### 3.684 $$\int (-\cosh (x)+\text{sech}(x))^{3/2} \, dx$$

Optimal. Leaf size=33 $-\frac{2}{3} \sinh (x) \sqrt{-\sinh (x) \tanh (x)}-\frac{8}{3} \text{csch}(x) \sqrt{-\sinh (x) \tanh (x)}$

[Out]

(-8*Csch[x]*Sqrt[-(Sinh[x]*Tanh[x])])/3 - (2*Sinh[x]*Sqrt[-(Sinh[x]*Tanh[x])])/3

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Rubi [A]  time = 0.0967737, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.364, Rules used = {4397, 4400, 2598, 2589} $-\frac{2}{3} \sinh (x) \sqrt{-\sinh (x) \tanh (x)}-\frac{8}{3} \text{csch}(x) \sqrt{-\sinh (x) \tanh (x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(-Cosh[x] + Sech[x])^(3/2),x]

[Out]

(-8*Csch[x]*Sqrt[-(Sinh[x]*Tanh[x])])/3 - (2*Sinh[x]*Sqrt[-(Sinh[x]*Tanh[x])])/3

Rule 4397

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

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

Mathematica [A]  time = 0.0831163, size = 24, normalized size = 0.73 $\frac{2}{3} \coth (x) \left (4 \text{csch}^2(x)+1\right ) (-\sinh (x) \tanh (x))^{3/2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-Cosh[x] + Sech[x])^(3/2),x]

[Out]

(2*Coth[x]*(1 + 4*Csch[x]^2)*(-(Sinh[x]*Tanh[x]))^(3/2))/3

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Maple [F]  time = 0.129, size = 0, normalized size = 0. \begin{align*} \int \left ( -\cosh \left ( x \right ) +{\rm sech} \left (x\right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-cosh(x)+sech(x))^(3/2),x)

[Out]

int((-cosh(x)+sech(x))^(3/2),x)

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Maxima [B]  time = 1.95826, size = 104, normalized size = 3.15 \begin{align*} -\frac{\sqrt{2} e^{\left (\frac{3}{2} \, x\right )}}{6 \,{\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}} - \frac{5 \, \sqrt{2} e^{\left (-\frac{1}{2} \, x\right )}}{2 \,{\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}} - \frac{5 \, \sqrt{2} e^{\left (-\frac{5}{2} \, x\right )}}{2 \,{\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}} - \frac{\sqrt{2} e^{\left (-\frac{9}{2} \, x\right )}}{6 \,{\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.84747, size = 354, normalized size = 10.73 \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 )} \sqrt{-\frac{1}{\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 )}}}{3 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(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(-1/(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 + 1)
*sinh(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((-cosh(x)+sech(x))**(3/2),x)

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))^(3/2),x, algorithm="giac")

[Out]

integrate((-cosh(x) + sech(x))^(3/2), x)