3.1.69 \(\int \frac {1}{x^{3/2} (a+b \text {csch}(c+d \sqrt {x}))^2} \, dx\) [69]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{x^{3/2} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{x^{3/2} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 43.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3/2} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 5.31, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{\frac {3}{2}} \left (a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x)/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {3}{2}} \left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{x^{3/2}\,{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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