3.69 \(\int (\frac{x}{\sinh ^{\frac{5}{2}}(x)}+\frac{x}{3 \sqrt{\sinh (x)}}) \, dx\)

Optimal. Leaf size=24 \[ -\frac{4}{3 \sqrt{\sinh (x)}}-\frac{2 x \cosh (x)}{3 \sinh ^{\frac{3}{2}}(x)} \]

[Out]

(-2*x*Cosh[x])/(3*Sinh[x]^(3/2)) - 4/(3*Sqrt[Sinh[x]])

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Rubi [A]  time = 0.0602881, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {3315} \[ -\frac{4}{3 \sqrt{\sinh (x)}}-\frac{2 x \cosh (x)}{3 \sinh ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Int[x/Sinh[x]^(5/2) + x/(3*Sqrt[Sinh[x]]),x]

[Out]

(-2*x*Cosh[x])/(3*Sinh[x]^(3/2)) - 4/(3*Sqrt[Sinh[x]])

Rule 3315

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)*Cos[e + f*x]*(b*Si
n[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + (Dist[(n + 2)/(b^2*(n + 1)), Int[(c + d*x)*(b*Sin[e + f*x])^(n + 2),
x], x] - Simp[(d*(b*Sin[e + f*x])^(n + 2))/(b^2*f^2*(n + 1)*(n + 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[
n, -1] && NeQ[n, -2]

Rubi steps

\begin{align*} \int \left (\frac{x}{\sinh ^{\frac{5}{2}}(x)}+\frac{x}{3 \sqrt{\sinh (x)}}\right ) \, dx &=\frac{1}{3} \int \frac{x}{\sqrt{\sinh (x)}} \, dx+\int \frac{x}{\sinh ^{\frac{5}{2}}(x)} \, dx\\ &=-\frac{2 x \cosh (x)}{3 \sinh ^{\frac{3}{2}}(x)}-\frac{4}{3 \sqrt{\sinh (x)}}\\ \end{align*}

Mathematica [A]  time = 0.0705882, size = 22, normalized size = 0.92 \[ \frac{1}{6} \sqrt{\sinh (x)} (-8 \text{csch}(x)-4 x \coth (x) \text{csch}(x)) \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sinh[x]^(5/2) + x/(3*Sqrt[Sinh[x]]),x]

[Out]

((-8*Csch[x] - 4*x*Coth[x]*Csch[x])*Sqrt[Sinh[x]])/6

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \sinh \left ( x \right ) \right ) ^{-{\frac{5}{2}}}}+{\frac{x}{3}{\frac{1}{\sqrt{\sinh \left ( x \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/sinh(x)^(5/2)+1/3*x/sinh(x)^(1/2),x)

[Out]

int(x/sinh(x)^(5/2)+1/3*x/sinh(x)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{3 \, \sqrt{\sinh \left (x\right )}} + \frac{x}{\sinh \left (x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sinh(x)^(5/2)+1/3*x/sinh(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/3*x/sqrt(sinh(x)) + x/sinh(x)^(5/2), x)

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Fricas [B]  time = 2.621, size = 375, normalized size = 15.62 \begin{align*} -\frac{4 \,{\left ({\left (x + 2\right )} \cosh \left (x\right )^{3} + 3 \,{\left (x + 2\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} +{\left (x + 2\right )} \sinh \left (x\right )^{3} +{\left (x - 2\right )} \cosh \left (x\right ) +{\left (3 \,{\left (x + 2\right )} \cosh \left (x\right )^{2} + x - 2\right )} \sinh \left (x\right )\right )} \sqrt{\sinh \left (x\right )}}{3 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sinh(x)^(5/2)+1/3*x/sinh(x)^(1/2),x, algorithm="fricas")

[Out]

-4/3*((x + 2)*cosh(x)^3 + 3*(x + 2)*cosh(x)*sinh(x)^2 + (x + 2)*sinh(x)^3 + (x - 2)*cosh(x) + (3*(x + 2)*cosh(
x)^2 + x - 2)*sinh(x))*sqrt(sinh(x))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x
)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{3 x}{\sinh ^{\frac{5}{2}}{\left (x \right )}}\, dx + \int \frac{x}{\sqrt{\sinh{\left (x \right )}}}\, dx}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sinh(x)**(5/2)+1/3*x/sinh(x)**(1/2),x)

[Out]

(Integral(3*x/sinh(x)**(5/2), x) + Integral(x/sqrt(sinh(x)), x))/3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{3 \, \sqrt{\sinh \left (x\right )}} + \frac{x}{\sinh \left (x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sinh(x)^(5/2)+1/3*x/sinh(x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/3*x/sqrt(sinh(x)) + x/sinh(x)^(5/2), x)