Optimal. Leaf size=109 \[ \frac{2 x \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{1}{4} \left (-3-\frac{2 i}{b n}\right ),\frac{1}{4} \left (1-\frac{2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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Rubi [A] time = 0.0678703, antiderivative size = 109, 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, 4492, 364} \[ \frac{2 x \, _2F_1\left (-\frac{3}{2},\frac{1}{4} \left (-3-\frac{2 i}{b n}\right );\frac{1}{4} \left (1-\frac{2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4484
Rule 4492
Rule 364
Rubi steps
\begin{align*} \int \cos ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \cos ^{\frac{3}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{\frac{3 i b}{2}-\frac{1}{n}} \cos ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \operatorname{Subst}\left (\int x^{-1-\frac{3 i b}{2}+\frac{1}{n}} \left (1+e^{2 i a} x^{2 i b}\right )^{3/2} \, dx,x,c x^n\right )}{n \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\\ &=\frac{2 x \cos ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (-\frac{3}{2},\frac{1}{4} \left (-3-\frac{2 i}{b n}\right );\frac{1}{4} \left (1-\frac{2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.66796, size = 163, normalized size = 1.5 \[ \frac{x \left ((b n-2 i) \left (3 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+4 \cos ^2\left (a+b \log \left (c x^n\right )\right )\right )-6 i b^2 n^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \text{Hypergeometric2F1}\left (1,\frac{3}{4}-\frac{i}{2 b n},\frac{5}{4}-\frac{i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{(b n-2 i) \left (9 b^2 n^2+4\right ) \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.162, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}\,{d x} \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 \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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