Optimal. Leaf size=41 \[ \frac {x^{-2 n} (e x)^{2 n} \text {Int}\left (x^{2 n-1} \left (a+b \sinh \left (c+d x^n\right )\right )^p,x\right )}{e} \]
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Rubi [A] time = 0.06, 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 = {} \[ \int (e x)^{-1+2 n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (e x)^{-1+2 n} \left (a+b \sinh \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 (a+b \sinh \left (c+d x^n\right )\right )^p \, dx}{e}\\ \end {align*}
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Mathematica [A] time = 8.45, size = 0, normalized size = 0.00 \[ \int (e x)^{-1+2 n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{2 \, n - 1} {\left (b \sinh \left (d x^{n} + c\right ) + a\right )}^{p}, x\right ) \]
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 \[ \int \left (e x\right )^{2 \, n - 1} {\left (b \sinh \left (d x^{n} + c\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{-1+2 n} \left (a +b \sinh \left (c +d \,x^{n}\right )\right )^{p}\, dx \]
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 \[ \int \left (e x\right )^{2 \, n - 1} {\left (b \sinh \left (d x^{n} + c\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (e\,x\right )}^{2\,n-1}\,{\left (a+b\,\mathrm {sinh}\left (c+d\,x^n\right )\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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