Optimal. Leaf size=107 \[ -\frac{2 \sqrt{2} a^{3/2} (B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{d}+\frac{2 a (B+i A) \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 B (a+i a \tan (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.0999415, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3527, 3478, 3480, 206} \[ -\frac{2 \sqrt{2} a^{3/2} (B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{d}+\frac{2 a (B+i A) \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 B (a+i a \tan (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3527
Rule 3478
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx &=\frac{2 B (a+i a \tan (c+d x))^{3/2}}{3 d}-(-A+i B) \int (a+i a \tan (c+d x))^{3/2} \, dx\\ &=\frac{2 a (i A+B) \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 B (a+i a \tan (c+d x))^{3/2}}{3 d}+(2 a (A-i B)) \int \sqrt{a+i a \tan (c+d x)} \, dx\\ &=\frac{2 a (i A+B) \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 B (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac{\left (4 a^2 (i A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{d}\\ &=-\frac{2 \sqrt{2} a^{3/2} (i A+B) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{d}+\frac{2 a (i A+B) \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 B (a+i a \tan (c+d x))^{3/2}}{3 d}\\ \end{align*}
Mathematica [A] time = 2.54725, size = 190, normalized size = 1.78 \[ \frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left (\frac{2}{3} (\cos (c)-i \sin (c)) \sqrt{\sec (c+d x)} (\sin (d x)+i \cos (d x)) (3 A+B \tan (c+d x)-4 i B)-\frac{2 i \sqrt{2} (A-i B) \sinh ^{-1}\left (e^{i (c+d x)}\right )}{\left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{3/2} \left (1+e^{2 i (c+d x)}\right )^{3/2}}\right )}{d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 99, normalized size = 0.9 \begin{align*}{\frac{2\,i}{d} \left ( -{\frac{i}{3}}B \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}-iBa\sqrt{a+ia\tan \left ( dx+c \right ) }+A\sqrt{a+ia\tan \left ( dx+c \right ) }a-{a}^{{\frac{3}{2}}} \left ( A-iB \right ) \sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt{a}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70316, size = 1017, normalized size = 9.5 \begin{align*} \frac{\sqrt{2}{\left ({\left (12 i \, A + 20 \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (12 i \, A + 12 \, B\right )} a\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )} - 3 \, \sqrt{-\frac{{\left (8 \, A^{2} - 16 i \, A B - 8 \, B^{2}\right )} a^{3}}{d^{2}}}{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac{{\left (\sqrt{2}{\left ({\left (2 i \, A + 2 \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (2 i \, A + 2 \, B\right )} a\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )} + \sqrt{-\frac{{\left (8 \, A^{2} - 16 i \, A B - 8 \, B^{2}\right )} a^{3}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (2 i \, A + 2 \, B\right )} a}\right ) + 3 \, \sqrt{-\frac{{\left (8 \, A^{2} - 16 i \, A B - 8 \, B^{2}\right )} a^{3}}{d^{2}}}{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac{{\left (\sqrt{2}{\left ({\left (2 i \, A + 2 \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (2 i \, A + 2 \, B\right )} a\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )} - \sqrt{-\frac{{\left (8 \, A^{2} - 16 i \, A B - 8 \, B^{2}\right )} a^{3}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (2 i \, A + 2 \, B\right )} a}\right )}{6 \,{\left (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*} \int \left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}} \left (A + B \tan{\left (c + d x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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.
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