Optimal. Leaf size=112 \[ \frac{x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \text{Hypergeometric2F1}\left (-p,-\frac{b n p+i}{2 b n},\frac{1}{2} \left (-\frac{i}{b n}-p+2\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^p\left (a+b \log \left (c x^n\right )\right )}{1-i b n p} \]
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Rubi [A] time = 0.0706092, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4484, 4492, 364} \[ \frac{x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \, _2F_1\left (-p,-\frac{b n p+i}{2 b n};\frac{1}{2} \left (-p-\frac{i}{b n}+2\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^p\left (a+b \log \left (c x^n\right )\right )}{1-i b n p} \]
Antiderivative was successfully verified.
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Rule 4484
Rule 4492
Rule 364
Rubi steps
\begin{align*} \int \cos ^p\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 ^p(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-\frac{1}{n}+i b p} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \cos ^p\left (a+b \log \left (c x^n\right )\right )\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}-i b p} \left (1+e^{2 i a} x^{2 i b}\right )^p \, dx,x,c x^n\right )}{n}\\ &=\frac{x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (-p,-\frac{i+b n p}{2 b n};\frac{1}{2} \left (2-\frac{i}{b n}-p\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1-i b n p}\\ \end{align*}
Mathematica [A] time = 0.556962, size = 102, normalized size = 0.91 \[ \frac{i x \left (1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \cos ^p\left (a+b \log \left (c x^n\right )\right ) \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (-\frac{i}{b n}+p+2\right ),-\frac{i}{2 b n}-\frac{p}{2}+1,-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{b n p+i} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.113, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{p}\, 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 )^{p}\,{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 (\cos \left (b \log \left (c x^{n}\right ) + a\right )^{p}, 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 \cos ^{p}{\left (a + b \log{\left (c x^{n} \right )} \right )}\, 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 \cos \left (b \log \left (c x^{n}\right ) + a\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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