3.7.84 \(\int (-\cosh (x)+\text {sech}(x))^{3/2} \, dx\) [684]

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

[Out]

-8/3*csch(x)*(-sinh(x)*tanh(x))^(1/2)-2/3*sinh(x)*(-sinh(x)*tanh(x))^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {4482, 4485, 2678, 2669} \begin {gather*} -\frac {2}{3} \sinh (x) \sqrt {-\sinh (x) \tanh (x)}-\frac {8}{3} \text {csch}(x) \sqrt {-\sinh (x) \tanh (x)} \end {gather*}

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 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 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 (-\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*}

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Mathematica [A]
time = 0.06, size = 24, normalized size = 0.73 \begin {gather*} \frac {2}{3} \coth (x) \left (1+4 \text {csch}^2(x)\right ) (-\sinh (x) \tanh (x))^{3/2} \end {gather*}

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 = 2.32, size = 0, normalized size = 0.00 \[\int \left (-\cosh \left (x \right )+\mathrm {sech}\left (x \right )\right )^{\frac {3}{2}}\, dx\]

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] Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (25) = 50\).
time = 0.48, size = 77, normalized size = 2.33 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (25) = 50\).
time = 0.37, size = 99, normalized size = 3.00 \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 )} \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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \cosh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-cosh(x) + sech(x))**(3/2), x)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cosh(x) - cosh(x))^(3/2),x)

[Out]

int((1/cosh(x) - cosh(x))^(3/2), x)

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