Integrand size = 28, antiderivative size = 125 \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac {\sqrt {\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {7 \sqrt {\tan (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}} \]
(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+7/6*tan(d*x+c)^(1/2)/a/d/(a+I*a*tan(d*x+c))^(1/2)+1/3*tan(d*x +c)^(1/2)/d/(a+I*a*tan(d*x+c))^(3/2)
Time = 1.22 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {-\frac {3 i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)}}{a^2}+\frac {4 \tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}}+\frac {14 \tan (c+d x)}{a \sqrt {a+i a \tan (c+d x)}}}{12 d \sqrt {\tan (c+d x)}} \]
(((-3*I)*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]])/a^2 + (4*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^(3/2) + (14*Tan[c + d*x])/(a*Sqrt[a + I*a*Tan[c + d*x]]))/(12*d*Sq rt[Tan[c + d*x]])
Time = 0.62 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {3042, 4042, 27, 3042, 4079, 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.
| \(\displaystyle \int \frac {1}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4042 |
| \(\displaystyle \frac {\int \frac {5 a-2 i a \tan (c+d x)}{2 \sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a}}dx}{3 a^2}+\frac {\sqrt {\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 a-2 i a \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a}}dx}{6 a^2}+\frac {\sqrt {\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 a-2 i a \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a}}dx}{6 a^2}+\frac {\sqrt {\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\int \frac {3 a^2 \sqrt {i \tan (c+d x) a+a}}{2 \sqrt {\tan (c+d x)}}dx}{a^2}+\frac {7 a \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {\sqrt {\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{2} \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx+\frac {7 a \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {\sqrt {\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{2} \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx+\frac {7 a \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {\sqrt {\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \frac {\frac {7 a \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}-\frac {3 i a^2 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}}{6 a^2}+\frac {\sqrt {\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\left (\frac {3}{2}-\frac {3 i}{2}\right ) \sqrt {a} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}+\frac {7 a \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {\sqrt {\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
Sqrt[Tan[c + d*x]]/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (((3/2 - (3*I)/2)* Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (7*a*Sqrt[Tan[c + d*x]])/(d*Sqrt[a + I*a*Tan[c + d*x]]))/(6*a^ 2)
3.3.22.3.1 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (98 ) = 196\).
Time = 1.10 (sec) , antiderivative size = 460, normalized size of antiderivative = 3.68
| method | result | size |
| derivativedivides | \(\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (3 i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{3}\left (d x +c \right )\right )-9 i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )+28 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{2}\left (d x +c \right )\right )+9 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{2}\left (d x +c \right )\right )-36 i \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}-3 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a +64 \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{24 d \,a^{2} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\, \left (-\tan \left (d x +c \right )+i\right )^{3}}\) | \(460\) |
| default | \(\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (3 i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{3}\left (d x +c \right )\right )-9 i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )+28 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{2}\left (d x +c \right )\right )+9 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{2}\left (d x +c \right )\right )-36 i \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}-3 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a +64 \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{24 d \,a^{2} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\, \left (-\tan \left (d x +c \right )+i\right )^{3}}\) | \(460\) |
1/24/d*tan(d*x+c)^(1/2)*(a*(1+I*tan(d*x+c)))^(1/2)*(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)^3-9*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*t an(d*x+c)+28*I*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)*tan(d*x+ c)^2+9*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)^2-36*I*(a*tan(d*x+c )*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)-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+64*tan(d*x+c)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2))/a^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. 320 vs. \(2 (91) = 182\).
Time = 0.26 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.56 \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {{\left (3 \, a^{2} d \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac {1}{2} \, a^{2} d \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right ) - 3 \, a^{2} d \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (-\frac {1}{2} \, a^{2} d \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (8 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 9 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{12 \, a^{2} d} \]
1/12*(3*a^2*d*sqrt(-1/2*I/(a^3*d^2))*e^(3*I*d*x + 3*I*c)*log(1/2*a^2*d*sqr t(-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*sqrt(-1/2*I/(a^3*d^2))*e^(3*I*d*x + 3*I* c)*log(-1/2*a^2*d*sqrt(-1/2*I/(a^3*d^2))*e^(I*d*x + I*c) + 1/4*sqrt(2)*sqr t(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)) + 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) )*(8*e^(4*I*d*x + 4*I*c) + 9*e^(2*I*d*x + 2*I*c) + 1))*e^(-3*I*d*x - 3*I*c )/(a^2*d)
\[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {1}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \sqrt {\tan {\left (c + d x \right )}}}\, dx \]
Exception generated. \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 751 vs. \(2 (91) = 182\).
Time = 1.60 (sec) , antiderivative size = 751, normalized size of antiderivative = 6.01 \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
-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*ta n(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)*(3*(sqrt(2)*sqrt(I*a*tan(d*x + c) + a)*(-I*abs(a)/a + 1)*ab s(a)^(3/2)/a^2 - sqrt(-2*(I*a*tan(d*x + c) + a)*a + 2*a^2)*(tan(d*x + c)/s qrt(((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)) + 3*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) - 72*I*(sqrt(2)*sqrt(I*a*t an(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)) + 72*( sqrt(2)*sqrt(I*a*tan(d*x + c) + a)*(-I*abs(a)/a + 1)*abs(a)^(3/2)/a^2 - sq rt(-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*abs (a)^(3/2) - 112*a*sqrt(abs(a)) - 112*I*abs(a)^(3/2))/(((sqrt(2)*sqrt(I*a*t an(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*...
Timed out. \[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {1}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]