\(\int (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}) \, dx\) [95]

Optimal result

Integrand size = 24, antiderivative size = 76 \[ \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx=-\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}+\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {16}{27} i \sqrt {\text {csch}(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)} \]

[Out]

-8/9*x/csch(x)^(3/2)+16/27*cosh(x)/csch(x)^(1/2)+2/3*x^2*cosh(x)/csch(x)^(1/2)-16/27*I*(sin(1/4*Pi+1/2*I*x)^2)
^(1/2)/sin(1/4*Pi+1/2*I*x)*EllipticF(cos(1/4*Pi+1/2*I*x),2^(1/2))*csch(x)^(1/2)*(I*sinh(x))^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4273, 4274, 3854, 3856, 2720} \[ \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx=\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}-\frac {16}{27} i \sqrt {i \sinh (x)} \sqrt {\text {csch}(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \]

[In]

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

[Out]

(-8*x)/(9*Csch[x]^(3/2)) + (16*Cosh[x])/(27*Sqrt[Csch[x]]) + (2*x^2*Cosh[x])/(3*Sqrt[Csch[x]]) - ((16*I)/27)*S
qrt[Csch[x]]*EllipticF[Pi/4 - (I/2)*x, 2]*Sqrt[I*Sinh[x]]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3854

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

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4273

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Csc[e + f*x])^n/(f^2*n^2)), x] + (Dist[(n + 1)/(b^2*n), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n + 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^n, x], x] + Simp[(c + d*x)^m*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] && GtQ[m, 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^2 \sqrt {\text {csch}(x)} \, dx+\int \frac {x^2}{\text {csch}^{\frac {3}{2}}(x)} \, dx \\ & = -\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {1}{3} \int x^2 \sqrt {\text {csch}(x)} \, dx+\frac {8}{9} \int \frac {1}{\text {csch}^{\frac {3}{2}}(x)} \, dx+\frac {1}{3} \left (\sqrt {\text {csch}(x)} \sqrt {-\sinh (x)}\right ) \int \frac {x^2}{\sqrt {-\sinh (x)}} \, dx \\ & = -\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}+\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {8}{27} \int \sqrt {\text {csch}(x)} \, dx \\ & = -\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}+\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {1}{27} \left (8 \sqrt {\text {csch}(x)} \sqrt {i \sinh (x)}\right ) \int \frac {1}{\sqrt {i \sinh (x)}} \, dx \\ & = -\frac {8 x}{9 \text {csch}^{\frac {3}{2}}(x)}+\frac {16 \cosh (x)}{27 \sqrt {\text {csch}(x)}}+\frac {2 x^2 \cosh (x)}{3 \sqrt {\text {csch}(x)}}-\frac {16}{27} i \sqrt {\text {csch}(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88 \[ \int \left (\frac {x^2}{\text {csch}^{\frac {3}{2}}(x)}+\frac {1}{3} x^2 \sqrt {\text {csch}(x)}\right ) \, dx=\frac {4 \left (3+\text {csch}^2(x)\right ) \left (-12 x+\left (8+9 x^2\right ) \coth (x)+\frac {8 \text {csch}(x) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right )}{\sqrt {i \sinh (x)}}\right )}{27 (-1+3 \cosh (2 x)) \text {csch}^{\frac {7}{2}}(x)} \]

[In]

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

[Out]

(4*(3 + Csch[x]^2)*(-12*x + (8 + 9*x^2)*Coth[x] + (8*Csch[x]*EllipticF[(Pi - (2*I)*x)/4, 2])/Sqrt[I*Sinh[x]]))
/(27*(-1 + 3*Cosh[2*x])*Csch[x]^(7/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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