\(\int \frac {1}{x^2 (a+b \text {csch}(c+d x^2))^2} \, dx\) [29]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/x^2/(a+b*csch(d*x^2+c))^2,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 = {} \[ \int \frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx \]

[In]

Int[1/(x^2*(a + b*Csch[c + d*x^2])^2),x]

[Out]

Defer[Int][1/(x^2*(a + b*Csch[c + d*x^2])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 19.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx \]

[In]

Integrate[1/(x^2*(a + b*Csch[c + d*x^2])^2),x]

[Out]

Integrate[1/(x^2*(a + b*Csch[c + d*x^2])^2), x]

Maple [N/A] (verified)

Not integrable

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

\[\int \frac {1}{x^{2} {\left (a +b \,\operatorname {csch}\left (d \,x^{2}+c \right )\right )}^{2}}d x\]

[In]

int(1/x^2/(a+b*csch(d*x^2+c))^2,x)

[Out]

int(1/x^2/(a+b*csch(d*x^2+c))^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.44 \[ \int \frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*csch(d*x^2+c))^2,x, algorithm="fricas")

[Out]

integral(1/(b^2*x^2*csch(d*x^2 + c)^2 + 2*a*b*x^2*csch(d*x^2 + c) + a^2*x^2), x)

Sympy [N/A]

Not integrable

Time = 0.83 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x^{2} \left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

[In]

integrate(1/x**2/(a+b*csch(d*x**2+c))**2,x)

[Out]

Integral(1/(x**2*(a + b*csch(c + d*x**2))**2), x)

Maxima [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 313, normalized size of antiderivative = 17.39 \[ \int \frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*csch(d*x^2+c))^2,x, algorithm="maxima")

[Out]

-((a^3*d*e^(2*c) + a*b^2*d*e^(2*c))*x^2*e^(2*d*x^2) + a*b^2 - (a^3*d + a*b^2*d)*x^2 - (b^3*e^c - 2*(a^2*b*d*e^
c + b^3*d*e^c)*x^2)*e^(d*x^2))/((a^5*d*e^(2*c) + a^3*b^2*d*e^(2*c))*x^3*e^(2*d*x^2) + 2*(a^4*b*d*e^c + a^2*b^3
*d*e^c)*x^3*e^(d*x^2) - (a^5*d + a^3*b^2*d)*x^3) - integrate((3*a*b^2 - (3*b^3*e^c - 2*(2*a^2*b*d*e^c + b^3*d*
e^c)*x^2)*e^(d*x^2))/((a^5*d*e^(2*c) + a^3*b^2*d*e^(2*c))*x^4*e^(2*d*x^2) + 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*x^
4*e^(d*x^2) - (a^5*d + a^3*b^2*d)*x^4), x)

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*csch(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate(1/((b*csch(d*x^2 + c) + a)^2*x^2), x)

Mupad [N/A]

Not integrable

Time = 2.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x^2\,{\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right )}^2} \,d x \]

[In]

int(1/(x^2*(a + b/sinh(c + d*x^2))^2),x)

[Out]

int(1/(x^2*(a + b/sinh(c + d*x^2))^2), x)