Optimal. Leaf size=122 \[ \frac{3 i \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{3 i \sec (c+d x)}{16 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}} \]
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Rubi [A] time = 0.115477, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3502, 3489, 206} \[ \frac{3 i \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{3 i \sec (c+d x)}{16 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3489
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\frac{i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}+\frac{3 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx}{8 a}\\ &=\frac{i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}+\frac{3 i \sec (c+d x)}{16 a d (a+i a \tan (c+d x))^{3/2}}+\frac{3 \int \frac{\sec (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx}{32 a^2}\\ &=\frac{i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}+\frac{3 i \sec (c+d x)}{16 a d (a+i a \tan (c+d x))^{3/2}}+\frac{(3 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 )}{16 a^2 d}\\ &=\frac{3 i \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}+\frac{3 i \sec (c+d x)}{16 a d (a+i a \tan (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.751814, size = 121, normalized size = 0.99 \[ -\frac{i \sec ^3(c+d x) \left (3 i \sin (2 (c+d x))+7 \cos (2 (c+d x))+3 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )+7\right )}{32 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.283, size = 346, normalized size = 2.8 \begin{align*}{\frac{1}{64\,d{a}^{3}} \left ( 64\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}+3\,i\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ({\frac{\sqrt{2} \left ( i\cos \left ( dx+c \right ) -i+\sin \left ( dx+c \right ) \right ) }{2\,\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}}} \right ) \sqrt{2}\cos \left ( dx+c \right ) +64\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+3\,i\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ({\frac{\sqrt{2} \left ( i\cos \left ( dx+c \right ) -i+\sin \left ( dx+c \right ) \right ) }{2\,\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}}} \right ) \sqrt{2}+3\,\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 ) -24\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -12\,i\cos \left ( dx+c \right ) \right ) \sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11828, size = 807, normalized size = 6.61 \begin{align*} \frac{{\left (3 i \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left ({\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} 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 ) - 3 i \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (-{\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} 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 ) + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (5 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{32 \, a^{3} d} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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