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

Optimal. Leaf size=39 \[ \frac {x^{-2 n} (e x)^{2 n} \text {Int}\left (x^{2 n-1} \left (b \sinh \left (c+d x^n\right )\right )^p,x\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.05, 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 = {} \[ \int (e x)^{-1+2 n} \left (b \sinh \left (c+d x^n\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int (e x)^{-1+2 n} \left (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 (b \sinh \left (c+d x^n\right )\right )^p \, dx}{e}\\ \end {align*}

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Mathematica [A]  time = 5.90, size = 0, normalized size = 0.00 \[ \int (e x)^{-1+2 n} \left (b \sinh \left (c+d x^n\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{2 \, n - 1} \left (b \sinh \left (d x^{n} + c\right )\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{2 \, n - 1} \left (b \sinh \left (d x^{n} + c\right )\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.88, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{-1+2 n} \left (b \sinh \left (c +d \,x^{n}\right )\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{2 \, n - 1} \left (b \sinh \left (d x^{n} + c\right )\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (b\,\mathrm {sinh}\left (c+d\,x^n\right )\right )}^p\,{\left (e\,x\right )}^{2\,n-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sinh {\left (c + d x^{n} \right )}\right )^{p} \left (e x\right )^{2 n - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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