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 = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 261
Rule 4482
Rule 4484
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*}
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Mathematica [A] time = 0.12, 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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fricas [A] time = 0.61, size = 39, normalized size = 0.81 \[ -\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {c^{4} x^{4} + e^{\left (-2 i \, a\right )}} e^{\left (-\frac {3}{2} i \, a\right )}}{c^{5} x^{4} + c e^{\left (-2 i \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cos \left (a - 2 i \, \log \left (c x\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cos \left (a -2 i \ln \left (c x \right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 187, normalized size = 3.90 \[ -\frac {{\left ({\left (\sqrt {2} \cos \left (\frac {3}{2} \, a\right ) + i \, \sqrt {2} \sin \left (\frac {3}{2} \, a\right )\right )} c^{4} x^{4} + \sqrt {2} \cos \left (\frac {1}{2} \, a\right ) - i \, \sqrt {2} \sin \left (\frac {1}{2} \, a\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (c^{4} x^{4} \sin \left (2 \, a\right ), c^{4} x^{4} \cos \left (2 \, a\right ) + 1\right )\right ) + {\left ({\left (-i \, \sqrt {2} \cos \left (\frac {3}{2} \, a\right ) + \sqrt {2} \sin \left (\frac {3}{2} \, a\right )\right )} c^{4} x^{4} - i \, \sqrt {2} \cos \left (\frac {1}{2} \, a\right ) - \sqrt {2} \sin \left (\frac {1}{2} \, a\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (c^{4} x^{4} \sin \left (2 \, a\right ), c^{4} x^{4} \cos \left (2 \, a\right ) + 1\right )\right )}{{\left ({\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} c^{8} x^{8} + 2 \, c^{4} x^{4} \cos \left (2 \, a\right ) + 1\right )}^{\frac {3}{4}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.79, size = 48, normalized size = 1.00 \[ -\frac {2\,x\,\sqrt {\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}}}{2\,c^2\,x^2}+\frac {c^2\,x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}}{2}}}{{\mathrm {e}}^{a\,2{}\mathrm {i}}\,c^4\,x^4+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cos ^{\frac {3}{2}}{\left (a - 2 i \log {\left (c x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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