Optimal. Leaf size=80 \[ \frac{2 a (B+i A)}{d \sqrt{\cot (c+d x)}}+\frac{2 \sqrt [4]{-1} a (A-i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\cot (c+d x)}\right )}{d}+\frac{2 i a B}{3 d \cot ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.187618, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {3581, 3591, 3529, 3533, 208} \[ \frac{2 a (B+i A)}{d \sqrt{\cot (c+d x)}}+\frac{2 \sqrt [4]{-1} a (A-i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\cot (c+d x)}\right )}{d}+\frac{2 i a B}{3 d \cot ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3581
Rule 3591
Rule 3529
Rule 3533
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx &=\int \frac{(i a+a \cot (c+d x)) (B+A \cot (c+d x))}{\cot ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 i a B}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\int \frac{a (i A+B)+a (A-i B) \cot (c+d x)}{\cot ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 i a B}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a (i A+B)}{d \sqrt{\cot (c+d x)}}+\int \frac{a (A-i B)-a (i A+B) \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx\\ &=\frac{2 i a B}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a (i A+B)}{d \sqrt{\cot (c+d x)}}+\frac{\left (2 a^2 (A-i B)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a (A-i B)-a (i A+B) x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}\\ &=\frac{2 \sqrt [4]{-1} a (A-i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\cot (c+d x)}\right )}{d}+\frac{2 i a B}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a (i A+B)}{d \sqrt{\cot (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.64355, size = 96, normalized size = 1.2 \[ \frac{2 a \left (3 (B+i A) \cot (c+d x)+\frac{3 (A-i B) \tanh ^{-1}\left (\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )}{(i \tan (c+d x))^{3/2}}+i B\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.501, size = 889, normalized size = 11.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55281, size = 239, normalized size = 2.99 \begin{align*} \frac{8 \,{\left (i \, B a - \frac{3 \,{\left (-i \, A - B\right )} a}{\tan \left (d x + c\right )}\right )} \tan \left (d x + c\right )^{\frac{3}{2}} + 3 \,{\left (2 \, \sqrt{2}{\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt{2}{\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) + \sqrt{2}{\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt{2}{\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (-\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right )\right )} a}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.47274, size = 1135, normalized size = 14.19 \begin{align*} -\frac{3 \,{\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt{\frac{{\left (4 i \, A^{2} + 8 \, A B - 4 i \, B^{2}\right )} a^{2}}{d^{2}}} \log \left (-\frac{{\left (2 \,{\left (A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} -{\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt{\frac{{\left (4 i \, A^{2} + 8 \, A B - 4 i \, B^{2}\right )} a^{2}}{d^{2}}} \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}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (i \, A + B\right )} a}\right ) - 3 \,{\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt{\frac{{\left (4 i \, A^{2} + 8 \, A B - 4 i \, B^{2}\right )} a^{2}}{d^{2}}} \log \left (-\frac{{\left (2 \,{\left (A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} -{\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt{\frac{{\left (4 i \, A^{2} + 8 \, A B - 4 i \, B^{2}\right )} a^{2}}{d^{2}}} \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}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (i \, A + B\right )} a}\right ) -{\left (8 \,{\left (3 \, A - 4 i \, B\right )} a e^{\left (4 i \, d x + 4 i \, c\right )} + 16 i \, B a e^{\left (2 i \, d x + 2 i \, c\right )} - 8 \,{\left (3 \, A - 2 i \, B\right )} a\right )} \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}}}{12 \,{\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \frac{A}{\sqrt{\cot{\left (c + d x \right )}}}\, dx + \int \frac{B \tan{\left (c + d x \right )}}{\sqrt{\cot{\left (c + d x \right )}}}\, dx + \int \frac{i A \tan{\left (c + d x \right )}}{\sqrt{\cot{\left (c + d x \right )}}}\, dx + \int \frac{i B \tan ^{2}{\left (c + d x \right )}}{\sqrt{\cot{\left (c + d x \right )}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}}{\sqrt{\cot \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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