3.79 \(\int (e x)^{-1+2 n} (a+b \sinh (c+d x^n))^p \, dx\)

Optimal. Leaf size=40 \[ \frac{x^{-2 n} (e x)^{2 n} \text{Unintegrable}\left (x^{2 n-1} \left (a+b \sinh \left (c+d x^n\right )\right )^p,x\right )}{e} \]

[Out]

((e*x)^(2*n)*Unintegrable[x^(-1 + 2*n)*(a + b*Sinh[c + d*x^n])^p, x])/(e*x^(2*n))

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Rubi [A]  time = 0.0595259, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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.

[In]

Int[(e*x)^(-1 + 2*n)*(a + b*Sinh[c + d*x^n])^p,x]

[Out]

((e*x)^(2*n)*Defer[Int][x^(-1 + 2*n)*(a + b*Sinh[c + d*x^n])^p, x])/(e*x^(2*n))

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*}

Mathematica [A]  time = 8.35159, size = 0, normalized size = 0. \[ \int (e x)^{-1+2 n} \left (a+b \sinh \left (c+d x^n\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e*x)^(-1 + 2*n)*(a + b*Sinh[c + d*x^n])^p,x]

[Out]

Integrate[(e*x)^(-1 + 2*n)*(a + b*Sinh[c + d*x^n])^p, x]

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Maple [A]  time = 0.674, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{-1+2\,n} \left ( a+b\sinh \left ( c+d{x}^{n} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(-1+2*n)*(a+b*sinh(c+d*x^n))^p,x)

[Out]

int((e*x)^(-1+2*n)*(a+b*sinh(c+d*x^n))^p,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{2 \, n - 1}{\left (b \sinh \left (d x^{n} + c\right ) + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+2*n)*(a+b*sinh(c+d*x^n))^p,x, algorithm="maxima")

[Out]

integrate((e*x)^(2*n - 1)*(b*sinh(d*x^n + c) + a)^p, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{2 \, n - 1}{\left (b \sinh \left (d x^{n} + c\right ) + a\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+2*n)*(a+b*sinh(c+d*x^n))^p,x, algorithm="fricas")

[Out]

integral((e*x)^(2*n - 1)*(b*sinh(d*x^n + c) + a)^p, x)

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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.

[In]

integrate((e*x)**(-1+2*n)*(a+b*sinh(c+d*x**n))**p,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{2 \, n - 1}{\left (b \sinh \left (d x^{n} + c\right ) + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+2*n)*(a+b*sinh(c+d*x^n))^p,x, algorithm="giac")

[Out]

integrate((e*x)^(2*n - 1)*(b*sinh(d*x^n + c) + a)^p, x)