3.4 \(\int \text{sech}^4(a+b x) \, dx\)

Optimal. Leaf size=26 \[ \frac{\tanh (a+b x)}{b}-\frac{\tanh ^3(a+b x)}{3 b} \]

[Out]

Tanh[a + b*x]/b - Tanh[a + b*x]^3/(3*b)

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Rubi [A]  time = 0.011633, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3767} \[ \frac{\tanh (a+b x)}{b}-\frac{\tanh ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*x]^4,x]

[Out]

Tanh[a + b*x]/b - Tanh[a + b*x]^3/(3*b)

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

Mathematica [A]  time = 0.0064148, size = 26, normalized size = 1. \[ \frac{\tanh (a+b x)}{b}-\frac{\tanh ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*x]^4,x]

[Out]

Tanh[a + b*x]/b - Tanh[a + b*x]^3/(3*b)

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Maple [A]  time = 0.012, size = 23, normalized size = 0.9 \begin{align*}{\frac{\tanh \left ( bx+a \right ) }{b} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{2}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x+a)^4,x)

[Out]

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

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Maxima [B]  time = 1.01786, size = 122, normalized size = 4.69 \begin{align*} \frac{4 \, e^{\left (-2 \, b x - 2 \, a\right )}}{b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} + \frac{4}{3 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^4,x, algorithm="maxima")

[Out]

4*e^(-2*b*x - 2*a)/(b*(3*e^(-2*b*x - 2*a) + 3*e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a) + 1)) + 4/3/(b*(3*e^(-2*b*x
- 2*a) + 3*e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a) + 1))

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Fricas [B]  time = 1.96322, size = 448, normalized size = 17.23 \begin{align*} -\frac{8 \,{\left (2 \, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{5} + 5 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + b \sinh \left (b x + a\right )^{5} + 3 \, b \cosh \left (b x + a\right )^{3} +{\left (10 \, b \cosh \left (b x + a\right )^{2} + 3 \, b\right )} \sinh \left (b x + a\right )^{3} +{\left (10 \, b \cosh \left (b x + a\right )^{3} + 9 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 4 \, b \cosh \left (b x + a\right ) +{\left (5 \, b \cosh \left (b x + a\right )^{4} + 9 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^4,x, algorithm="fricas")

[Out]

-8/3*(2*cosh(b*x + a) + sinh(b*x + a))/(b*cosh(b*x + a)^5 + 5*b*cosh(b*x + a)*sinh(b*x + a)^4 + b*sinh(b*x + a
)^5 + 3*b*cosh(b*x + a)^3 + (10*b*cosh(b*x + a)^2 + 3*b)*sinh(b*x + a)^3 + (10*b*cosh(b*x + a)^3 + 9*b*cosh(b*
x + a))*sinh(b*x + a)^2 + 4*b*cosh(b*x + a) + (5*b*cosh(b*x + a)^4 + 9*b*cosh(b*x + a)^2 + 2*b)*sinh(b*x + a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)**4,x)

[Out]

Integral(sech(a + b*x)**4, x)

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Giac [A]  time = 1.15398, size = 42, normalized size = 1.62 \begin{align*} -\frac{4 \,{\left (3 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^4,x, algorithm="giac")

[Out]

-4/3*(3*e^(2*b*x + 2*a) + 1)/(b*(e^(2*b*x + 2*a) + 1)^3)