3.1.83 \(\int (e x)^m \text {csch}^2(a+b x^n) \, dx\) [83]

Optimal. Leaf size=28 \[ x^{-m} (e x)^m \text {Int}\left (x^m \text {csch}^2\left (a+b x^n\right ),x\right ) \]

[Out]

(e*x)^m*Unintegrable(x^m*csch(a+b*x^n)^2,x)/(x^m)

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Rubi [A]
time = 0.03, 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)^m \text {csch}^2\left (a+b x^n\right ) \, dx \end {gather*}

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*}

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Mathematica [A]
time = 16.54, size = 0, normalized size = 0.00 \begin {gather*} \int (e x)^m \text {csch}^2\left (a+b x^n\right ) \, dx \end {gather*}

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 = 0.52, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m}}{\sinh \left (a +b \,x^{n}\right )^{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(m*e^m - (n - 1)*e^m)*integrate(1/4*x^m/(b*n*x^n + b*n*e^(b*x^n + n*log(x) + a)), x) + 4*(m*e^m - (n - 1)*e
^m)*integrate(-1/4*x^m/(b*n*x^n - b*n*e^(b*x^n + n*log(x) + a)), x) + 2*x*e^(m*log(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e x\right )^{m}}{\sinh ^{2}{\left (a + b x^{n} \right )}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (e\,x\right )}^m}{{\mathrm {sinh}\left (a+b\,x^n\right )}^2} \,d x \end {gather*}

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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