3.1018 \(\int \text {sech}(x) (5-11 \text {sech}^2(x)) \tanh (x) \, dx\)

Optimal. Leaf size=13 \[ \frac {11 \text {sech}^3(x)}{3}-5 \text {sech}(x) \]

[Out]

-5*sech(x)+11/3*sech(x)^3

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Rubi [A]  time = 0.04, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4339, 14} \[ \frac {11 \text {sech}^3(x)}{3}-5 \text {sech}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]*(5 - 11*Sech[x]^2)*Tanh[x],x]

[Out]

-5*Sech[x] + (11*Sech[x]^3)/3

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 4339

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[(b*
c)^(-1), Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[
c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])

Rubi steps

\begin {align*} \int \text {sech}(x) \left (5-11 \text {sech}^2(x)\right ) \tanh (x) \, dx &=\operatorname {Subst}\left (\int \frac {-11+5 x^2}{x^4} \, dx,x,\cosh (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {11}{x^4}+\frac {5}{x^2}\right ) \, dx,x,\cosh (x)\right )\\ &=-5 \text {sech}(x)+\frac {11 \text {sech}^3(x)}{3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \[ \frac {11 \text {sech}^3(x)}{3}-5 \text {sech}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]*(5 - 11*Sech[x]^2)*Tanh[x],x]

[Out]

-5*Sech[x] + (11*Sech[x]^3)/3

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fricas [B]  time = 0.41, size = 87, normalized size = 6.69 \[ -\frac {2 \, {\left (15 \, \cosh \relax (x)^{3} + 45 \, \cosh \relax (x) \sinh \relax (x)^{2} + 15 \, \sinh \relax (x)^{3} + {\left (45 \, \cosh \relax (x)^{2} - 29\right )} \sinh \relax (x) + \cosh \relax (x)\right )}}{3 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 2\right )} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)*(5-11*sech(x)^2)*tanh(x),x, algorithm="fricas")

[Out]

-2/3*(15*cosh(x)^3 + 45*cosh(x)*sinh(x)^2 + 15*sinh(x)^3 + (45*cosh(x)^2 - 29)*sinh(x) + cosh(x))/(cosh(x)^4 +
4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 2)*sinh(x)^2 + 4*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(
x) + 3)

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giac [B]  time = 0.13, size = 24, normalized size = 1.85 \[ -\frac {2 \, {\left (15 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 44\right )}}{3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)*(5-11*sech(x)^2)*tanh(x),x, algorithm="giac")

[Out]

-2/3*(15*(e^(-x) + e^x)^2 - 44)/(e^(-x) + e^x)^3

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maple [A]  time = 0.09, size = 12, normalized size = 0.92 \[ -5 \,\mathrm {sech}\relax (x )+\frac {11 \mathrm {sech}\relax (x )^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)*(5-11*sech(x)^2)*tanh(x),x)

[Out]

-5*sech(x)+11/3*sech(x)^3

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maxima [B]  time = 0.31, size = 23, normalized size = 1.77 \[ -\frac {10}{e^{\left (-x\right )} + e^{x}} + \frac {88}{3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)*(5-11*sech(x)^2)*tanh(x),x, algorithm="maxima")

[Out]

-10/(e^(-x) + e^x) + 88/3/(e^(-x) + e^x)^3

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mupad [B]  time = 1.86, size = 26, normalized size = 2.00 \[ -\frac {2\,{\mathrm {e}}^x\,\left (15\,{\mathrm {e}}^{4\,x}-14\,{\mathrm {e}}^{2\,x}+15\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(tanh(x)*(11/cosh(x)^2 - 5))/cosh(x),x)

[Out]

-(2*exp(x)*(15*exp(4*x) - 14*exp(2*x) + 15))/(3*(exp(2*x) + 1)^3)

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sympy [A]  time = 0.50, size = 12, normalized size = 0.92 \[ \frac {11 \operatorname {sech}^{3}{\relax (x )}}{3} - 5 \operatorname {sech}{\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)*(5-11*sech(x)**2)*tanh(x),x)

[Out]

11*sech(x)**3/3 - 5*sech(x)

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