3.64 \(\int \frac{1}{x^{3/2} (a+b \text{csch}(c+d \sqrt{x}))} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{x^{3/2} \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )},x\right ) \]

[Out]

Unintegrable[1/(x^(3/2)*(a + b*Csch[c + d*Sqrt[x]])), x]

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Rubi [A]  time = 0.0279057, 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 = {} \[ \int \frac{1}{x^{3/2} \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^(3/2)*(a + b*Csch[c + d*Sqrt[x]])),x]

[Out]

Defer[Int][1/(x^(3/2)*(a + b*Csch[c + d*Sqrt[x]])), x]

Rubi steps

\begin{align*} \int \frac{1}{x^{3/2} \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )} \, dx &=\int \frac{1}{x^{3/2} \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 7.22908, size = 0, normalized size = 0. \[ \int \frac{1}{x^{3/2} \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^(3/2)*(a + b*Csch[c + d*Sqrt[x]])),x]

[Out]

Integrate[1/(x^(3/2)*(a + b*Csch[c + d*Sqrt[x]])), x]

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Maple [A]  time = 0.095, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-{\frac{3}{2}}} \left ( a+b{\rm csch} \left (c+d\sqrt{x}\right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(a+b*csch(c+d*x^(1/2))),x)

[Out]

int(1/x^(3/2)/(a+b*csch(c+d*x^(1/2))),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(a+b*csch(c+d*x^(1/2))),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(a+b*csch(c+d*x^(1/2))),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{3}{2}} \left (a + b \operatorname{csch}{\left (c + d \sqrt{x} \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(3/2)/(a+b*csch(c+d*x**(1/2))),x)

[Out]

Integral(1/(x**(3/2)*(a + b*csch(c + d*sqrt(x)))), 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 )} x^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(a+b*csch(c+d*x^(1/2))),x, algorithm="giac")

[Out]

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