3.338 $$\int \tanh (a+b x) \, dx$$

Optimal. Leaf size=11 $\frac{\log (\cosh (a+b x))}{b}$

[Out]

Log[Cosh[a + b*x]]/b

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Rubi [A]  time = 0.0062648, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {3475} $\frac{\log (\cosh (a+b x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Tanh[a + b*x],x]

[Out]

Log[Cosh[a + b*x]]/b

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 (a+b x) \, dx &=\frac{\log (\cosh (a+b x))}{b}\\ \end{align*}

Mathematica [A]  time = 0.0076157, size = 11, normalized size = 1. $\frac{\log (\cosh (a+b x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Tanh[a + b*x],x]

[Out]

Log[Cosh[a + b*x]]/b

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Maple [A]  time = 0.005, size = 13, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ({\rm sech} \left (bx+a\right ) \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b*ln(sech(b*x+a))

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Maxima [A]  time = 1.00375, size = 28, normalized size = 2.55 \begin{align*} \frac{\log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e^(b*x + a) + e^(-b*x - a))/b

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Fricas [B]  time = 1.86847, size = 88, normalized size = 8. \begin{align*} -\frac{b x - \log \left (\frac{2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x - log(2*cosh(b*x + a)/(cosh(b*x + a) - sinh(b*x + a))))/b

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x)*sech(a + b*x), x)

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Giac [B]  time = 1.26725, size = 36, normalized size = 3.27 \begin{align*} -\frac{b x + a}{b} + \frac{\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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