3.83 \(\int (e x)^m \text{csch}^2(a+b x^n) \, dx\)

Optimal. Leaf size=27 \[ x^{-m} (e x)^m \text{Unintegrable}\left (x^m \text{csch}^2\left (a+b x^n\right ),x\right ) \]

[Out]

((e*x)^m*Unintegrable[x^m*Csch[a + b*x^n]^2, x])/x^m

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Rubi [A]  time = 0.0377175, 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 = {} \[ \int (e x)^m \text{csch}^2\left (a+b x^n\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Csch[a + b*x^n]^2,x]

[Out]

((e*x)^m*Defer[Int][x^m*Csch[a + b*x^n]^2, x])/x^m

Rubi steps

\begin{align*} \int (e x)^m \text{csch}^2\left (a+b x^n\right ) \, dx &=\left (x^{-m} (e x)^m\right ) \int x^m \text{csch}^2\left (a+b x^n\right ) \, dx\\ \end{align*}

Mathematica [A]  time = 24.3329, size = 0, normalized size = 0. \[ \int (e x)^m \text{csch}^2\left (a+b x^n\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e*x)^m*Csch[a + b*x^n]^2,x]

[Out]

Integrate[(e*x)^m*Csch[a + b*x^n]^2, x]

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Maple [A]  time = 0.13, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( \sinh \left ( a+b{x}^{n} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m/sinh(a+b*x^n)^2,x)

[Out]

int((e*x)^m/sinh(a+b*x^n)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -4 \, e^{m}{\left (m - n + 1\right )} \int \frac{x^{m}}{4 \,{\left (b n x^{n} + b n e^{\left (b x^{n} + n \log \left (x\right ) + a\right )}\right )}}\,{d x} + 4 \, e^{m}{\left (m - n + 1\right )} \int -\frac{x^{m}}{4 \,{\left (b n x^{n} - b n e^{\left (b x^{n} + n \log \left (x\right ) + a\right )}\right )}}\,{d x} + \frac{2 \, e^{m} x x^{m}}{b n x^{n} - b n e^{\left (2 \, b x^{n} + n \log \left (x\right ) + 2 \, a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/sinh(a+b*x^n)^2,x, algorithm="maxima")

[Out]

-4*e^m*(m - n + 1)*integrate(1/4*x^m/(b*n*x^n + b*n*e^(b*x^n + n*log(x) + a)), x) + 4*e^m*(m - n + 1)*integrat
e(-1/4*x^m/(b*n*x^n - b*n*e^(b*x^n + n*log(x) + a)), x) + 2*e^m*x*x^m/(b*n*x^n - b*n*e^(2*b*x^n + n*log(x) + 2
*a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/sinh(a+b*x^n)^2,x, algorithm="fricas")

[Out]

integral((e*x)^m/sinh(b*x^n + a)^2, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m}}{\sinh ^{2}{\left (a + b x^{n} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m/sinh(a+b*x**n)**2,x)

[Out]

Integral((e*x)**m/sinh(a + b*x**n)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/sinh(a+b*x^n)^2,x, algorithm="giac")

[Out]

integrate((e*x)^m/sinh(b*x^n + a)^2, x)