3.87 \(\int \text {sech}(a+b x) \tanh ^3(a+b x) \, dx\)

Optimal. Leaf size=27 \[ \frac {\text {sech}^3(a+b x)}{3 b}-\frac {\text {sech}(a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2606} \[ \frac {\text {sech}^3(a+b x)}{3 b}-\frac {\text {sech}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*x]*Tanh[a + b*x]^3,x]

[Out]

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

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 27, normalized size = 1.00 \[ \frac {\text {sech}^3(a+b x)}{3 b}-\frac {\text {sech}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*x]*Tanh[a + b*x]^3,x]

[Out]

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

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fricas [B]  time = 0.44, size = 172, normalized size = 6.37 \[ -\frac {2 \, {\left (3 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 3 \, \sinh \left (b x + a\right )^{3} + {\left (9 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) + 5 \, \cosh \left (b x + a\right )\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3 \, b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)*tanh(b*x+a)^3,x, algorithm="fricas")

[Out]

-2/3*(3*cosh(b*x + a)^3 + 9*cosh(b*x + a)*sinh(b*x + a)^2 + 3*sinh(b*x + a)^3 + (9*cosh(b*x + a)^2 - 1)*sinh(b
*x + a) + 5*cosh(b*x + a))/(b*cosh(b*x + a)^4 + 4*b*cosh(b*x + a)*sinh(b*x + a)^3 + b*sinh(b*x + a)^4 + 4*b*co
sh(b*x + a)^2 + 2*(3*b*cosh(b*x + a)^2 + 2*b)*sinh(b*x + a)^2 + 4*(b*cosh(b*x + a)^3 + b*cosh(b*x + a))*sinh(b
*x + a) + 3*b)

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giac [A]  time = 0.16, size = 49, normalized size = 1.81 \[ -\frac {2 \, {\left (3 \, e^{\left (5 \, b x + 5 \, a\right )} + 2 \, e^{\left (3 \, b x + 3 \, a\right )} + 3 \, e^{\left (b x + a\right )}\right )}}{3 \, b {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)*tanh(b*x+a)^3,x, algorithm="giac")

[Out]

-2/3*(3*e^(5*b*x + 5*a) + 2*e^(3*b*x + 3*a) + 3*e^(b*x + a))/(b*(e^(2*b*x + 2*a) + 1)^3)

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maple [A]  time = 0.16, size = 34, normalized size = 1.26 \[ \frac {-\frac {\sinh ^{2}\left (b x +a \right )}{\cosh \left (b x +a \right )^{3}}-\frac {2}{3 \cosh \left (b x +a \right )^{3}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x+a)*tanh(b*x+a)^3,x)

[Out]

1/b*(-sinh(b*x+a)^2/cosh(b*x+a)^3-2/3/cosh(b*x+a)^3)

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maxima [B]  time = 0.45, size = 148, normalized size = 5.48 \[ -\frac {2 \, e^{\left (-b x - a\right )}}{b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} - \frac {4 \, e^{\left (-3 \, b x - 3 \, a\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} - \frac {2 \, e^{\left (-5 \, b x - 5 \, a\right )}}{b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)*tanh(b*x+a)^3,x, algorithm="maxima")

[Out]

-2*e^(-b*x - 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*e^(-3*b*x - 3*a)/(b
*(3*e^(-2*b*x - 2*a) + 3*e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a) + 1)) - 2*e^(-5*b*x - 5*a)/(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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mupad [B]  time = 1.41, size = 48, normalized size = 1.78 \[ -\frac {2\,{\mathrm {e}}^{a+b\,x}\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}+3\right )}{3\,b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + b*x)^3/cosh(a + b*x),x)

[Out]

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

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sympy [A]  time = 0.85, size = 41, normalized size = 1.52 \[ \begin {cases} - \frac {\tanh ^{2}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}}{3 b} - \frac {2 \operatorname {sech}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \tanh ^{3}{\relax (a )} \operatorname {sech}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)*tanh(b*x+a)**3,x)

[Out]

Piecewise((-tanh(a + b*x)**2*sech(a + b*x)/(3*b) - 2*sech(a + b*x)/(3*b), Ne(b, 0)), (x*tanh(a)**3*sech(a), Tr
ue))

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