\(\int \frac {x^2}{a+b \text {csch}(c+d x^2)} \, dx\) [19]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {x^2}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\text {Int}\left (\frac {x^2}{a+b \text {csch}\left (c+d x^2\right )},x\right ) \]

[Out]

Unintegrable(x^2/(a+b*csch(d*x^2+c)),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {x^2}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int \frac {x^2}{a+b \text {csch}\left (c+d x^2\right )} \, dx \]

[In]

Int[x^2/(a + b*Csch[c + d*x^2]),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int \frac {x^2}{a+b \text {csch}\left (c+d x^2\right )} \, dx \]

[In]

Integrate[x^2/(a + b*Csch[c + d*x^2]),x]

[Out]

Integrate[x^2/(a + b*Csch[c + d*x^2]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

\[\int \frac {x^{2}}{a +b \,\operatorname {csch}\left (d \,x^{2}+c \right )}d x\]

[In]

int(x^2/(a+b*csch(d*x^2+c)),x)

[Out]

int(x^2/(a+b*csch(d*x^2+c)),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int { \frac {x^{2}}{b \operatorname {csch}\left (d x^{2} + c\right ) + a} \,d x } \]

[In]

integrate(x^2/(a+b*csch(d*x^2+c)),x, algorithm="fricas")

[Out]

integral(x^2/(b*csch(d*x^2 + c) + a), x)

Sympy [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {x^2}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int \frac {x^{2}}{a + b \operatorname {csch}{\left (c + d x^{2} \right )}}\, dx \]

[In]

integrate(x**2/(a+b*csch(d*x**2+c)),x)

[Out]

Integral(x**2/(a + b*csch(c + d*x**2)), x)

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.39 \[ \int \frac {x^2}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int { \frac {x^{2}}{b \operatorname {csch}\left (d x^{2} + c\right ) + a} \,d x } \]

[In]

integrate(x^2/(a+b*csch(d*x^2+c)),x, algorithm="maxima")

[Out]

1/3*x^3/a - 2*b*integrate(x^2*e^(d*x^2 + c)/(a^2*e^(2*d*x^2 + 2*c) + 2*a*b*e^(d*x^2 + c) - a^2), x)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int { \frac {x^{2}}{b \operatorname {csch}\left (d x^{2} + c\right ) + a} \,d x } \]

[In]

integrate(x^2/(a+b*csch(d*x^2+c)),x, algorithm="giac")

[Out]

integrate(x^2/(b*csch(d*x^2 + c) + a), x)

Mupad [N/A]

Not integrable

Time = 2.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {x^2}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int \frac {x^2}{a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}} \,d x \]

[In]

int(x^2/(a + b/sinh(c + d*x^2)),x)

[Out]

int(x^2/(a + b/sinh(c + d*x^2)), x)