Optimal. Leaf size=53 \[ -\frac{e^{-2 i a} c x^3}{8 \sqrt{c x^2}}-\frac{e^{2 i a} x \log (x)}{4 \sqrt{c x^2}}+\frac{x}{2} \]
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Rubi [A] time = 0.0452294, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4483, 4489} \[ -\frac{e^{-2 i a} c x^3}{8 \sqrt{c x^2}}-\frac{e^{2 i a} x \log (x)}{4 \sqrt{c x^2}}+\frac{x}{2} \]
Antiderivative was successfully verified.
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Rule 4483
Rule 4489
Rubi steps
\begin{align*} \int \sin ^2\left (a+\frac{1}{4} i \log \left (c x^2\right )\right ) \, dx &=\frac{x \operatorname{Subst}\left (\int \frac{\sin ^2\left (a+\frac{1}{4} i \log (x)\right )}{\sqrt{x}} \, dx,x,c x^2\right )}{2 \sqrt{c x^2}}\\ &=-\frac{x \operatorname{Subst}\left (\int \left (e^{-2 i a}+\frac{e^{2 i a}}{x}-\frac{2}{\sqrt{x}}\right ) \, dx,x,c x^2\right )}{8 \sqrt{c x^2}}\\ &=\frac{x}{2}-\frac{c e^{-2 i a} x^3}{8 \sqrt{c x^2}}-\frac{e^{2 i a} x \log (x)}{4 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0995453, size = 60, normalized size = 1.13 \[ \frac{x \left (i \sin (2 a) \left (c x^2-2 \log (x)\right )-\cos (2 a) \left (c x^2+2 \log (x)\right )+4 \sqrt{c x^2}\right )}{8 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 173, normalized size = 3.3 \begin{align*}{ \left ({\frac{x}{4}}+{\frac{5\,x}{2} \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{8}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{2}}+{\frac{x}{4} \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{8}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{4}}-{\frac{x\ln \left ( c{x}^{2} \right ) }{8}}+{\frac{3\,x\ln \left ( c{x}^{2} \right ) }{4} \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{8}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{2}}-{\frac{x\ln \left ( c{x}^{2} \right ) }{8} \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{8}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{4}}-{\frac{i}{2}}x\ln \left ( c{x}^{2} \right ) \tan \left ({\frac{a}{2}}+{\frac{i}{8}}\ln \left ( c{x}^{2} \right ) \right ) +{\frac{i}{2}}x\ln \left ( c{x}^{2} \right ) \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{8}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{3} \right ) \left ( 1+ \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{8}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{2} \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02164, size = 65, normalized size = 1.23 \begin{align*} \frac{4 \, c x -{\left (c x^{2}{\left (\cos \left (2 \, a\right ) - i \, \sin \left (2 \, a\right )\right )} +{\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \log \left (x\right )\right )} \sqrt{c}}{8 \, 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 ^{2}{\left (a + \frac{i \log{\left (c x^{2} \right )}}{4} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26885, size = 43, normalized size = 0.81 \begin{align*} \frac{1}{2} \, x - \frac{c^{\frac{3}{2}} x^{2} e^{\left (-2 i \, a\right )} + 2 \, \sqrt{c} e^{\left (2 i \, a\right )} \log \left (x\right )}{8 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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