\(\int (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}) \, dx\) [68]

Optimal result

Integrand size = 18, antiderivative size = 20 \[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=-\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}}+4 \sqrt {\sinh (x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {3396} \[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=4 \sqrt {\sinh (x)}-\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}} \]

[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 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*} \text {integral}& = \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] (verified)

Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=\frac {-2 x \cosh (x)+4 \sinh (x)}{\sqrt {\sinh (x)}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (\frac {x}{\sinh \left (x \right )^{\frac {3}{2}}}-x \sqrt {\sinh \left (x \right )}\right )d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [F]

\[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=- \int \left (- \frac {x}{\sinh ^{\frac {3}{2}}{\left (x \right )}}\right )\, dx - \int x \sqrt {\sinh {\left (x \right )}}\, dx \]

[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)

Maxima [F]

\[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=\int { -x \sqrt {\sinh \left (x\right )} + \frac {x}{\sinh \left (x\right )^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=\int { -x \sqrt {\sinh \left (x\right )} + \frac {x}{\sinh \left (x\right )^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=-\frac {2\,\sqrt {\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}}\,\left (x-2\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}+2\right )}{{\mathrm {e}}^{2\,x}-1} \]

[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)