3.96 \(\int \text{sech}(x) \tanh ^5(x) \, dx\)

Optimal. Leaf size=21 \[ -\frac{1}{5} \text{sech}^5(x)+\frac{2 \text{sech}^3(x)}{3}-\text{sech}(x) \]

[Out]

-Sech[x] + (2*Sech[x]^3)/3 - Sech[x]^5/5

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Rubi [A]  time = 0.0186715, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2606, 194} \[ -\frac{1}{5} \text{sech}^5(x)+\frac{2 \text{sech}^3(x)}{3}-\text{sech}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]*Tanh[x]^5,x]

[Out]

-Sech[x] + (2*Sech[x]^3)/3 - Sech[x]^5/5

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \text{sech}(x) \tanh ^5(x) \, dx &=-\operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\text{sech}(x)\right )\\ &=-\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\text{sech}(x)\right )\\ &=-\text{sech}(x)+\frac{2 \text{sech}^3(x)}{3}-\frac{\text{sech}^5(x)}{5}\\ \end{align*}

Mathematica [A]  time = 0.012073, size = 21, normalized size = 1. \[ -\frac{1}{5} \text{sech}^5(x)+\frac{2 \text{sech}^3(x)}{3}-\text{sech}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]*Tanh[x]^5,x]

[Out]

-Sech[x] + (2*Sech[x]^3)/3 - Sech[x]^5/5

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Maple [B]  time = 0.007, size = 46, normalized size = 2.2 \begin{align*} -{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{4}}{ \left ( \cosh \left ( x \right ) \right ) ^{5}}}-{\frac{4\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{5\, \left ( \cosh \left ( x \right ) \right ) ^{5}}}+{\frac{8\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{15\, \left ( \cosh \left ( x \right ) \right ) ^{3}}}+{\frac{8\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{15\,\cosh \left ( x \right ) }}-{\frac{8\,\cosh \left ( x \right ) }{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)*tanh(x)^5,x)

[Out]

-sinh(x)^4/cosh(x)^5-4/5*sinh(x)^2/cosh(x)^5+8/15*sinh(x)^2/cosh(x)^3+8/15*sinh(x)^2/cosh(x)-8/15*cosh(x)

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Maxima [B]  time = 1.28342, size = 258, normalized size = 12.29 \begin{align*} -\frac{2 \, e^{\left (-x\right )}}{5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1} - \frac{8 \, e^{\left (-3 \, x\right )}}{3 \,{\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} - \frac{116 \, e^{\left (-5 \, x\right )}}{15 \,{\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} - \frac{8 \, e^{\left (-7 \, x\right )}}{3 \,{\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} - \frac{2 \, e^{\left (-9 \, x\right )}}{5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*e^(-x)/(5*e^(-2*x) + 10*e^(-4*x) + 10*e^(-6*x) + 5*e^(-8*x) + e^(-10*x) + 1) - 8/3*e^(-3*x)/(5*e^(-2*x) + 1
0*e^(-4*x) + 10*e^(-6*x) + 5*e^(-8*x) + e^(-10*x) + 1) - 116/15*e^(-5*x)/(5*e^(-2*x) + 10*e^(-4*x) + 10*e^(-6*
x) + 5*e^(-8*x) + e^(-10*x) + 1) - 8/3*e^(-7*x)/(5*e^(-2*x) + 10*e^(-4*x) + 10*e^(-6*x) + 5*e^(-8*x) + e^(-10*
x) + 1) - 2*e^(-9*x)/(5*e^(-2*x) + 10*e^(-4*x) + 10*e^(-6*x) + 5*e^(-8*x) + e^(-10*x) + 1)

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Fricas [B]  time = 1.83942, size = 620, normalized size = 29.52 \begin{align*} -\frac{2 \,{\left (15 \, \cosh \left (x\right )^{5} + 75 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + 15 \, \sinh \left (x\right )^{5} + 5 \,{\left (30 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{3} + 35 \, \cosh \left (x\right )^{3} + 15 \,{\left (10 \, \cosh \left (x\right )^{3} + 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (75 \, \cosh \left (x\right )^{4} + 15 \, \cosh \left (x\right )^{2} + 38\right )} \sinh \left (x\right ) + 78 \, \cosh \left (x\right )\right )}}{15 \,{\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \,{\left (5 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} + 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right )^{2} + 15 \, \cosh \left (x\right )^{2} + 2 \,{\left (3 \, \cosh \left (x\right )^{5} + 8 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 10\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(15*cosh(x)^5 + 75*cosh(x)*sinh(x)^4 + 15*sinh(x)^5 + 5*(30*cosh(x)^2 + 1)*sinh(x)^3 + 35*cosh(x)^3 + 15
*(10*cosh(x)^3 + 7*cosh(x))*sinh(x)^2 + (75*cosh(x)^4 + 15*cosh(x)^2 + 38)*sinh(x) + 78*cosh(x))/(cosh(x)^6 +
6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 2)*sinh(x)^4 + 6*cosh(x)^4 + 4*(5*cosh(x)^3 + 4*cosh(x))*si
nh(x)^3 + 3*(5*cosh(x)^4 + 12*cosh(x)^2 + 5)*sinh(x)^2 + 15*cosh(x)^2 + 2*(3*cosh(x)^5 + 8*cosh(x)^3 + 5*cosh(
x))*sinh(x) + 10)

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Sympy [A]  time = 2.50233, size = 29, normalized size = 1.38 \begin{align*} - \frac{\tanh ^{4}{\left (x \right )} \operatorname{sech}{\left (x \right )}}{5} - \frac{4 \tanh ^{2}{\left (x \right )} \operatorname{sech}{\left (x \right )}}{15} - \frac{8 \operatorname{sech}{\left (x \right )}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)*tanh(x)**5,x)

[Out]

-tanh(x)**4*sech(x)/5 - 4*tanh(x)**2*sech(x)/15 - 8*sech(x)/15

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Giac [B]  time = 1.21732, size = 47, normalized size = 2.24 \begin{align*} -\frac{2 \,{\left (15 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 40 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 48\right )}}{15 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(15*(e^(-x) + e^x)^4 - 40*(e^(-x) + e^x)^2 + 48)/(e^(-x) + e^x)^5