Optimal. Leaf size=86 \[ \frac{2 i \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac{2 i \sec (c+d x)}{a d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.101564, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3491, 3489, 206} \[ \frac{2 i \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac{2 i \sec (c+d x)}{a d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3491
Rule 3489
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=-\frac{2 i \sec (c+d x)}{a d \sqrt{a+i a \tan (c+d x)}}+\frac{2 \int \frac{\sec (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx}{a}\\ &=-\frac{2 i \sec (c+d x)}{a d \sqrt{a+i a \tan (c+d x)}}+\frac{(4 i) \operatorname{Subst}\left (\int \frac{1}{2-a x^2} \, dx,x,\frac{\sec (c+d x)}{\sqrt{a+i a \tan (c+d x)}}\right )}{a d}\\ &=\frac{2 i \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac{2 i \sec (c+d x)}{a d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.72189, size = 101, normalized size = 1.17 \[ \frac{8 e^{3 i (c+d x)} \left (-1+\sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )\right )}{a d \left (1+e^{2 i (c+d x)}\right )^2 (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.238, size = 158, normalized size = 1.8 \begin{align*} -2\,{\frac{1}{{a}^{2}d \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) -1 \right ) }\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}} \left ( \sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ( 1/2\,{\frac{\sqrt{2} \left ( i\cos \left ( dx+c \right ) -i+\sin \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}}} \right ) \sqrt{2}\sin \left ( dx+c \right ) -i\cos \left ( dx+c \right ) +i-\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.0252, size = 1099, normalized size = 12.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10683, size = 690, normalized size = 8.02 \begin{align*} \frac{{\left (i \, \sqrt{2} a^{2} d \sqrt{\frac{1}{a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} \log \left ({\left (\sqrt{2} a^{2} d \sqrt{\frac{1}{a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - i \, \sqrt{2} a^{2} d \sqrt{\frac{1}{a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} \log \left (-{\left (\sqrt{2} a^{2} d \sqrt{\frac{1}{a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} - \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 2 i \, \sqrt{2} \sqrt{\frac{a}{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 )}}{a^{2} d} \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 \frac{\sec ^{3}{\left (c + d x \right )}}{\left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \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{\sec \left (d x + c\right )^{3}}{{\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.
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