Optimal. Leaf size=86 \[ \frac{\cos \left (a+b \log \left (c x^n\right )\right ) \sin ^{p+1}\left (a+b \log \left (c x^n\right )\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{p+1}{2},\frac{p+3}{2},\sin ^2\left (a+b \log \left (c x^n\right )\right )\right )}{b n (p+1) \sqrt{\cos ^2\left (a+b \log \left (c x^n\right )\right )}} \]
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Rubi [A] time = 0.0601127, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2643} \[ \frac{\cos \left (a+b \log \left (c x^n\right )\right ) \sin ^{p+1}\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\sin ^2\left (a+b \log \left (c x^n\right )\right )\right )}{b n (p+1) \sqrt{\cos ^2\left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int \frac{\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sin ^p(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\cos \left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (\frac{1}{2},\frac{1+p}{2};\frac{3+p}{2};\sin ^2\left (a+b \log \left (c x^n\right )\right )\right ) \sin ^{1+p}\left (a+b \log \left (c x^n\right )\right )}{b n (1+p) \sqrt{\cos ^2\left (a+b \log \left (c x^n\right )\right )}}\\ \end{align*}
Mathematica [A] time = 0.149624, size = 86, normalized size = 1. \[ \frac{\sec \left (a+b \log \left (c x^n\right )\right ) \sqrt{\cos ^2\left (a+b \log \left (c x^n\right )\right )} \sin ^{p+1}\left (a+b \log \left (c x^n\right )\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{p+1}{2},\frac{p+3}{2},\sin ^2\left (a+b \log \left (c x^n\right )\right )\right )}{b n (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.13, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{p}}{x}}\, 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 \frac{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x}, x\right ) \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 \frac{\sin ^{p}{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x}\, dx \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 \frac{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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