Optimal. Leaf size=135 \[ -\frac{2 (3 B+i A) \sqrt{a+i a \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}+\frac{(1+i) \sqrt{a} (B+i A) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.342461, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3598, 12, 3544, 205} \[ -\frac{2 (3 B+i A) \sqrt{a+i a \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}+\frac{(1+i) \sqrt{a} (B+i A) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3598
Rule 12
Rule 3544
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac{5}{2}}(c+d x)} \, dx &=-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \int \frac{\sqrt{a+i a \tan (c+d x)} \left (\frac{1}{2} a (i A+3 B)-a A \tan (c+d x)\right )}{\tan ^{\frac{3}{2}}(c+d x)} \, dx}{3 a}\\ &=-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 (i A+3 B) \sqrt{a+i a \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}+\frac{4 \int -\frac{3 a^2 (A-i B) \sqrt{a+i a \tan (c+d x)}}{4 \sqrt{\tan (c+d x)}} \, dx}{3 a^2}\\ &=-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 (i A+3 B) \sqrt{a+i a \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}+(-A+i B) \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx\\ &=-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 (i A+3 B) \sqrt{a+i a \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}+\frac{\left (2 a^2 (i A+B)\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 )}{d}\\ &=\frac{(1+i) \sqrt{a} (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 )}{d}-\frac{2 A \sqrt{a+i a \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 (i A+3 B) \sqrt{a+i a \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 6.45787, size = 174, normalized size = 1.29 \[ \frac{(B+i A) e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right ) \sqrt{a+i a \tan (c+d x)}}{d \sqrt{-\frac{i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}}}-\frac{2 \sqrt{a+i a \tan (c+d x)} (A \cot (c+d x)+i A+3 B)}{3 d \sqrt{\tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 553, normalized size = 4.1 \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 [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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Fricas [B] time = 1.81017, size = 1327, normalized size = 9.83 \begin{align*} \frac{\sqrt{2}{\left (4 \,{\left (2 \, A - 3 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, A e^{\left (2 i \, d x + 2 i \, c\right )} + 12 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 )} - 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 (-2 i \, A^{2} - 4 \, A B + 2 i \, B^{2}\right )} a}{d^{2}}} \log \left (\frac{{\left (\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 )} + i \, d \sqrt{\frac{{\left (-2 i \, A^{2} - 4 \, A B + 2 i \, B^{2}\right )} a}{d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\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 (-2 i \, A^{2} - 4 \, A B + 2 i \, B^{2}\right )} a}{d^{2}}} \log \left (\frac{{\left (\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 )} - i \, d \sqrt{\frac{{\left (-2 i \, A^{2} - 4 \, A B + 2 i \, B^{2}\right )} a}{d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right )}{6 \,{\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(-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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Giac [A] time = 1.42187, size = 240, normalized size = 1.78 \begin{align*} \frac{\left (i + 1\right ) \, \sqrt{-2 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )} a + 2 \, a^{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} +{\left (-\left (2 i - 2\right ) \,{\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{3} + \left (2 i - 2\right ) \, a^{4}\right )} \sqrt{-2 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )} a + 2 \, a^{2}} \sqrt{i \, a \tan \left (d x + c\right ) + a} B}{2 \,{\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a - 5 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{2} + 9 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{3} - 7 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} + 2 \, a^{5}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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