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

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int (e x)^m \text {csch}^2\left (a+b x^n\right ) \, dx=x^{-m} (e x)^m \text {Int}\left (x^m \text {csch}^2\left (a+b x^n\right ),x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (e x)^m \text {csch}^2\left (a+b x^n\right ) \, dx=\int (e x)^m \text {csch}^2\left (a+b x^n\right ) \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 18.35 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (e x)^m \text {csch}^2\left (a+b x^n\right ) \, dx=\int (e x)^m \text {csch}^2\left (a+b x^n\right ) \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int \frac {\left (e x \right )^{m}}{\sinh \left (a +b \,x^{n}\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (e x)^m \text {csch}^2\left (a+b x^n\right ) \, dx=\int { \frac {\left (e x\right )^{m}}{\sinh \left (b x^{n} + a\right )^{2}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 1.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int (e x)^m \text {csch}^2\left (a+b x^n\right ) \, dx=\int \frac {\left (e x\right )^{m}}{\sinh ^{2}{\left (a + b x^{n} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 7.69 \[ \int (e x)^m \text {csch}^2\left (a+b x^n\right ) \, dx=\int { \frac {\left (e x\right )^{m}}{\sinh \left (b x^{n} + a\right )^{2}} \,d x } \]

[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))

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (e x)^m \text {csch}^2\left (a+b x^n\right ) \, dx=\int { \frac {\left (e x\right )^{m}}{\sinh \left (b x^{n} + a\right )^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 1.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (e x)^m \text {csch}^2\left (a+b x^n\right ) \, dx=\int \frac {{\left (e\,x\right )}^m}{{\mathrm {sinh}\left (a+b\,x^n\right )}^2} \,d x \]

[In]

int((e*x)^m/sinh(a + b*x^n)^2,x)

[Out]

int((e*x)^m/sinh(a + b*x^n)^2, x)