3.1.6 \(\int \tanh (a+b x) \, dx\) [6]

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 = 0.00, 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 = {3556} \begin {gather*} \frac {\log (\cosh (a+b x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[a + b*x],x]

[Out]

Log[Cosh[a + b*x]]/b

Rule 3556

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 \begin {gather*} \frac {\log (\cosh (a+b x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[a + b*x],x]

[Out]

Log[Cosh[a + b*x]]/b

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(27\) vs. \(2(11)=22\).
time = 0.25, size = 28, normalized size = 2.55

method	result	size
risch	\(-x -\frac {2 a}{b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}\)	\(27\)
derivativedivides	\(\frac {-\frac {\ln \left (-1+\tanh \left (b x +a \right )\right )}{2}-\frac {\ln \left (\tanh \left (b x +a \right )+1\right )}{2}}{b}\)	\(28\)
default	\(\frac {-\frac {\ln \left (-1+\tanh \left (b x +a \right )\right )}{2}-\frac {\ln \left (\tanh \left (b x +a \right )+1\right )}{2}}{b}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 11, normalized size = 1.00 \begin {gather*} \frac {\log \left (\cosh \left (b x + a\right )\right )}{b} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (11) = 22\).
time = 0.66, size = 37, normalized size = 3.36 \begin {gather*} -\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 {gather*}

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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (8) = 16\).
time = 0.06, size = 17, normalized size = 1.55 \begin {gather*} \begin {cases} x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} & \text {for}\: b \neq 0 \\x \tanh {\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (11) = 22\).
time = 0.40, size = 24, normalized size = 2.18 \begin {gather*} -\frac {b x + a - \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} \end {gather*}

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

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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