Optimal. Leaf size=50 \[ \frac {a x}{a^2-b^2}-\frac {b \log (a \sinh (c+d x)+b \cosh (c+d x))}{d \left (a^2-b^2\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3484, 3530} \[ \frac {a x}{a^2-b^2}-\frac {b \log (a \sinh (c+d x)+b \cosh (c+d x))}{d \left (a^2-b^2\right )} \]
Antiderivative was successfully verified.
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Rule 3484
Rule 3530
Rubi steps
\begin {align*} \int \frac {1}{a+b \coth (c+d x)} \, dx &=\frac {a x}{a^2-b^2}-\frac {(i b) \int \frac {-i b-i a \coth (c+d x)}{a+b \coth (c+d x)} \, dx}{a^2-b^2}\\ &=\frac {a x}{a^2-b^2}-\frac {b \log (b \cosh (c+d x)+a \sinh (c+d x))}{\left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 64, normalized size = 1.28 \[ \frac {(b-a) \log (1-\coth (c+d x))+(a+b) \log (\coth (c+d x)+1)-2 b \log (a+b \coth (c+d x))}{2 d (a-b) (a+b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 62, normalized size = 1.24 \[ \frac {{\left (a + b\right )} d x - b \log \left (\frac {2 \, {\left (b \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{{\left (a^{2} - b^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 62, normalized size = 1.24 \[ -\frac {\frac {b \log \left ({\left | a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b \right |}\right )}{a^{2} - b^{2}} - \frac {d x + c}{a - b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 76, normalized size = 1.52 \[ -\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{d \left (2 b +2 a \right )}+\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{d \left (2 a -2 b \right )}-\frac {b \ln \left (a +b \coth \left (d x +c \right )\right )}{d \left (a -b \right ) \left (a +b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 52, normalized size = 1.04 \[ -\frac {b \log \left (-{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a + b\right )}{{\left (a^{2} - b^{2}\right )} d} + \frac {d x + c}{{\left (a + b\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 55, normalized size = 1.10 \[ \frac {x}{a-b}-\frac {b\,\ln \left (b-a+a\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )}{a^2\,d-b^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.56, size = 236, normalized size = 4.72 \[ \begin {cases} \frac {\tilde {\infty } x}{\coth {\relax (c )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\- \frac {d x \tanh {\left (c + d x \right )}}{2 b d \tanh {\left (c + d x \right )} - 2 b d} + \frac {d x}{2 b d \tanh {\left (c + d x \right )} - 2 b d} - \frac {1}{2 b d \tanh {\left (c + d x \right )} - 2 b d} & \text {for}\: a = - b \\\frac {d x \tanh {\left (c + d x \right )}}{2 b d \tanh {\left (c + d x \right )} + 2 b d} + \frac {d x}{2 b d \tanh {\left (c + d x \right )} + 2 b d} + \frac {1}{2 b d \tanh {\left (c + d x \right )} + 2 b d} & \text {for}\: a = b \\\frac {x}{a + b \coth {\relax (c )}} & \text {for}\: d = 0 \\\frac {x - \frac {\log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d}}{b} & \text {for}\: a = 0 \\\frac {a d x}{a^{2} d - b^{2} d} - \frac {b d x}{a^{2} d - b^{2} d} + \frac {b \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{a^{2} d - b^{2} d} - \frac {b \log {\left (\tanh {\left (c + d x \right )} + \frac {b}{a} \right )}}{a^{2} d - b^{2} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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