3.7 \(\int \coth (a+b x) \, dx\)

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

[Out]

ln(sinh(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} \[ \frac {\log (\sinh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*x],x]

[Out]

Log[Sinh[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 \coth (a+b x) \, dx &=\frac {\log (\sinh (a+b x))}{b}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.73 \[ \frac {\log (\tanh (a+b x))+\log (\cosh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + b*x],x]

[Out]

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

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fricas [B]  time = 0.47, size = 37, normalized size = 3.36 \[ -\frac {b x - \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.12, size = 25, normalized size = 2.27 \[ -\frac {b x + a - \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 11, normalized size = 1.00 \[ \frac {\log \left (\sinh \left (b x + a\right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sinh(b*x + a))/b

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mupad [B]  time = 0.04, size = 21, normalized size = 1.91 \[ \frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + b*x),x)

[Out]

log(exp(2*a)*exp(2*b*x) - 1)/b - x

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sympy [A]  time = 0.36, size = 27, normalized size = 2.45 \[ \begin {cases} x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} + \frac {\log {\left (\tanh {\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\x \coth {\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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