\(\int (\frac {x}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x \sqrt {\text {csch}(x)}) \, dx\) [92]

Optimal result

Integrand size = 20, antiderivative size = 24 \[ \int \left (\frac {x}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x \sqrt {\text {csch}(x)}\right ) \, dx=-\frac {4}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {2 x \cosh (x)}{3 \sqrt {\text {csch}(x)}} \]

[Out]

-4/9/csch(x)^(3/2)+2/3*x*cosh(x)/csch(x)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4272, 4274} \[ \int \left (\frac {x}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x \sqrt {\text {csch}(x)}\right ) \, dx=\frac {2 x \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {4}{9 \text {csch}^{\frac {3}{2}}(x)} \]

[In]

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

[Out]

-4/(9*Csch[x]^(3/2)) + (2*x*Cosh[x])/(3*Sqrt[Csch[x]])

Rule 4272

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

Rule 4274

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[(b*Sin[e + f*x])^n*(b*C
sc[e + f*x])^n, Int[(c + d*x)^m/(b*Sin[e + f*x])^n, x], x] /; FreeQ[{b, c, d, e, f, m, n}, x] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int x \sqrt {\text {csch}(x)} \, dx+\int \frac {x}{\text {csch}^{\frac {3}{2}}(x)} \, dx \\ & = -\frac {4}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {2 x \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {1}{3} \int x \sqrt {\text {csch}(x)} \, dx+\frac {1}{3} \left (\sqrt {\text {csch}(x)} \sqrt {-\sinh (x)}\right ) \int \frac {x}{\sqrt {-\sinh (x)}} \, dx \\ & = -\frac {4}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {2 x \cosh (x)}{3 \sqrt {\text {csch}(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \left (\frac {x}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x \sqrt {\text {csch}(x)}\right ) \, dx=\frac {2 (-2+3 x \coth (x))}{9 \text {csch}^{\frac {3}{2}}(x)} \]

[In]

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

[Out]

(2*(-2 + 3*x*Coth[x]))/(9*Csch[x]^(3/2))

Maple [F]

\[\int \left (\frac {x}{\operatorname {csch}\left (x \right )^{\frac {3}{2}}}+\frac {x \sqrt {\operatorname {csch}\left (x \right )}}{3}\right )d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x \sqrt {\text {csch}(x)}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \left (\frac {x}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x \sqrt {\text {csch}(x)}\right ) \, dx=\frac {\int \frac {3 x}{\operatorname {csch}^{\frac {3}{2}}{\left (x \right )}}\, dx + \int x \sqrt {\operatorname {csch}{\left (x \right )}}\, dx}{3} \]

[In]

integrate(x/csch(x)**(3/2)+1/3*x*csch(x)**(1/2),x)

[Out]

(Integral(3*x/csch(x)**(3/2), x) + Integral(x*sqrt(csch(x)), x))/3

Maxima [F]

\[ \int \left (\frac {x}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x \sqrt {\text {csch}(x)}\right ) \, dx=\int { \frac {1}{3} \, x \sqrt {\operatorname {csch}\left (x\right )} + \frac {x}{\operatorname {csch}\left (x\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \left (\frac {x}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x \sqrt {\text {csch}(x)}\right ) \, dx=\int { \frac {1}{3} \, x \sqrt {\operatorname {csch}\left (x\right )} + \frac {x}{\operatorname {csch}\left (x\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \left (\frac {x}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x \sqrt {\text {csch}(x)}\right ) \, dx=\int \frac {x\,\sqrt {\frac {1}{\mathrm {sinh}\left (x\right )}}}{3}+\frac {x}{{\left (\frac {1}{\mathrm {sinh}\left (x\right )}\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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