Optimal. Leaf size=20 \[ \frac {\log (\cosh (x))}{a}-\frac {\log (a+b \cosh (x))}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2721, 36, 29, 31} \[ \frac {\log (\cosh (x))}{a}-\frac {\log (a+b \cosh (x))}{a} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2721
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{a+b \cosh (x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,b \cosh (x)\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,b \cosh (x)\right )}{a}-\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cosh (x)\right )}{a}\\ &=\frac {\log (\cosh (x))}{a}-\frac {\log (a+b \cosh (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \[ \frac {\log (\cosh (x))}{a}-\frac {\log (a+b \cosh (x))}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.43, size = 40, normalized size = 2.00 \[ -\frac {\log \left (\frac {2 \, {\left (b \cosh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) - \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 33, normalized size = 1.65 \[ \frac {\log \left (e^{\left (-x\right )} + e^{x}\right )}{a} - \frac {\log \left ({\left | b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 21, normalized size = 1.05 \[ \frac {\ln \left (\cosh \relax (x )\right )}{a}-\frac {\ln \left (a +b \cosh \relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 33, normalized size = 1.65 \[ -\frac {\log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{a} + \frac {\log \left (e^{\left (-2 \, x\right )} + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 201, normalized size = 10.05 \[ \frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {-a^2}+b\,{\mathrm {e}}^x\,\sqrt {-a^2}+2\,a\,{\mathrm {e}}^{2\,x}\,\sqrt {-a^2}+b\,{\mathrm {e}}^{3\,x}\,\sqrt {-a^2}}{a^2}\right )}{\sqrt {-a^2}}-\frac {2\,\mathrm {atan}\left (\left (4\,a^4\,b\,\sqrt {-a^2}-4\,a^2\,b^3\,\sqrt {-a^2}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {1}{16\,b^2\,{\left (a^2-b^2\right )}^2}-\frac {{\left (a^2-2\,b^2\right )}^2}{16\,a^4\,b^2\,{\left (a^2-b^2\right )}^2}\right )+\frac {1}{8\,a\,b\,{\left (a^2-b^2\right )}^2}+\frac {a^2-2\,b^2}{8\,a^3\,b\,{\left (a^2-b^2\right )}^2}\right )\right )}{\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh {\relax (x )}}{a + b \cosh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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