3.7.78 \(\int (\text {csch}(x)+\sinh (x))^{3/2} \, dx\) [678]

Optimal. Leaf size=31 \[ \frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}-\frac {8}{3} \sqrt {\cosh (x) \coth (x)} \text {sech}(x) \]

[Out]

2/3*cosh(x)*(cosh(x)*coth(x))^(1/2)-8/3*sech(x)*(cosh(x)*coth(x))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4482, 4483, 4485, 2678, 2669} \begin {gather*} \frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}-\frac {8}{3} \text {sech}(x) \sqrt {\cosh (x) \coth (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Csch[x] + Sinh[x])^(3/2),x]

[Out]

(2*Cosh[x]*Sqrt[Cosh[x]*Coth[x]])/3 - (8*Sqrt[Cosh[x]*Coth[x]]*Sech[x])/3

Rule 2669

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*(a*Sin[e
+ f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*m)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]

Rule 2678

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[(-b)*(a*Sin
[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*m)), x] + Dist[a^2*((m + n - 1)/m), Int[(a*Sin[e + f*x])^(m - 2)*(b*
Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, 1/2])) && IntegersQ
[2*m, 2*n]

Rule 4482

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 4483

Int[(u_.)*((a_)*(v_))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v]}, Dist[a^IntPart[p]*
((a*vv)^FracPart[p]/vv^FracPart[p]), Int[uu*vv^p, x], x]] /; FreeQ[{a, p}, x] &&  !IntegerQ[p] &&  !InertTrigF
reeQ[v]

Rule 4485

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = Ac
tivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])), Int[uu*vv^(m*p)*ww^(n*p)
, x], x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  !InertTrigFreeQ[w])

Rubi steps

\begin {align*} \int (\text {csch}(x)+\sinh (x))^{3/2} \, dx &=\int (\cosh (x) \coth (x))^{3/2} \, dx\\ &=\frac {\left (i \sqrt {\cosh (x) \coth (x)}\right ) \int (-i \cosh (x) \coth (x))^{3/2} \, dx}{\sqrt {-i \cosh (x) \coth (x)}}\\ &=\frac {\left (i \sqrt {\cosh (x) \coth (x)}\right ) \int \cosh ^{\frac {3}{2}}(x) (-i \coth (x))^{3/2} \, dx}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\\ &=\frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}+\frac {\left (4 i \sqrt {\cosh (x) \coth (x)}\right ) \int \frac {(-i \coth (x))^{3/2}}{\sqrt {\cosh (x)}} \, dx}{3 \sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\\ &=\frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}-\frac {8}{3} \sqrt {\cosh (x) \coth (x)} \text {sech}(x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 21, normalized size = 0.68 \begin {gather*} \frac {2}{3} \left (-4+\cosh ^2(x)\right ) \sqrt {\cosh (x) \coth (x)} \text {sech}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-4 + Cosh[x]^2)*Sqrt[Cosh[x]*Coth[x]]*Sech[x])/3

________________________________________________________________________________________

Maple [F]
time = 2.35, size = 0, normalized size = 0.00 \[\int \left (\mathrm {csch}\left (x \right )+\sinh \left (x \right )\right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((csch(x)+sinh(x))^(3/2),x)

[Out]

int((csch(x)+sinh(x))^(3/2),x)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (23) = 46\).
time = 0.49, size = 109, normalized size = 3.52 \begin {gather*} \frac {\sqrt {2} e^{\left (\frac {3}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} e^{\left (-\frac {1}{2} \, x\right )}}{2 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {5 \, \sqrt {2} e^{\left (-\frac {5}{2} \, x\right )}}{2 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (-\frac {9}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(2)*e^(3/2*x)/((e^(-x) + 1)^(3/2)*(-e^(-x) + 1)^(3/2)) - 5/2*sqrt(2)*e^(-1/2*x)/((e^(-x) + 1)^(3/2)*(-
e^(-x) + 1)^(3/2)) + 5/2*sqrt(2)*e^(-5/2*x)/((e^(-x) + 1)^(3/2)*(-e^(-x) + 1)^(3/2)) - 1/6*sqrt(2)*e^(-9/2*x)/
((e^(-x) + 1)^(3/2)*(-e^(-x) + 1)^(3/2))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (23) = 46\).
time = 0.39, size = 97, normalized size = 3.13 \begin {gather*} \frac {\sqrt {\frac {1}{2}} {\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} - 7\right )} \sinh \left (x\right )^{2} - 14 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}}{3 \, \sqrt {\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(1/2)*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 7)*sinh(x)^2 - 14*cosh(x)^2 + 4*
(cosh(x)^3 - 7*cosh(x))*sinh(x) + 1)/(sqrt(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 - 1)*sin
h(x) - cosh(x))*(cosh(x) + sinh(x)))

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

________________________________________________________________________________________

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((csch(x)+sinh(x))^(3/2),x, algorithm="giac")

[Out]

integrate((csch(x) + sinh(x))^(3/2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (\mathrm {sinh}\left (x\right )+\frac {1}{\mathrm {sinh}\left (x\right )}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sinh(x) + 1/sinh(x))^(3/2),x)

[Out]

int((sinh(x) + 1/sinh(x))^(3/2), x)

________________________________________________________________________________________