\(\int (e x)^m \text {csch}(a+b x^2) \, dx\) [27]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int (e x)^m \text {csch}\left (a+b x^2\right ) \, dx=x^{-m} (e x)^m \text {Int}\left (x^m \text {csch}\left (a+b x^2\right ),x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 14, 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 = {} \[ \int (e x)^m \text {csch}\left (a+b x^2\right ) \, dx=\int (e x)^m \text {csch}\left (a+b x^2\right ) \, dx \]

[In]

Int[(e*x)^m*Csch[a + b*x^2],x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int (e x)^m \text {csch}\left (a+b x^2\right ) \, dx=\int (e x)^m \text {csch}\left (a+b x^2\right ) \, dx \]

[In]

Integrate[(e*x)^m*Csch[a + b*x^2],x]

[Out]

Integrate[(e*x)^m*Csch[a + b*x^2], x]

Maple [N/A] (verified)

Not integrable

Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int (e x)^m \text {csch}\left (a+b x^2\right ) \, dx=\int { \frac {\left (e x\right )^{m}}{\sinh \left (b x^{2} + a\right )} \,d x } \]

[In]

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

[Out]

integral((e*x)^m/sinh(b*x^2 + a), x)

Sympy [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (e x)^m \text {csch}\left (a+b x^2\right ) \, dx=\int \frac {\left (e x\right )^{m}}{\sinh {\left (a + b x^{2} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int (e x)^m \text {csch}\left (a+b x^2\right ) \, dx=\int { \frac {\left (e x\right )^{m}}{\sinh \left (b x^{2} + a\right )} \,d x } \]

[In]

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

[Out]

integrate((e*x)^m/sinh(b*x^2 + a), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int (e x)^m \text {csch}\left (a+b x^2\right ) \, dx=\int { \frac {\left (e x\right )^{m}}{\sinh \left (b x^{2} + a\right )} \,d x } \]

[In]

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

[Out]

integrate((e*x)^m/sinh(b*x^2 + a), x)

Mupad [N/A]

Not integrable

Time = 1.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int (e x)^m \text {csch}\left (a+b x^2\right ) \, dx=\int \frac {{\left (e\,x\right )}^m}{\mathrm {sinh}\left (b\,x^2+a\right )} \,d x \]

[In]

int((e*x)^m/sinh(a + b*x^2),x)

[Out]

int((e*x)^m/sinh(a + b*x^2), x)