3.585 \(\int \frac{1}{a+b \tanh (x)} \, dx\)

Optimal. Leaf size=39 \[ \frac{a x}{a^2-b^2}-\frac{b \log (a \cosh (x)+b \sinh (x))}{a^2-b^2} \]

[Out]

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

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Rubi [A]  time = 0.0790553, antiderivative size = 39, 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 \[ \frac{a x}{a^2-b^2}-\frac{b \log (a \cosh (x)+b \sinh (x))}{a^2-b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Tanh[x])^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 59.3487, size = 42, normalized size = 1.08 \[ - \frac{b \log{\left (a + b \tanh{\left (x \right )} \right )}}{a^{2} - b^{2}} - \frac{\log{\left (- \tanh{\left (x \right )} + 1 \right )}}{2 \left (a + b\right )} + \frac{\log{\left (\tanh{\left (x \right )} + 1 \right )}}{2 \left (a - b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*tanh(x)),x)

[Out]

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

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Mathematica [A]  time = 0.0620722, size = 29, normalized size = 0.74 \[ \frac{a x-b \log (a \cosh (x)+b \sinh (x))}{a^2-b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Tanh[x])^(-1),x]

[Out]

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

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Maple [A]  time = 0.024, size = 55, normalized size = 1.4 \[ -{\frac{b\ln \left ( a+b\tanh \left ( x \right ) \right ) }{ \left ( a+b \right ) \left ( a-b \right ) }}+{\frac{\ln \left ( 1+\tanh \left ( x \right ) \right ) }{2\,a-2\,b}}-{\frac{\ln \left ( -1+\tanh \left ( x \right ) \right ) }{2\,a+2\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*tanh(x)),x)

[Out]

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

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Maxima [A]  time = 1.36427, size = 55, normalized size = 1.41 \[ -\frac{b \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{2} - b^{2}} + \frac{x}{a + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*tanh(x) + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.223995, size = 57, normalized size = 1.46 \[ \frac{{\left (a + b\right )} x - b \log \left (\frac{2 \,{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} - b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*tanh(x) + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.14998, size = 146, normalized size = 3.74 \[ \begin{cases} \tilde{\infty } \left (x - \log{\left (\tanh{\left (x \right )} + 1 \right )} + \log{\left (\tanh{\left (x \right )} \right )}\right ) & \text{for}\: a = 0 \wedge b = 0 \\- \frac{x \tanh{\left (x \right )}}{2 b \tanh{\left (x \right )} - 2 b} + \frac{x}{2 b \tanh{\left (x \right )} - 2 b} + \frac{1}{2 b \tanh{\left (x \right )} - 2 b} & \text{for}\: a = - b \\\frac{x \tanh{\left (x \right )}}{2 b \tanh{\left (x \right )} + 2 b} + \frac{x}{2 b \tanh{\left (x \right )} + 2 b} - \frac{1}{2 b \tanh{\left (x \right )} + 2 b} & \text{for}\: a = b \\\frac{x}{a} & \text{for}\: b = 0 \\\frac{a x}{a^{2} - b^{2}} - \frac{b x}{a^{2} - b^{2}} - \frac{b \log{\left (\frac{a}{b} + \tanh{\left (x \right )} \right )}}{a^{2} - b^{2}} + \frac{b \log{\left (\tanh{\left (x \right )} + 1 \right )}}{a^{2} - b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*tanh(x)),x)

[Out]

Piecewise((zoo*(x - log(tanh(x) + 1) + log(tanh(x))), Eq(a, 0) & Eq(b, 0)), (-x*
tanh(x)/(2*b*tanh(x) - 2*b) + x/(2*b*tanh(x) - 2*b) + 1/(2*b*tanh(x) - 2*b), Eq(
a, -b)), (x*tanh(x)/(2*b*tanh(x) + 2*b) + x/(2*b*tanh(x) + 2*b) - 1/(2*b*tanh(x)
+ 2*b), Eq(a, b)), (x/a, Eq(b, 0)), (a*x/(a**2 - b**2) - b*x/(a**2 - b**2) - b*
log(a/b + tanh(x))/(a**2 - b**2) + b*log(tanh(x) + 1)/(a**2 - b**2), True))

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GIAC/XCAS [A]  time = 0.211538, size = 58, normalized size = 1.49 \[ -\frac{b{\rm ln}\left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{2} - b^{2}} + \frac{x}{a - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*tanh(x) + a),x, algorithm="giac")

[Out]

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