3.7.85 \(\int (-\cosh (x)+\text {sech}(x))^{5/2} \, dx\) [685]

Optimal. Leaf size=53 \[ -\frac {64}{15} \coth (x) \sqrt {-\sinh (x) \tanh (x)}+\frac {16}{15} \tanh (x) \sqrt {-\sinh (x) \tanh (x)}+\frac {2}{5} \sinh ^2(x) \tanh (x) \sqrt {-\sinh (x) \tanh (x)} \]

[Out]

-64/15*coth(x)*(-sinh(x)*tanh(x))^(1/2)+16/15*(-sinh(x)*tanh(x))^(1/2)*tanh(x)+2/5*sinh(x)^2*(-sinh(x)*tanh(x)
)^(1/2)*tanh(x)

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Rubi [A]
time = 0.10, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4482, 4485, 2678, 2674, 2669} \begin {gather*} \frac {2}{5} \sinh ^2(x) \tanh (x) \sqrt {-\sinh (x) \tanh (x)}+\frac {16}{15} \tanh (x) \sqrt {-\sinh (x) \tanh (x)}-\frac {64}{15} \coth (x) \sqrt {-\sinh (x) \tanh (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-Cosh[x] + Sech[x])^(5/2),x]

[Out]

(-64*Coth[x]*Sqrt[-(Sinh[x]*Tanh[x])])/15 + (16*Tanh[x]*Sqrt[-(Sinh[x]*Tanh[x])])/15 + (2*Sinh[x]^2*Tanh[x]*Sq
rt[-(Sinh[x]*Tanh[x])])/5

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 2674

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*(n - 1))), x] - Dist[b^2*((m + n - 1)/(n - 1)), Int[(a*Sin[e + f*x])^m*(
b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n] &&  !(GtQ[m,
1] &&  !IntegerQ[(m - 1)/2])

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))^{5/2} \, dx &=\int (-\sinh (x) \tanh (x))^{5/2} \, dx\\ &=\frac {\sqrt {-\sinh (x) \tanh (x)} \int (i \sinh (x))^{5/2} (i \tanh (x))^{5/2} \, dx}{\sqrt {i \sinh (x)} \sqrt {i \tanh (x)}}\\ &=\frac {2}{5} \sinh ^2(x) \tanh (x) \sqrt {-\sinh (x) \tanh (x)}+\frac {\left (8 \sqrt {-\sinh (x) \tanh (x)}\right ) \int \sqrt {i \sinh (x)} (i \tanh (x))^{5/2} \, dx}{5 \sqrt {i \sinh (x)} \sqrt {i \tanh (x)}}\\ &=\frac {16}{15} \tanh (x) \sqrt {-\sinh (x) \tanh (x)}+\frac {2}{5} \sinh ^2(x) \tanh (x) \sqrt {-\sinh (x) \tanh (x)}-\frac {\left (32 \sqrt {-\sinh (x) \tanh (x)}\right ) \int \sqrt {i \sinh (x)} \sqrt {i \tanh (x)} \, dx}{15 \sqrt {i \sinh (x)} \sqrt {i \tanh (x)}}\\ &=-\frac {64}{15} \coth (x) \sqrt {-\sinh (x) \tanh (x)}+\frac {16}{15} \tanh (x) \sqrt {-\sinh (x) \tanh (x)}+\frac {2}{5} \sinh ^2(x) \tanh (x) \sqrt {-\sinh (x) \tanh (x)}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 30, normalized size = 0.57 \begin {gather*} \frac {2}{15} \left (-5-3 \cosh ^2(x)+32 \coth ^2(x)\right ) \text {csch}(x) (-\sinh (x) \tanh (x))^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-Cosh[x] + Sech[x])^(5/2),x]

[Out]

(2*(-5 - 3*Cosh[x]^2 + 32*Coth[x]^2)*Csch[x]*(-(Sinh[x]*Tanh[x]))^(3/2))/15

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Maple [F]
time = 2.28, size = 0, normalized size = 0.00 \[\int \left (-\cosh \left (x \right )+\mathrm {sech}\left (x \right )\right )^{\frac {5}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-cosh(x)+sech(x))^(5/2),x)

