Optimal. Leaf size=98 \[ -\frac{i e^{-3 i a} c x^3}{16 \sqrt{c x^2}}+\frac{9}{32} i e^{-i a} x \sqrt [6]{c x^2}-\frac{9 i e^{i a} x}{16 \sqrt [6]{c x^2}}+\frac{i e^{3 i a} x \log (x)}{8 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0609948, antiderivative size = 98, 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} \[ -\frac{i e^{-3 i a} c x^3}{16 \sqrt{c x^2}}+\frac{9}{32} i e^{-i a} x \sqrt [6]{c x^2}-\frac{9 i e^{i a} x}{16 \sqrt [6]{c x^2}}+\frac{i e^{3 i a} x \log (x)}{8 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 4483
Rule 4489
Rubi steps
\begin{align*} \int \sin ^3\left (a+\frac{1}{6} i \log \left (c x^2\right )\right ) \, dx &=\frac{x \operatorname{Subst}\left (\int \frac{\sin ^3\left (a+\frac{1}{6} 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^{-3 i a}+\frac{e^{3 i a}}{x}-\frac{3 e^{i a}}{x^{2/3}}+\frac{3 e^{-i a}}{\sqrt [3]{x}}\right ) \, dx,x,c x^2\right )}{16 \sqrt{c x^2}}\\ &=-\frac{i c e^{-3 i a} x^3}{16 \sqrt{c x^2}}-\frac{9 i e^{i a} x}{16 \sqrt [6]{c x^2}}+\frac{9}{32} i e^{-i a} x \sqrt [6]{c x^2}+\frac{i e^{3 i a} x \log (x)}{8 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.121187, size = 103, normalized size = 1.05 \[ \frac{x \left (-2 c x^2 \sin (3 a)+9 \sin (a) \left (c x^2\right )^{2/3}+18 \sin (a) \sqrt [3]{c x^2}+9 i \cos (a) \sqrt [3]{c x^2} \left (\sqrt [3]{c x^2}-2\right )-2 i \cos (3 a) \left (c x^2-2 \log (x)\right )-4 \sin (3 a) \log (x)\right )}{32 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.081, size = 284, normalized size = 2.9 \begin{align*}{ \left ( -{\frac{23\,i}{40}}x+{\frac{27\,x}{10}\tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) }+{\frac{27\,x}{10} \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{5}}+{\frac{33\,i}{8}}x \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{2}+{\frac{23\,i}{40}}x \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{6}-{\frac{33\,i}{8}}x \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{4}-{\frac{3\,x\ln \left ( c{x}^{2} \right ) }{8}\tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) }+{\frac{5\,x\ln \left ( c{x}^{2} \right ) }{4} \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{3}}-{\frac{3\,x\ln \left ( c{x}^{2} \right ) }{8} \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{5}}+{\frac{i}{16}}x\ln \left ( c{x}^{2} \right ) -{\frac{15\,i}{16}}x\ln \left ( c{x}^{2} \right ) \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{2}+{\frac{15\,i}{16}}x\ln \left ( c{x}^{2} \right ) \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{4}-{\frac{i}{16}}x\ln \left ( c{x}^{2} \right ) \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{6} \right ) \left ( 1+ \left ( \tan \left ({\frac{a}{2}}+{\frac{i}{12}}\ln \left ( c{x}^{2} \right ) \right ) \right ) ^{2} \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08594, size = 101, normalized size = 1.03 \begin{align*} -\frac{9 \, c^{\frac{4}{3}} x^{\frac{4}{3}}{\left (-i \, \cos \left (a\right ) - \sin \left (a\right )\right )} + 18 \, c x^{\frac{2}{3}}{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} + 2 \,{\left (c x^{2}{\left (i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} + 2 \,{\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \log \left (x\right )\right )} c^{\frac{2}{3}}}{32 \, c^{\frac{7}{6}}} \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(-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 \sin \left (a + \frac{1}{6} i \, \log \left (c x^{2}\right )\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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