Optimal. Leaf size=214 \[ -\frac{(25 A+7 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{6 a^2 d}+\frac{\left (\frac{1}{4}+\frac{i}{4}\right ) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac{(A+i B) \sqrt{\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(11 A+5 i B) \sqrt{\cot (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.725347, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4241, 3596, 3598, 12, 3544, 205} \[ -\frac{(25 A+7 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{6 a^2 d}+\frac{\left (\frac{1}{4}+\frac{i}{4}\right ) (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac{(A+i B) \sqrt{\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(11 A+5 i B) \sqrt{\cot (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4241
Rule 3596
Rule 3598
Rule 12
Rule 3544
Rule 205
Rubi steps
\begin{align*} \int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{A+B \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx\\ &=\frac{(A+i B) \sqrt{\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{1}{2} a (7 A+i B)-2 a (i A-B) \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=\frac{(A+i B) \sqrt{\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(11 A+5 i B) \sqrt{\cot (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)} \left (\frac{1}{4} a^2 (25 A+7 i B)-\frac{1}{2} a^2 (11 i A-5 B) \tan (c+d x)\right )}{\tan ^{\frac{3}{2}}(c+d x)} \, dx}{3 a^4}\\ &=\frac{(A+i B) \sqrt{\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(11 A+5 i B) \sqrt{\cot (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}}-\frac{(25 A+7 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{6 a^2 d}+\frac{\left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{3 a^3 (i A+B) \sqrt{a+i a \tan (c+d x)}}{8 \sqrt{\tan (c+d x)}} \, dx}{3 a^5}\\ &=\frac{(A+i B) \sqrt{\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(11 A+5 i B) \sqrt{\cot (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}}-\frac{(25 A+7 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{6 a^2 d}+\frac{\left ((i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx}{4 a^2}\\ &=\frac{(A+i B) \sqrt{\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(11 A+5 i B) \sqrt{\cot (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}}-\frac{(25 A+7 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{6 a^2 d}-\frac{\left (i (i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-i a-2 a^2 x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{2 d}\\ &=\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) (i A+B) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{a^{3/2} d}+\frac{(A+i B) \sqrt{\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(11 A+5 i B) \sqrt{\cot (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}}-\frac{(25 A+7 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{6 a^2 d}\\ \end{align*}
Mathematica [A] time = 4.73117, size = 195, normalized size = 0.91 \[ -\frac{\cot ^{\frac{3}{2}}(c+d x) \left (-3 (A-i B) e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )+A \left (-13 e^{2 i (c+d x)}+38 e^{4 i (c+d x)}-1\right )+i B \left (-7 e^{2 i (c+d x)}+8 e^{4 i (c+d x)}-1\right )\right )}{3 a d \left (1+e^{2 i (c+d x)}\right )^2 (\cot (c+d x)+i) \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.673, size = 648, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.96903, size = 1299, normalized size = 6.07 \begin{align*} \frac{{\left (3 \, \sqrt{\frac{1}{2}} a^{2} d \sqrt{\frac{i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac{{\left (2 i \, \sqrt{\frac{1}{2}} a^{2} d \sqrt{\frac{i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2}{\left ({\left (i \, A + B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, A - B\right )} \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}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{4 i \, A + 4 \, B}\right ) - 3 \, \sqrt{\frac{1}{2}} a^{2} d \sqrt{\frac{i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac{{\left (-2 i \, \sqrt{\frac{1}{2}} a^{2} d \sqrt{\frac{i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2}{\left ({\left (i \, A + B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, A - B\right )} \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}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{4 i \, A + 4 \, B}\right ) - \sqrt{2}{\left (2 \,{\left (19 \, A + 4 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} -{\left (13 \, A + 7 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - A - i \, B\right )} \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}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{12 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac{3}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]