Optimal. Leaf size=48 \[ -\frac{e^{-2 i a} \left (1+e^{2 i a} c^4 x^4\right )}{2 c^4 x^3 \cos ^{\frac{3}{2}}(a-2 i \log (c x))} \]
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Rubi [A] time = 0.0361138, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4484, 4482, 261} \[ -\frac{e^{-2 i a} \left (1+e^{2 i a} c^4 x^4\right )}{2 c^4 x^3 \cos ^{\frac{3}{2}}(a-2 i \log (c x))} \]
Antiderivative was successfully verified.
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Rule 4484
Rule 4482
Rule 261
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{3}{2}}(a-2 i \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\cos ^{\frac{3}{2}}(a-2 i \log (x))} \, dx,x,c x\right )}{c}\\ &=\frac{\left (1+c^4 e^{2 i a} x^4\right )^{3/2} \operatorname{Subst}\left (\int \frac{x^3}{\left (1+e^{2 i a} x^4\right )^{3/2}} \, dx,x,c x\right )}{c^4 x^3 \cos ^{\frac{3}{2}}(a-2 i \log (c x))}\\ &=-\frac{e^{-2 i a} \left (1+c^4 e^{2 i a} x^4\right )}{2 c^4 x^3 \cos ^{\frac{3}{2}}(a-2 i \log (c x))}\\ \end{align*}
Mathematica [A] time = 0.115488, size = 82, normalized size = 1.71 \[ -\frac{x (\cos (a)-i \sin (a)) \sqrt{\frac{2 \cos (a) \left (c^4 x^4+1\right )+2 i \sin (a) \left (c^4 x^4-1\right )}{c^2 x^2}}}{\cos (a) \left (c^4 x^4+1\right )+i \sin (a) \left (c^4 x^4-1\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.343, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( a-2\,i\ln \left ( cx \right ) \right ) \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.80503, size = 252, normalized size = 5.25 \begin{align*} -\frac{{\left ({\left (\sqrt{2} \cos \left (\frac{3}{2} \, a\right ) + i \, \sqrt{2} \sin \left (\frac{3}{2} \, a\right )\right )} c^{4} x^{4} + \sqrt{2} \cos \left (\frac{1}{2} \, a\right ) - i \, \sqrt{2} \sin \left (\frac{1}{2} \, a\right )\right )} \cos \left (\frac{3}{2} \, \arctan \left (c^{4} x^{4} \sin \left (2 \, a\right ), c^{4} x^{4} \cos \left (2 \, a\right ) + 1\right )\right ) +{\left ({\left (-i \, \sqrt{2} \cos \left (\frac{3}{2} \, a\right ) + \sqrt{2} \sin \left (\frac{3}{2} \, a\right )\right )} c^{4} x^{4} - i \, \sqrt{2} \cos \left (\frac{1}{2} \, a\right ) - \sqrt{2} \sin \left (\frac{1}{2} \, a\right )\right )} \sin \left (\frac{3}{2} \, \arctan \left (c^{4} x^{4} \sin \left (2 \, a\right ), c^{4} x^{4} \cos \left (2 \, a\right ) + 1\right )\right )}{{\left ({\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} c^{8} x^{8} + 2 \, c^{4} x^{4} \cos \left (2 \, a\right ) + 1\right )}^{\frac{3}{4}} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.456619, size = 103, normalized size = 2.15 \begin{align*} -\frac{2 \, \sqrt{\frac{1}{2}} x e^{\left (-\frac{3}{2} i \, a - 3 \, \log \left (c x\right )\right )}}{\sqrt{e^{\left (-2 i \, a - 4 \, \log \left (c x\right )\right )} + 1}} \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{1}{\cos ^{\frac{3}{2}}{\left (a - 2 i \log{\left (c x \right )} \right )}}\, 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{1}{\cos \left (a - 2 i \, \log \left (c x\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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