3.1.54 \(\int \frac {a+b \text {csch}(c+d \sqrt {x})}{x^{3/2}} \, dx\) [54]

Optimal. Leaf size=30 \[ -\frac {2 a}{\sqrt {x}}+b \text {Int}\left (\frac {\text {csch}\left (c+d \sqrt {x}\right )}{x^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, 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 {a+b \text {csch}\left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 15.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \text {csch}\left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 3.10, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )}{x^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}}{x^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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