3.7 \(\int \text{sech}^4(7 x) \, dx\)

Optimal. Leaf size=19 \[ \frac{1}{7} \tanh (7 x)-\frac{1}{21} \tanh ^3(7 x) \]

[Out]

Tanh[7*x]/7 - Tanh[7*x]^3/21

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Rubi [A]  time = 0.0096991, antiderivative size = 19, 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} \[ \frac{1}{7} \tanh (7 x)-\frac{1}{21} \tanh ^3(7 x) \]

Antiderivative was successfully verified.

[In]

Int[Sech[7*x]^4,x]

[Out]

Tanh[7*x]/7 - Tanh[7*x]^3/21

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}^4(7 x) \, dx &=\frac{1}{7} i \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (7 x)\right )\\ &=\frac{1}{7} \tanh (7 x)-\frac{1}{21} \tanh ^3(7 x)\\ \end{align*}

Mathematica [A]  time = 0.0036122, size = 19, normalized size = 1. \[ \frac{1}{7} \tanh (7 x)-\frac{1}{21} \tanh ^3(7 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[7*x]^4,x]

[Out]

Tanh[7*x]/7 - Tanh[7*x]^3/21

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Maple [A]  time = 0.007, size = 17, normalized size = 0.9 \begin{align*}{\frac{\tanh \left ( 7\,x \right ) }{7} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (7\,x\right ) \right ) ^{2}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(7*x)^4,x)

[Out]

1/7*(2/3+1/3*sech(7*x)^2)*tanh(7*x)

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Maxima [B]  time = 1.04548, size = 66, normalized size = 3.47 \begin{align*} \frac{4 \, e^{\left (-14 \, x\right )}}{7 \,{\left (3 \, e^{\left (-14 \, x\right )} + 3 \, e^{\left (-28 \, x\right )} + e^{\left (-42 \, x\right )} + 1\right )}} + \frac{4}{21 \,{\left (3 \, e^{\left (-14 \, x\right )} + 3 \, e^{\left (-28 \, x\right )} + e^{\left (-42 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(7*x)^4,x, algorithm="maxima")

[Out]

4/7*e^(-14*x)/(3*e^(-14*x) + 3*e^(-28*x) + e^(-42*x) + 1) + 4/21/(3*e^(-14*x) + 3*e^(-28*x) + e^(-42*x) + 1)

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Fricas [B]  time = 2.10968, size = 331, normalized size = 17.42 \begin{align*} -\frac{8 \,{\left (2 \, \cosh \left (7 \, x\right ) + \sinh \left (7 \, x\right )\right )}}{21 \,{\left (\cosh \left (7 \, x\right )^{5} + 5 \, \cosh \left (7 \, x\right ) \sinh \left (7 \, x\right )^{4} + \sinh \left (7 \, x\right )^{5} +{\left (10 \, \cosh \left (7 \, x\right )^{2} + 3\right )} \sinh \left (7 \, x\right )^{3} + 3 \, \cosh \left (7 \, x\right )^{3} +{\left (10 \, \cosh \left (7 \, x\right )^{3} + 9 \, \cosh \left (7 \, x\right )\right )} \sinh \left (7 \, x\right )^{2} +{\left (5 \, \cosh \left (7 \, x\right )^{4} + 9 \, \cosh \left (7 \, x\right )^{2} + 2\right )} \sinh \left (7 \, x\right ) + 4 \, \cosh \left (7 \, x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(7*x)^4,x, algorithm="fricas")

[Out]

-8/21*(2*cosh(7*x) + sinh(7*x))/(cosh(7*x)^5 + 5*cosh(7*x)*sinh(7*x)^4 + sinh(7*x)^5 + (10*cosh(7*x)^2 + 3)*si
nh(7*x)^3 + 3*cosh(7*x)^3 + (10*cosh(7*x)^3 + 9*cosh(7*x))*sinh(7*x)^2 + (5*cosh(7*x)^4 + 9*cosh(7*x)^2 + 2)*s
inh(7*x) + 4*cosh(7*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}^{4}{\left (7 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(7*x)**4,x)

[Out]

Integral(sech(7*x)**4, x)

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Giac [A]  time = 1.13372, size = 24, normalized size = 1.26 \begin{align*} -\frac{4 \,{\left (3 \, e^{\left (14 \, x\right )} + 1\right )}}{21 \,{\left (e^{\left (14 \, x\right )} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(7*x)^4,x, algorithm="giac")

[Out]

-4/21*(3*e^(14*x) + 1)/(e^(14*x) + 1)^3