Optimal. Leaf size=39 \[ \frac {x^{-2 n} (e x)^{2 n} \text {Int}\left (x^{-1+2 n} \left (b \sin \left (c+d x^n\right )\right )^p,x\right )}{e} \]
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Rubi [A]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int (e x)^{-1+2 n} \left (b \sin \left (c+d x^n\right )\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int (e x)^{-1+2 n} \left (b \sin \left (c+d x^n\right )\right )^p \, dx &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int x^{-1+2 n} \left (b \sin \left (c+d x^n\right )\right )^p \, dx}{e}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int (e x)^{-1+2 n} \left (b \sin \left (c+d x^n\right )\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{-1+2 n} \left (b \sin \left (c +d \,x^{n}\right )\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \sin {\left (c + d x^{n} \right )}\right )^{p} \left (e x\right )^{2 n - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (b\,\sin \left (c+d\,x^n\right )\right )}^p\,{\left (e\,x\right )}^{2\,n-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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