Optimal. Leaf size=52 \[ \frac{i e^{-i a} c x^3}{4 \sqrt{c x^2}}-\frac{i e^{i a} x \log (x)}{2 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0354132, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4483, 4489} \[ \frac{i e^{-i a} c x^3}{4 \sqrt{c x^2}}-\frac{i e^{i a} x \log (x)}{2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 4483
Rule 4489
Rubi steps
\begin{align*} \int \sin \left (a+\frac{1}{2} i \log \left (c x^2\right )\right ) \, dx &=\frac{x \operatorname{Subst}\left (\int \frac{\sin \left (a+\frac{1}{2} i \log (x)\right )}{\sqrt{x}} \, dx,x,c x^2\right )}{2 \sqrt{c x^2}}\\ &=-\frac{(i x) \operatorname{Subst}\left (\int \left (-e^{-i a}+\frac{e^{i a}}{x}\right ) \, dx,x,c x^2\right )}{4 \sqrt{c x^2}}\\ &=\frac{i c e^{-i a} x^3}{4 \sqrt{c x^2}}-\frac{i e^{i a} x \log (x)}{2 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.058512, size = 44, normalized size = 0.85 \[ \frac{x \left (\sin (a) \left (c x^2+2 \log (x)\right )+i \cos (a) \left (c x^2-2 \log (x)\right )\right )}{4 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 106, normalized size = 2. \begin{align*}{ \left ({\frac{i}{2}}x-{\frac{i}{2}}x \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{4}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{2}+{\frac{x\ln \left ( c{x}^{2} \right ) }{2}\tan \left ({\frac{a}{2}}+{\frac{i}{4}}\ln \left ( c{x}^{2} \right ) \right ) }-{\frac{i}{4}}x\ln \left ( c{x}^{2} \right ) +{\frac{i}{4}}x\ln \left ( c{x}^{2} \right ) \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{4}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{2} \right ) \left ( 1+ \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{4}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{2} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00925, size = 42, normalized size = 0.81 \begin{align*} \frac{c x^{2}{\left (i \, \cos \left (a\right ) + \sin \left (a\right )\right )} - 2 \,{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \log \left (x\right )}{4 \, \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 \sin{\left (a + \frac{i \log{\left (c x^{2} \right )}}{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18503, size = 39, normalized size = 0.75 \begin{align*} -\frac{-i \, c^{\frac{3}{2}} x^{2} e^{\left (-i \, a\right )} + 2 i \, \sqrt{c} e^{\left (i \, a\right )} \log \left (x\right )}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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