3.1.69 \(\int (\frac {x}{\sinh ^{\frac {5}{2}}(x)}+\frac {x}{3 \sqrt {\sinh (x)}}) \, dx\) [69]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3396} \begin {gather*} -\frac {4}{3 \sqrt {\sinh (x)}}-\frac {2 x \cosh (x)}{3 \sinh ^{\frac {3}{2}}(x)} \end {gather*}

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 3396

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*}

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Mathematica [A]
time = 0.04, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{6} (-8 \text {csch}(x)-4 x \coth (x) \text {csch}(x)) \sqrt {\sinh (x)} \end {gather*}

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 = 1.65, size = 0, normalized size = 0.00 \[\int \frac {x}{\sinh \left (x \right )^{\frac {5}{2}}}+\frac {x}{3 \sqrt {\sinh \left (x \right )}}\, dx\]

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 = 0.00, 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(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] Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (16) = 32\).
time = 0.35, size = 108, normalized size = 4.50 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {3 x}{\sinh ^{\frac {5}{2}}{\left (x \right )}}\, dx + \int \frac {x}{\sqrt {\sinh {\left (x \right )}}}\, dx}{3} \end {gather*}

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 = 0.00, 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(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)

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Mupad [B]
time = 0.15, size = 40, normalized size = 1.67 \begin {gather*} -\frac {4\,{\mathrm {e}}^x\,\sqrt {\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}}\,\left (x+2\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}-2\right )}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*exp(x)*(exp(x)/2 - exp(-x)/2)^(1/2)*(x + 2*exp(2*x) + x*exp(2*x) - 2))/(3*(exp(2*x) - 1)^2)

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