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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{x^2 \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02599, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x^2 \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 71.2205, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.151, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b{\rm csch} \left (c+d\sqrt{x}\right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{4 \, a b^{2} \sqrt{x} +{\left (a^{3} d e^{\left (2 \, c\right )} + a b^{2} d e^{\left (2 \, c\right )}\right )} x e^{\left (2 \, d \sqrt{x}\right )} -{\left (a^{3} d + a b^{2} d\right )} x - 2 \,{\left (2 \, b^{3} \sqrt{x} e^{c} -{\left (a^{2} b d e^{c} + b^{3} d e^{c}\right )} x\right )} e^{\left (d \sqrt{x}\right )}}{{\left (a^{5} d e^{\left (2 \, c\right )} + a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} x^{2} e^{\left (2 \, d \sqrt{x}\right )} + 2 \,{\left (a^{4} b d e^{c} + a^{2} b^{3} d e^{c}\right )} x^{2} e^{\left (d \sqrt{x}\right )} -{\left (a^{5} d + a^{3} b^{2} d\right )} x^{2}} + \int -\frac{2 \,{\left (3 \, a b^{2} \sqrt{x} -{\left (3 \, b^{3} \sqrt{x} e^{c} -{\left (2 \, a^{2} b d e^{c} + b^{3} d e^{c}\right )} x\right )} e^{\left (d \sqrt{x}\right )}\right )}}{{\left (a^{5} d e^{\left (2 \, c\right )} + a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} x^{3} e^{\left (2 \, d \sqrt{x}\right )} + 2 \,{\left (a^{4} b d e^{c} + a^{2} b^{3} d e^{c}\right )} x^{3} e^{\left (d \sqrt{x}\right )} -{\left (a^{5} d + a^{3} b^{2} d\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{2} x^{2} \operatorname{csch}\left (d \sqrt{x} + c\right )^{2} + 2 \, a b x^{2} \operatorname{csch}\left (d \sqrt{x} + c\right ) + a^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (a + b \operatorname{csch}{\left (c + d \sqrt{x} \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{csch}\left (d \sqrt{x} + c\right ) + a\right )}^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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