3.8 \(\int \text{sech}^6(\pi x) \, dx\)

Optimal. Leaf size=35 \[ \frac{\tanh ^5(\pi x)}{5 \pi }-\frac{2 \tanh ^3(\pi x)}{3 \pi }+\frac{\tanh (\pi x)}{\pi } \]

[Out]

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

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Rubi [A]  time = 0.013558, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3767} \[ \frac{\tanh ^5(\pi x)}{5 \pi }-\frac{2 \tanh ^3(\pi x)}{3 \pi }+\frac{\tanh (\pi x)}{\pi } \]

Antiderivative was successfully verified.

[In]

Int[Sech[Pi*x]^6,x]

[Out]

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \text{sech}^6(\pi x) \, dx &=\frac{i \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \tanh (\pi x)\right )}{\pi }\\ &=\frac{\tanh (\pi x)}{\pi }-\frac{2 \tanh ^3(\pi x)}{3 \pi }+\frac{\tanh ^5(\pi x)}{5 \pi }\\ \end{align*}

Mathematica [A]  time = 0.0040945, size = 35, normalized size = 1. \[ \frac{\tanh ^5(\pi x)}{5 \pi }-\frac{2 \tanh ^3(\pi x)}{3 \pi }+\frac{\tanh (\pi x)}{\pi } \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[Pi*x]^6,x]

[Out]

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

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Maple [A]  time = 0.009, size = 27, normalized size = 0.8 \begin{align*}{\frac{\tanh \left ( \pi \,x \right ) }{\pi } \left ({\frac{8}{15}}+{\frac{ \left ({\rm sech} \left (\pi \,x\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm sech} \left (\pi \,x\right ) \right ) ^{2}}{15}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(Pi*x)^6,x)

[Out]

1/Pi*(8/15+1/5*sech(Pi*x)^4+4/15*sech(Pi*x)^2)*tanh(Pi*x)

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Maxima [B]  time = 1.02779, size = 185, normalized size = 5.29 \begin{align*} \frac{16 \, e^{\left (-2 \, \pi x\right )}}{3 \, \pi{\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi x\right )} + 1\right )}} + \frac{32 \, e^{\left (-4 \, \pi x\right )}}{3 \, \pi{\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi x\right )} + 1\right )}} + \frac{16}{15 \, \pi{\left (5 \, e^{\left (-2 \, \pi x\right )} + 10 \, e^{\left (-4 \, \pi x\right )} + 10 \, e^{\left (-6 \, \pi x\right )} + 5 \, e^{\left (-8 \, \pi x\right )} + e^{\left (-10 \, \pi x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(pi*x)^6,x, algorithm="maxima")

[Out]

16/3*e^(-2*pi*x)/(pi*(5*e^(-2*pi*x) + 10*e^(-4*pi*x) + 10*e^(-6*pi*x) + 5*e^(-8*pi*x) + e^(-10*pi*x) + 1)) + 3
2/3*e^(-4*pi*x)/(pi*(5*e^(-2*pi*x) + 10*e^(-4*pi*x) + 10*e^(-6*pi*x) + 5*e^(-8*pi*x) + e^(-10*pi*x) + 1)) + 16
/15/(pi*(5*e^(-2*pi*x) + 10*e^(-4*pi*x) + 10*e^(-6*pi*x) + 5*e^(-8*pi*x) + e^(-10*pi*x) + 1))

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Fricas [B]  time = 2.12901, size = 859, normalized size = 24.54 \begin{align*} -\frac{16 \,{\left (11 \, \cosh \left (\pi x\right )^{2} + 18 \, \cosh \left (\pi x\right ) \sinh \left (\pi x\right ) + 11 \, \sinh \left (\pi x\right )^{2} + 5\right )}}{15 \,{\left (5 \, \pi + \pi \cosh \left (\pi x\right )^{8} + 8 \, \pi \cosh \left (\pi x\right ) \sinh \left (\pi x\right )^{7} + \pi \sinh \left (\pi x\right )^{8} + 5 \, \pi \cosh \left (\pi x\right )^{6} +{\left (5 \, \pi + 28 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{6} + 2 \,{\left (28 \, \pi \cosh \left (\pi x\right )^{3} + 15 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )^{5} + 10 \, \pi \cosh \left (\pi x\right )^{4} + 5 \,{\left (2 \, \pi + 14 \, \pi \cosh \left (\pi x\right )^{4} + 15 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{4} + 4 \,{\left (14 \, \pi \cosh \left (\pi x\right )^{5} + 25 \, \pi \cosh \left (\pi x\right )^{3} + 10 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )^{3} + 11 \, \pi \cosh \left (\pi x\right )^{2} +{\left (11 \, \pi + 28 \, \pi \cosh \left (\pi x\right )^{6} + 75 \, \pi \cosh \left (\pi x\right )^{4} + 60 \, \pi \cosh \left (\pi x\right )^{2}\right )} \sinh \left (\pi x\right )^{2} + 2 \,{\left (4 \, \pi \cosh \left (\pi x\right )^{7} + 15 \, \pi \cosh \left (\pi x\right )^{5} + 20 \, \pi \cosh \left (\pi x\right )^{3} + 9 \, \pi \cosh \left (\pi x\right )\right )} \sinh \left (\pi x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(pi*x)^6,x, algorithm="fricas")

[Out]

-16/15*(11*cosh(pi*x)^2 + 18*cosh(pi*x)*sinh(pi*x) + 11*sinh(pi*x)^2 + 5)/(5*pi + pi*cosh(pi*x)^8 + 8*pi*cosh(
pi*x)*sinh(pi*x)^7 + pi*sinh(pi*x)^8 + 5*pi*cosh(pi*x)^6 + (5*pi + 28*pi*cosh(pi*x)^2)*sinh(pi*x)^6 + 2*(28*pi
*cosh(pi*x)^3 + 15*pi*cosh(pi*x))*sinh(pi*x)^5 + 10*pi*cosh(pi*x)^4 + 5*(2*pi + 14*pi*cosh(pi*x)^4 + 15*pi*cos
h(pi*x)^2)*sinh(pi*x)^4 + 4*(14*pi*cosh(pi*x)^5 + 25*pi*cosh(pi*x)^3 + 10*pi*cosh(pi*x))*sinh(pi*x)^3 + 11*pi*
cosh(pi*x)^2 + (11*pi + 28*pi*cosh(pi*x)^6 + 75*pi*cosh(pi*x)^4 + 60*pi*cosh(pi*x)^2)*sinh(pi*x)^2 + 2*(4*pi*c
osh(pi*x)^7 + 15*pi*cosh(pi*x)^5 + 20*pi*cosh(pi*x)^3 + 9*pi*cosh(pi*x))*sinh(pi*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}^{6}{\left (\pi x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(pi*x)**6,x)

[Out]

Integral(sech(pi*x)**6, x)

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Giac [A]  time = 1.14291, size = 41, normalized size = 1.17 \begin{align*} -\frac{16 \,{\left (10 \, e^{\left (4 \, \pi x\right )} + 5 \, e^{\left (2 \, \pi x\right )} + 1\right )}}{15 \, \pi{\left (e^{\left (2 \, \pi x\right )} + 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(pi*x)^6,x, algorithm="giac")

[Out]

-16/15*(10*e^(4*pi*x) + 5*e^(2*pi*x) + 1)/(pi*(e^(2*pi*x) + 1)^5)