### 3.394 $$\int \tanh ^3(a+b x) \, dx$$

Optimal. Leaf size=27 $\frac{\log (\cosh (a+b x))}{b}-\frac{\tanh ^2(a+b x)}{2 b}$

[Out]

Log[Cosh[a + b*x]]/b - Tanh[a + b*x]^2/(2*b)

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Rubi [A]  time = 0.0180643, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {3473, 3475} $\frac{\log (\cosh (a+b x))}{b}-\frac{\tanh ^2(a+b x)}{2 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Tanh[a + b*x]^3,x]

[Out]

Log[Cosh[a + b*x]]/b - Tanh[a + b*x]^2/(2*b)

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \tanh ^3(a+b x) \, dx &=-\frac{\tanh ^2(a+b x)}{2 b}+\int \tanh (a+b x) \, dx\\ &=\frac{\log (\cosh (a+b x))}{b}-\frac{\tanh ^2(a+b x)}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0120026, size = 27, normalized size = 1. $\frac{\log (\cosh (a+b x))}{b}-\frac{\tanh ^2(a+b x)}{2 b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Tanh[a + b*x]^3,x]

[Out]

Log[Cosh[a + b*x]]/b - Tanh[a + b*x]^2/(2*b)

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Maple [A]  time = 0.017, size = 26, normalized size = 1. \begin{align*}{\frac{\ln \left ( \cosh \left ( bx+a \right ) \right ) }{b}}-{\frac{ \left ( \tanh \left ( bx+a \right ) \right ) ^{2}}{2\,b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(cosh(b*x+a))/b-1/2*tanh(b*x+a)^2/b

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Maxima [B]  time = 1.72886, size = 82, normalized size = 3.04 \begin{align*} x + \frac{a}{b} + \frac{\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} + \frac{2 \, e^{\left (-2 \, b x - 2 \, a\right )}}{b{\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.16612, size = 930, normalized size = 34.44 \begin{align*} -\frac{b x \cosh \left (b x + a\right )^{4} + 4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b x \sinh \left (b x + a\right )^{4} + 2 \,{\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b x \cosh \left (b x + a\right )^{2} + b x - 1\right )} \sinh \left (b x + a\right )^{2} + b x -{\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \,{\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac{2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \,{\left (b x \cosh \left (b x + a\right )^{3} +{\left (b x - 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{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} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b \cosh \left (b x + a\right )^{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 ) + b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 + 2*(b*x - 1)*cosh(b*x + a)^
2 + 2*(3*b*x*cosh(b*x + a)^2 + b*x - 1)*sinh(b*x + a)^2 + b*x - (cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x +
a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 +
cosh(b*x + a))*sinh(b*x + a) + 1)*log(2*cosh(b*x + a)/(cosh(b*x + a) - sinh(b*x + a))) + 4*(b*x*cosh(b*x + a)^
3 + (b*x - 1)*cosh(b*x + a))*sinh(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 + 2*b*cosh(b*x + a)^2 + 2*(3*b*cosh(b*x + a)^2 + b)*sinh(b*x + a)^2 + 4*(b*cosh(b*x + a)^3 + b*cosh(b
*x + a))*sinh(b*x + a) + b)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.13556, size = 93, normalized size = 3.44 \begin{align*} -\frac{b x + a}{b} + \frac{\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} - \frac{3 \, e^{\left (4 \, b x + 4 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 3}{2 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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