Integrand size = 28, antiderivative size = 201 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac {5}{2 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}-\frac {7 \sqrt {a+i a \tan (c+d x)}}{2 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {13 i \sqrt {a+i a \tan (c+d x)}}{2 a^2 d \sqrt {\tan (c+d x)}} \]
(-1/4+1/4*I)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/ 2))/a^(3/2)/d+13/2*I*(a+I*a*tan(d*x+c))^(1/2)/a^2/d/tan(d*x+c)^(1/2)+5/2/a /d/(a+I*a*tan(d*x+c))^(1/2)/tan(d*x+c)^(3/2)-7/2*(a+I*a*tan(d*x+c))^(1/2)/ a^2/d/tan(d*x+c)^(3/2)+1/3/d/tan(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^(3/2)
Time = 1.39 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) (1+i \tan (c+d x)) \tan (c+d x) \sqrt {i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}+2 a \left (4 i+12 \tan (c+d x)+57 i \tan ^2(c+d x)-39 \tan ^3(c+d x)\right )}{12 a^2 d \tan ^{\frac {3}{2}}(c+d x) (-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)}} \]
(3*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[I*a*Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d *x]]]*(1 + I*Tan[c + d*x])*Tan[c + d*x]*Sqrt[I*a*Tan[c + d*x]]*Sqrt[a + I* a*Tan[c + d*x]] + 2*a*(4*I + 12*Tan[c + d*x] + (57*I)*Tan[c + d*x]^2 - 39* Tan[c + d*x]^3))/(12*a^2*d*Tan[c + d*x]^(3/2)*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])
Time = 1.17 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.05, 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, 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.
| \(\displaystyle \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^{5/2} (a+i a \tan (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4042 |
| \(\displaystyle \frac {\int \frac {3 (3 a-2 i a \tan (c+d x))}{2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {i \tan (c+d x) a+a}}dx}{3 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a-2 i a \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {i \tan (c+d x) a+a}}dx}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a-2 i a \tan (c+d x)}{\tan (c+d x)^{5/2} \sqrt {i \tan (c+d x) a+a}}dx}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (21 a^2-20 i a^2 \tan (c+d x)\right )}{2 \tan ^{\frac {5}{2}}(c+d x)}dx}{a^2}+\frac {5 a}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (21 a^2-20 i a^2 \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)}dx}{2 a^2}+\frac {5 a}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (21 a^2-20 i a^2 \tan (c+d x)\right )}{\tan (c+d x)^{5/2}}dx}{2 a^2}+\frac {5 a}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {3 \sqrt {i \tan (c+d x) a+a} \left (14 \tan (c+d x) a^3+13 i a^3\right )}{2 \tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {14 a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {5 a}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (14 \tan (c+d x) a^3+13 i a^3\right )}{\tan ^{\frac {3}{2}}(c+d x)}dx}{a}-\frac {14 a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {5 a}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (14 \tan (c+d x) a^3+13 i a^3\right )}{\tan (c+d x)^{3/2}}dx}{a}-\frac {14 a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {5 a}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 \int \frac {a^4 \sqrt {i \tan (c+d x) a+a}}{2 \sqrt {\tan (c+d x)}}dx}{a}-\frac {26 i a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{a}-\frac {14 a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {5 a}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {a^3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {26 i a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{a}-\frac {14 a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {5 a}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {a^3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {26 i a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{a}-\frac {14 a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {5 a}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {2 i a^5 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {26 i a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{a}-\frac {14 a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}}{2 a^2}+\frac {5 a}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {-\frac {14 a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\frac {(1-i) a^{7/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {26 i a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{a}}{2 a^2}+\frac {5 a}{d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}}{2 a^2}+\frac {1}{3 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}\) |
1/(3*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + ((5*a)/(d*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) + ((-14*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Tan[c + d*x]^(3/2)) - (((1 - I)*a^(7/2)*ArcTanh[((1 + I)*Sqrt[a] *Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - ((26*I)*a^3*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]))/a)/(2*a^2))/(2*a^2)
3.3.24.3.1 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/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. 544 vs. \(2 (159 ) = 318\).
Time = 1.00 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.71
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (3 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 \left (\tan ^{5}\left (d x +c \right )\right )-9 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 \left (\tan ^{3}\left (d x +c \right )\right )+9 \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 \left (\tan ^{4}\left (d x +c \right )\right )+384 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{3}\left (d x +c \right )\right )+156 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{4}\left (d x +c \right )\right )-3 \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 \left (\tan ^{2}\left (d x +c \right )\right )-32 \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}-276 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{2}\left (d x +c \right )\right )-16 i \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{24 d \,a^{2} \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 )}\, \sqrt {-i a}\, \left (-\tan \left (d x +c \right )+i\right )^{3}}\) | \(545\) |
| default | \(-\frac {\sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (3 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 \left (\tan ^{5}\left (d x +c \right )\right )-9 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 \left (\tan ^{3}\left (d x +c \right )\right )+9 \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 \left (\tan ^{4}\left (d x +c \right )\right )+384 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{3}\left (d x +c \right )\right )+156 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{4}\left (d x +c \right )\right )-3 \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 \left (\tan ^{2}\left (d x +c \right )\right )-32 \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}-276 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{2}\left (d x +c \right )\right )-16 i \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{24 d \,a^{2} \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 )}\, \sqrt {-i a}\, \left (-\tan \left (d x +c \right )+i\right )^{3}}\) | \(545\) |
-1/24/d*(a*(1+I*tan(d*x+c)))^(1/2)/a^2/tan(d*x+c)^(3/2)*(3*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)^5-9*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)^3+9*2^(1/2)*ln((2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*t an(d*x+c)))^(1/2)-I*a+3*a*tan(d*x+c))/(tan(d*x+c)+I))*a*tan(d*x+c)^4+384*( -I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*tan(d*x+c)^3+156*I*(-I*a )^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*tan(d*x+c)^4-3*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)^2-32*tan(d*x+c)*(a*tan(d*x+c)*(1+I*tan( d*x+c)))^(1/2)*(-I*a)^(1/2)-276*I*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+ c)))^(1/2)*tan(d*x+c)^2-16*I*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^ (1/2))/(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)/(-I*a)^(1/2)/(-tan(d*x+c)+I)^ 3
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (149) = 298\).
