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

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

[Out]

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

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Rubi [A]  time = 0.025713, 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 = {} \[ \int \frac{\left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 60.2659, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.135, 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((a+b*csch(c+d*x^(1/2)))^2/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integral((b^2*csch(d*sqrt(x) + c)^2 + 2*a*b*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{\left (a + b \operatorname{csch}{\left (c + d \sqrt{x} \right )}\right )^{2}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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((a+b*csch(c+d*x^(1/2)))^2/x^2,x, algorithm="giac")

[Out]

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