Optimal. Leaf size=483 \[ -\frac{i \sqrt{2} \sqrt{a} e^{3/2} \sec (c+d x) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i \sqrt{2} \sqrt{a} e^{3/2} \sec (c+d x) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i \sqrt{a} e^{3/2} \sec (c+d x) \log \left (-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i \sqrt{a} e^{3/2} \sec (c+d x) \log \left (\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.352719, antiderivative size = 483, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3499, 3495, 297, 1162, 617, 204, 1165, 628} \[ -\frac{i \sqrt{2} \sqrt{a} e^{3/2} \sec (c+d x) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i \sqrt{2} \sqrt{a} e^{3/2} \sec (c+d x) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i \sqrt{a} e^{3/2} \sec (c+d x) \log \left (-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i \sqrt{a} e^{3/2} \sec (c+d x) \log \left (\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3499
Rule 3495
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(e \sec (c+d x))^{3/2}}{\sqrt{a+i a \tan (c+d x)}} \, dx &=\frac{(e \sec (c+d x)) \int \sqrt{e \sec (c+d x)} \sqrt{a-i a \tan (c+d x)} \, dx}{\sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{\left (4 i a e^3 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a^2+e^2 x^4} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=-\frac{\left (2 i a e^2 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{a-e x^2}{a^2+e^2 x^4} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (2 i a e^2 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{a+e x^2}{a^2+e^2 x^4} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{(i a e \sec (c+d x)) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{e}-\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}+x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{(i a e \sec (c+d x)) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{e}+\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}+x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (i \sqrt{a} e^{3/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{a}}{\sqrt{e}}+2 x}{-\frac{a}{e}-\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}-x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (i \sqrt{a} e^{3/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{a}}{\sqrt{e}}-2 x}{-\frac{a}{e}+\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}-x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{i \sqrt{a} e^{3/2} \log \left (a-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i \sqrt{a} e^{3/2} \log \left (a+\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (i \sqrt{2} \sqrt{a} e^{3/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{\left (i \sqrt{2} \sqrt{a} e^{3/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=-\frac{i \sqrt{2} \sqrt{a} e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right ) \sec (c+d x)}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i \sqrt{2} \sqrt{a} e^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right ) \sec (c+d x)}{d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i \sqrt{a} e^{3/2} \log \left (a-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{i \sqrt{a} e^{3/2} \log \left (a+\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{\sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.10315, size = 302, normalized size = 0.63 \[ -\frac{2 e \sqrt{\tan \left (\frac{d x}{2}\right )+i} (\cos (d x)+i \sin (d x)) \sqrt{e \sec (c+d x)} \left (\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\sin (c)-i \cos (c)-1} \tan ^{-1}\left (\frac{\sqrt{\sin (c)-i \cos (c)-1} \sqrt{-\tan \left (\frac{d x}{2}\right )+i}}{\sqrt{\sin (c)+i \cos (c)-1} \sqrt{\tan \left (\frac{d x}{2}\right )+i}}\right )-\sqrt{-\sin (c)+i \cos (c)-1} \sqrt{\sin (c)+i \cos (c)-1} \tan ^{-1}\left (\frac{\sqrt{-\sin (c)+i \cos (c)-1} \sqrt{-\tan \left (\frac{d x}{2}\right )+i}}{\sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\tan \left (\frac{d x}{2}\right )+i}}\right )\right )}{d \sqrt{-\sin (c)-i \cos (c)-1} \sqrt{\sin (c)+i \cos (c)-1} \sqrt{-\tan \left (\frac{d x}{2}\right )+i} \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.333, size = 237, normalized size = 0.5 \begin{align*} -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2}}{ad \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) -1 \right ) } \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}} \left ( i{\it Artanh} \left ({\frac{\cos \left ( dx+c \right ) +1-\sin \left ( dx+c \right ) }{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \right ) -i{\it Artanh} \left ({\frac{\cos \left ( dx+c \right ) +1+\sin \left ( dx+c \right ) }{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \right ) -{\it Artanh} \left ({\frac{\cos \left ( dx+c \right ) +1-\sin \left ( dx+c \right ) }{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \right ) -{\it Artanh} \left ({\frac{\cos \left ( dx+c \right ) +1+\sin \left ( dx+c \right ) }{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \right ) \right ) \left ( \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.04854, size = 980, normalized size = 2.03 \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 [A] time = 2.16165, size = 1239, normalized size = 2.57 \begin{align*} -\frac{1}{2} \, \sqrt{\frac{4 i \, e^{3}}{a d^{2}}} \log \left (\frac{{\left (a d \sqrt{\frac{4 i \, e^{3}}{a d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \,{\left (e e^{\left (2 i \, d x + 2 i \, c\right )} + e\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{3}{2} i \, d x + \frac{3}{2} i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{e}\right ) + \frac{1}{2} \, \sqrt{\frac{4 i \, e^{3}}{a d^{2}}} \log \left (-\frac{{\left (a d \sqrt{\frac{4 i \, e^{3}}{a d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \,{\left (e e^{\left (2 i \, d x + 2 i \, c\right )} + e\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{3}{2} i \, d x + \frac{3}{2} i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{e}\right ) + \frac{1}{2} \, \sqrt{-\frac{4 i \, e^{3}}{a d^{2}}} \log \left (\frac{{\left (a d \sqrt{-\frac{4 i \, e^{3}}{a d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \,{\left (e e^{\left (2 i \, d x + 2 i \, c\right )} + e\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{3}{2} i \, d x + \frac{3}{2} i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{e}\right ) - \frac{1}{2} \, \sqrt{-\frac{4 i \, e^{3}}{a d^{2}}} \log \left (-\frac{{\left (a d \sqrt{-\frac{4 i \, e^{3}}{a d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \,{\left (e e^{\left (2 i \, d x + 2 i \, c\right )} + e\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{3}{2} i \, d x + \frac{3}{2} i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{e}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}}}{\sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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