Time = 0.27 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.16 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (52 \, e^{\left (8 i \, d x + 8 i \, c\right )} - 35 \, e^{\left (6 i \, d x + 6 i \, c\right )} - 69 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 19 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} + 3 \, {\left (a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} - 2 \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} + a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} \log \left (\frac {1}{2} \, a^{2} d \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right ) - 3 \, {\left (a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} - 2 \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} + a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} \log \left (-\frac {1}{2} \, a^{2} d \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right )}{12 \, {\left (a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} - 2 \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} + a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \]
-1/12*(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))*(52*e^(8*I*d*x + 8*I*c) - 35*e^(6*I*d* x + 6*I*c) - 69*e^(4*I*d*x + 4*I*c) + 19*e^(2*I*d*x + 2*I*c) + 1) + 3*(a^2 *d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I*d*x + 3*I*c))*sqrt(-1/2*I/(a^3*d^2))*log(1/2*a^2*d*sqrt(-1/2*I/(a^3*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)) - 3*(a^2*d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I* d*x + 3*I*c))*sqrt(-1/2*I/(a^3*d^2))*log(-1/2*a^2*d*sqrt(-1/2*I/(a^3*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^2*d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^( 3*I*d*x + 3*I*c))
\[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {1}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \tan ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]
Exception generated. \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 875 vs. \(2 (149) = 298\).
Time = 8.81 (sec) , antiderivative size = 875, normalized size of antiderivative = 4.35 \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
1/3*sqrt(-2*(I*a*tan(d*x + c) + a)*a + 2*a^2)*sqrt(I*a*tan(d*x + c) + a)*( tan(d*x + c)/sqrt(((I*a*tan(d*x + c) + a)^2 - 2*(I*a*tan(d*x + c) + a)*a + a^2)/a^2) + 1)*(5*(I*a*tan(d*x + c) + a)/(a^3*d) - 6/(a^2*d))*abs(a)/(a^2 *tan(d*x + c)^2) + 1/4*(a*sqrt(abs(a)) + I*abs(a)^(3/2))*arctan(1/16*sqrt( 2)*((sqrt(2)*sqrt(I*a*tan(d*x + c) + a)*(-I*abs(a)/a + 1)*abs(a)^(3/2)/a^2 - sqrt(-2*(I*a*tan(d*x + c) + a)*a + 2*a^2)*(tan(d*x + c)/sqrt(((I*a*tan( d*x + c) + a)^2 - 2*(I*a*tan(d*x + c) + a)*a + a^2)/a^2) + 1)*abs(a)/a^2)^ 2 + 12*I))/(a^3*d) + 2/3*sqrt(2)*(15*(sqrt(2)*sqrt(I*a*tan(d*x + c) + a)*( -I*abs(a)/a + 1)*abs(a)^(3/2)/a^2 - sqrt(-2*(I*a*tan(d*x + c) + a)*a + 2*a ^2)*(tan(d*x + c)/sqrt(((I*a*tan(d*x + c) + a)^2 - 2*(I*a*tan(d*x + c) + a )*a + a^2)/a^2) + 1)*abs(a)/a^2)^4*a*sqrt(abs(a)) + 15*I*(sqrt(2)*sqrt(I*a *tan(d*x + c) + a)*(-I*abs(a)/a + 1)*abs(a)^(3/2)/a^2 - sqrt(-2*(I*a*tan(d *x + c) + a)*a + 2*a^2)*(tan(d*x + c)/sqrt(((I*a*tan(d*x + c) + a)^2 - 2*( I*a*tan(d*x + c) + a)*a + a^2)/a^2) + 1)*abs(a)/a^2)^4*abs(a)^(3/2) - 168* I*(sqrt(2)*sqrt(I*a*tan(d*x + c) + a)*(-I*abs(a)/a + 1)*abs(a)^(3/2)/a^2 - sqrt(-2*(I*a*tan(d*x + c) + a)*a + 2*a^2)*(tan(d*x + c)/sqrt(((I*a*tan(d* x + c) + a)^2 - 2*(I*a*tan(d*x + c) + a)*a + a^2)/a^2) + 1)*abs(a)/a^2)^2* a*sqrt(abs(a)) + 168*(sqrt(2)*sqrt(I*a*tan(d*x + c) + a)*(-I*abs(a)/a + 1) *abs(a)^(3/2)/a^2 - sqrt(-2*(I*a*tan(d*x + c) + a)*a + 2*a^2)*(tan(d*x + c )/sqrt(((I*a*tan(d*x + c) + a)^2 - 2*(I*a*tan(d*x + c) + a)*a + a^2)/a^...
Timed out. \[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {1}{{\mathrm {tan}\left (c+d\,x\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]