3.67 \(\int \frac{\sinh ^{\frac{3}{2}}(x)}{x^3} \, dx\)

Optimal. Leaf size=61 \[ \frac{9}{8} \text{Unintegrable}\left (\frac{\sinh ^{\frac{3}{2}}(x)}{x},x\right )+\frac{3}{8} \text{Unintegrable}\left (\frac{1}{x \sqrt{\sinh (x)}},x\right )-\frac{\sinh ^{\frac{3}{2}}(x)}{2 x^2}-\frac{3 \sqrt{\sinh (x)} \cosh (x)}{4 x} \]

[Out]

(-3*Cosh[x]*Sqrt[Sinh[x]])/(4*x) - Sinh[x]^(3/2)/(2*x^2) + (3*Unintegrable[1/(x*Sqrt[Sinh[x]]), x])/8 + (9*Uni
ntegrable[Sinh[x]^(3/2)/x, x])/8

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Rubi [A]  time = 0.107143, 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 = {} \[ \int \frac{\sinh ^{\frac{3}{2}}(x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sinh[x]^(3/2)/x^3,x]

[Out]

(-3*Cosh[x]*Sqrt[Sinh[x]])/(4*x) - Sinh[x]^(3/2)/(2*x^2) + (3*Defer[Int][1/(x*Sqrt[Sinh[x]]), x])/8 + (9*Defer
[Int][Sinh[x]^(3/2)/x, x])/8

Rubi steps

\begin{align*} \int \frac{\sinh ^{\frac{3}{2}}(x)}{x^3} \, dx &=-\frac{3 \cosh (x) \sqrt{\sinh (x)}}{4 x}-\frac{\sinh ^{\frac{3}{2}}(x)}{2 x^2}+\frac{3}{8} \int \frac{1}{x \sqrt{\sinh (x)}} \, dx+\frac{9}{8} \int \frac{\sinh ^{\frac{3}{2}}(x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 6.01493, size = 0, normalized size = 0. \[ \int \frac{\sinh ^{\frac{3}{2}}(x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sinh[x]^(3/2)/x^3,x]

[Out]

Integrate[Sinh[x]^(3/2)/x^3, x]

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Maple [A]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( \sinh \left ( x \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^(3/2)/x^3,x)

[Out]

int(sinh(x)^(3/2)/x^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (x\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^(3/2)/x^3,x, algorithm="maxima")

[Out]

integrate(sinh(x)^(3/2)/x^3, x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^(3/2)/x^3,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{\frac{3}{2}}{\left (x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**(3/2)/x**3,x)

[Out]

Integral(sinh(x)**(3/2)/x**3, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (x\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^(3/2)/x^3,x, algorithm="giac")

[Out]

integrate(sinh(x)^(3/2)/x^3, x)