3.68 \(\int (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}) \, dx\)

Optimal. Leaf size=20 \[ 4 \sqrt {\sinh (x)}-\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {3315} \[ 4 \sqrt {\sinh (x)}-\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*Cosh[x])/Sqrt[Sinh[x]] + 4*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 {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx &=\int \frac {x}{\sinh ^{\frac {3}{2}}(x)} \, dx-\int x \sqrt {\sinh (x)} \, dx\\ &=-\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}}+4 \sqrt {\sinh (x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 17, normalized size = 0.85 \[ \frac {4 \sinh (x)-2 x \cosh (x)}{\sqrt {\sinh (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -x \sqrt {\sinh \relax (x)} + \frac {x}{\sinh \relax (x)^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sinh \relax (x )^{\frac {3}{2}}}-x \left (\sqrt {\sinh }\relax (x )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -x \sqrt {\sinh \relax (x)} + \frac {x}{\sinh \relax (x)^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.17, size = 38, normalized size = 1.90 \[ -\frac {2\,\sqrt {\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}}\,\left (x-2\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}+2\right )}{{\mathrm {e}}^{2\,x}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {x}{\sinh ^{\frac {3}{2}}{\relax (x )}}\right )\, dx - \int x \sqrt {\sinh {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sinh(x)**(3/2)-x*sinh(x)**(1/2),x)

[Out]

-Integral(-x/sinh(x)**(3/2), x) - Integral(x*sqrt(sinh(x)), x)

________________________________________________________________________________________