 6*I*c) + 381*e^(4*I*d*x + 4*I*c) + 36* e^(2*I*d*x + 2*I*c) + 3) + 30*(a^3*d*e^(9*I*d*x + 9*I*c) - 2*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(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)) - 30*(a^3*d*e^(9*I*d*x + 9*I*c) - 2*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(-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^(9*I*d*x + 9*I*c) - 2*a^3*d*e^(7 *I*d*x + 7*I*c) + a^3*d*e^(5*I*d*x + 5*I*c))
Timed out. \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/tan(d*x+c)**(5/2)/(a+I*a*tan(d*x+c))**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/tan(d*x+c)^(5/2)/(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. \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/tan(d*x+c)^(5/2)/(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. \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {1}{{\mathrm {tan}\left (c+d\,x\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \] Input:
int(1/(tan(c + d*x)^(5/2)*(a + a*tan(c + d*x)*1i)^(5/2)),x)
Output:
int(1/(tan(c + d*x)^(5/2)*(a + a*tan(c + d*x)*1i)^(5/2)), x)
\[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {\int \frac {1}{\sqrt {\tan \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{4}-2 \sqrt {\tan \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{3} i -\sqrt {\tan \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{2}}d x}{\sqrt {a}\, a^{2}} \] Input:
int(1/tan(d*x+c)^(5/2)/(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)**4 - 2 *sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)**3*i - sqrt(tan( c + d*x))*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)**2),x))/(sqrt(a)*a**2)