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

Optimal result

Integrand size = 6, antiderivative size = 19 \[ \int \text {sech}^4(7 x) \, dx=\frac {1}{7} \tanh (7 x)-\frac {1}{21} \tanh ^3(7 x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3852} \[ \int \text {sech}^4(7 x) \, dx=\frac {1}{7} \tanh (7 x)-\frac {1}{21} \tanh ^3(7 x) \]

[In]

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

[Out]

Tanh[7*x]/7 - Tanh[7*x]^3/21

Rule 3852

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*} \text {integral}& = \frac {1}{7} i \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \text {sech}^4(7 x) \, dx=\frac {1}{7} \tanh (7 x)-\frac {1}{21} \tanh ^3(7 x) \]

[In]

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

[Out]

Tanh[7*x]/7 - Tanh[7*x]^3/21

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {\left (\frac {2}{3}+\frac {\operatorname {sech}\left (7 x \right )^{2}}{3}\right ) \tanh \left (7 x \right )}{7}\)	\(17\)
default	\(\frac {\left (\frac {2}{3}+\frac {\operatorname {sech}\left (7 x \right )^{2}}{3}\right ) \tanh \left (7 x \right )}{7}\)	\(17\)
risch	\(-\frac {4 \left (3 \,{\mathrm e}^{14 x}+1\right )}{21 \left ({\mathrm e}^{14 x}+1\right )^{3}}\)	\(19\)
parallelrisch	\(\frac {6 \tanh \left (\frac {7 x}{2}\right )^{5}+4 \tanh \left (\frac {7 x}{2}\right )^{3}+6 \tanh \left (\frac {7 x}{2}\right )}{21 \left (1+\tanh \left (\frac {7 x}{2}\right )^{2}\right )^{3}}\)	\(36\)

[In]

int(sech(7*x)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (15) = 30\).

Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 6.11 \[ \int \text {sech}^4(7 x) \, dx=-\frac {8 \, {\left (2 \, \cosh \left (7 \, x\right ) + \sinh \left (7 \, x\right )\right )}}{21 \, {\left (\cosh \left (7 \, x\right )^{5} + 5 \, \cosh \left (7 \, x\right ) \sinh \left (7 \, x\right )^{4} + \sinh \left (7 \, x\right )^{5} + {\left (10 \, \cosh \left (7 \, x\right )^{2} + 3\right )} \sinh \left (7 \, x\right )^{3} + 3 \, \cosh \left (7 \, x\right )^{3} + {\left (10 \, \cosh \left (7 \, x\right )^{3} + 9 \, \cosh \left (7 \, x\right )\right )} \sinh \left (7 \, x\right )^{2} + {\left (5 \, \cosh \left (7 \, x\right )^{4} + 9 \, \cosh \left (7 \, x\right )^{2} + 2\right )} \sinh \left (7 \, x\right ) + 4 \, \cosh \left (7 \, x\right )\right )}} \]

[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))

Sympy [F]

\[ \int \text {sech}^4(7 x) \, dx=\int \operatorname {sech}^{4}{\left (7 x \right )}\, dx \]

[In]

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

[Out]

Integral(sech(7*x)**4, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (15) = 30\).

Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.58 \[ \int \text {sech}^4(7 x) \, dx=\frac {4 \, e^{\left (-14 \, x\right )}}{7 \, {\left (3 \, e^{\left (-14 \, x\right )} + 3 \, e^{\left (-28 \, x\right )} + e^{\left (-42 \, x\right )} + 1\right )}} + \frac {4}{21 \, {\left (3 \, e^{\left (-14 \, x\right )} + 3 \, e^{\left (-28 \, x\right )} + e^{\left (-42 \, x\right )} + 1\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \text {sech}^4(7 x) \, dx=-\frac {4 \, {\left (3 \, e^{\left (14 \, x\right )} + 1\right )}}{21 \, {\left (e^{\left (14 \, x\right )} + 1\right )}^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \text {sech}^4(7 x) \, dx=-\frac {2\,\left (3\,{\mathrm {e}}^{14\,x}-3\,{\mathrm {e}}^{28\,x}-{\mathrm {e}}^{42\,x}+1\right )}{21\,{\left ({\mathrm {e}}^{14\,x}+1\right )}^3} \]

[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)