3.6 \(\int \tanh (a+b x) \, dx\)

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

[Out]

ln(cosh(b*x+a))/b

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 1.00 \[ \frac {\log (\cosh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[a + b*x],x]

[Out]

Log[Cosh[a + b*x]]/b

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fricas [B]  time = 0.56, size = 37, normalized size = 3.36 \[ -\frac {b x - \log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(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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giac [B]  time = 0.12, size = 24, normalized size = 2.18 \[ -\frac {b x + a - \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 30, normalized size = 2.73 \[ -\frac {\ln \left (-1+\tanh \left (b x +a \right )\right )}{2 b}-\frac {\ln \left (1+\tanh \left (b x +a \right )\right )}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(b*x+a),x)

[Out]

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

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maxima [A]  time = 0.30, size = 11, normalized size = 1.00 \[ \frac {\log \left (\cosh \left (b x + a\right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(cosh(b*x + a))/b

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mupad [B]  time = 1.02, size = 16, normalized size = 1.45 \[ x-\frac {\ln \left (\mathrm {tanh}\left (a+b\,x\right )+1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + b*x),x)

[Out]

x - log(tanh(a + b*x) + 1)/b

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sympy [A]  time = 0.13, size = 17, normalized size = 1.55 \[ \begin {cases} x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} & \text {for}\: b \neq 0 \\x \tanh {\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x - log(tanh(a + b*x) + 1)/b, Ne(b, 0)), (x*tanh(a), True))

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