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]

1/7*tanh(7*x)-1/21*tanh(7*x)^3

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \[ \frac {1}{7} \tanh (7 x)-\frac {1}{21} \tanh ^3(7 x) \]

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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fricas [B]  time = 0.59, size = 116, normalized size = 6.11 \[ -\frac {8 \, {\left (2 \, \cosh \left (7 \, x\right ) + \sinh \left (7 \, x\right )\right )}}{21 \, {\left (\cosh \left (7 \, x\right )^{5} + 5 \, \cosh \left (7 \, x\right ) \sinh \left (7 \, x\right )^{4} + \sinh \left (7 \, x\right )^{5} + {\left (10 \, \cosh \left (7 \, x\right )^{2} + 3\right )} \sinh \left (7 \, x\right )^{3} + 3 \, \cosh \left (7 \, x\right )^{3} + {\left (10 \, \cosh \left (7 \, x\right )^{3} + 9 \, \cosh \left (7 \, x\right )\right )} \sinh \left (7 \, x\right )^{2} + {\left (5 \, \cosh \left (7 \, x\right )^{4} + 9 \, \cosh \left (7 \, x\right )^{2} + 2\right )} \sinh \left (7 \, x\right ) + 4 \, \cosh \left (7 \, x\right )\right )}} \]

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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giac [A]  time = 0.11, size = 18, normalized size = 0.95 \[ -\frac {4 \, {\left (3 \, e^{\left (14 \, x\right )} + 1\right )}}{21 \, {\left (e^{\left (14 \, x\right )} + 1\right )}^{3}} \]

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

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maple [A]  time = 0.23, size = 17, normalized size = 0.89 \[ \frac {\left (\frac {2}{3}+\frac {\mathrm {sech}\left (7 x \right )^{2}}{3}\right ) \tanh \left (7 x \right )}{7} \]

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 = 0.31, size = 49, normalized size = 2.58 \[ \frac {4 \, e^{\left (-14 \, x\right )}}{7 \, {\left (3 \, e^{\left (-14 \, x\right )} + 3 \, e^{\left (-28 \, x\right )} + e^{\left (-42 \, x\right )} + 1\right )}} + \frac {4}{21 \, {\left (3 \, e^{\left (-14 \, x\right )} + 3 \, e^{\left (-28 \, x\right )} + e^{\left (-42 \, x\right )} + 1\right )}} \]

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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mupad [B]  time = 0.10, size = 30, normalized size = 1.58 \[ -\frac {2\,\left (3\,{\mathrm {e}}^{14\,x}-3\,{\mathrm {e}}^{28\,x}-{\mathrm {e}}^{42\,x}+1\right )}{21\,{\left ({\mathrm {e}}^{14\,x}+1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cosh(7*x)^4,x)

[Out]

-(2*(3*exp(14*x) - 3*exp(28*x) - exp(42*x) + 1))/(21*(exp(14*x) + 1)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {sech}^{4}{\left (7 x \right )}\, dx \]

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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