[Out]

int((-cosh(x)+sech(x))^(5/2),x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (41) = 82\).
time = 0.47, size = 115, normalized size = 2.17 \begin {gather*} -\frac {\sqrt {2} e^{\left (\frac {5}{2} \, x\right )}}{20 \, {\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac {5}{2}}} + \frac {7 \, \sqrt {2} e^{\left (\frac {1}{2} \, x\right )}}{4 \, {\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac {5}{2}}} + \frac {41 \, \sqrt {2} e^{\left (-\frac {3}{2} \, x\right )}}{6 \, {\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac {5}{2}}} + \frac {41 \, \sqrt {2} e^{\left (-\frac {7}{2} \, x\right )}}{6 \, {\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac {5}{2}}} + \frac {7 \, \sqrt {2} e^{\left (-\frac {11}{2} \, x\right )}}{4 \, {\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac {5}{2}}} - \frac {\sqrt {2} e^{\left (-\frac {15}{2} \, x\right )}}{20 \, {\left (-e^{\left (-2 \, x\right )} - 1\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))^(5/2),x, algorithm="maxima")

[Out]

-1/20*sqrt(2)*e^(5/2*x)/(-e^(-2*x) - 1)^(5/2) + 7/4*sqrt(2)*e^(1/2*x)/(-e^(-2*x) - 1)^(5/2) + 41/6*sqrt(2)*e^(
-3/2*x)/(-e^(-2*x) - 1)^(5/2) + 41/6*sqrt(2)*e^(-7/2*x)/(-e^(-2*x) - 1)^(5/2) + 7/4*sqrt(2)*e^(-11/2*x)/(-e^(-
2*x) - 1)^(5/2) - 1/20*sqrt(2)*e^(-15/2*x)/(-e^(-2*x) - 1)^(5/2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (41) = 82\).
time = 0.40, size = 257, normalized size = 4.85 \begin {gather*} \frac {\sqrt {\frac {1}{2}} {\left (3 \, \cosh \left (x\right )^{8} + 24 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + 3 \, \sinh \left (x\right )^{8} + 12 \, {\left (7 \, \cosh \left (x\right )^{2} - 9\right )} \sinh \left (x\right )^{6} - 108 \, \cosh \left (x\right )^{6} + 24 \, {\left (7 \, \cosh \left (x\right )^{3} - 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (105 \, \cosh \left (x\right )^{4} - 810 \, \cosh \left (x\right )^{2} - 151\right )} \sinh \left (x\right )^{4} - 302 \, \cosh \left (x\right )^{4} + 8 \, {\left (21 \, \cosh \left (x\right )^{5} - 270 \, \cosh \left (x\right )^{3} - 151 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 12 \, {\left (7 \, \cosh \left (x\right )^{6} - 135 \, \cosh \left (x\right )^{4} - 151 \, \cosh \left (x\right )^{2} - 9\right )} \sinh \left (x\right )^{2} - 108 \, \cosh \left (x\right )^{2} + 8 \, {\left (3 \, \cosh \left (x\right )^{7} - 81 \, \cosh \left (x\right )^{5} - 151 \, \cosh \left (x\right )^{3} - 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3\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 )}}}{30 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))^(5/2),x, algorithm="fricas")

[Out]

1/30*sqrt(1/2)*(3*cosh(x)^8 + 24*cosh(x)*sinh(x)^7 + 3*sinh(x)^8 + 12*(7*cosh(x)^2 - 9)*sinh(x)^6 - 108*cosh(x
)^6 + 24*(7*cosh(x)^3 - 27*cosh(x))*sinh(x)^5 + 2*(105*cosh(x)^4 - 810*cosh(x)^2 - 151)*sinh(x)^4 - 302*cosh(x
)^4 + 8*(21*cosh(x)^5 - 270*cosh(x)^3 - 151*cosh(x))*sinh(x)^3 + 12*(7*cosh(x)^6 - 135*cosh(x)^4 - 151*cosh(x)
^2 - 9)*sinh(x)^2 - 108*cosh(x)^2 + 8*(3*cosh(x)^7 - 81*cosh(x)^5 - 151*cosh(x)^3 - 27*cosh(x))*sinh(x) + 3)*s
qrt(-1/(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 + 1)*sinh(x) + cosh(x)))/(cosh(x)^4 + 4*cos
h(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 + 1)*sinh(x)^2 + cosh(x)^2 + 2*(2*cosh(x)^3 + cosh(x))*sinh(x))

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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((-cosh(x)+sech(x))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8010 deep

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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))^(5/2),x, algorithm="giac")

[Out]

integrate((-cosh(x) + sech(x))^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (\frac {1}{\mathrm {cosh}\left (x\right )}-\mathrm {cosh}\left (x\right )\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cosh(x) - cosh(x))^(5/2),x)

[Out]

int((1/cosh(x) - cosh(x))^(5/2), x)

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