### 3.678 $$\int (\text{csch}(x)+\sinh (x))^{3/2} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.122285, antiderivative size = 31, normalized size of antiderivative = 1., 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 = {4397, 4398, 4400, 2598, 2589} $\frac{2}{3} \cosh (x) \sqrt{\cosh (x) \coth (x)}-\frac{8}{3} \text{sech}(x) \sqrt{\cosh (x) \coth (x)}$

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 4397

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

Rule 4398

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 4400

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])

Rule 2598

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*Ta
n[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 2589

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]

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 = 0.0497481, size = 21, normalized size = 0.68 $\frac{2}{3} \left (\cosh ^2(x)-4\right ) \text{sech}(x) \sqrt{\cosh (x) \coth (x)}$

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

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm csch} \left (x\right )+\sinh \left ( x \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

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)

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Maxima [B]  time = 1.84759, size = 147, normalized size = 4.74 \begin{align*} \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{align*}

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))

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Fricas [B]  time = 1.9067, size = 348, normalized size = 11.23 \begin{align*} \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{align*}

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)))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\operatorname{csch}\left (x\right ) + \sinh \left (x\right )\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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)