Optimal. Leaf size=65 \[ \frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}-\frac {B \log (a+b \cosh (x))}{a}+\frac {B \log (\cosh (x))}{a} \]
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Rubi [A] time = 0.15, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {4401, 2659, 208, 2721, 36, 29, 31} \[ \frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}-\frac {B \log (a+b \cosh (x))}{a}+\frac {B \log (\cosh (x))}{a} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 208
Rule 2659
Rule 2721
Rule 4401
Rubi steps
\begin {align*} \int \frac {A+B \tanh (x)}{a+b \cosh (x)} \, dx &=\int \left (\frac {A}{a+b \cosh (x)}+\frac {B \tanh (x)}{a+b \cosh (x)}\right ) \, dx\\ &=A \int \frac {1}{a+b \cosh (x)} \, dx+B \int \frac {\tanh (x)}{a+b \cosh (x)} \, dx\\ &=(2 A) \operatorname {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )+B \operatorname {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,b \cosh (x)\right )\\ &=\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {B \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,b \cosh (x)\right )}{a}-\frac {B \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cosh (x)\right )}{a}\\ &=\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {B \log (\cosh (x))}{a}-\frac {B \log (a+b \cosh (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 61, normalized size = 0.94 \[ \frac {B (\log (\cosh (x))-\log (a+b \cosh (x)))}{a}-\frac {2 A \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 315, normalized size = 4.85 \[ \left [\frac {\sqrt {a^{2} - b^{2}} A a \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) + b}\right ) - {\left (B a^{2} - B b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + {\left (B a^{2} - B b^{2}\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{3} - a b^{2}}, -\frac {2 \, \sqrt {-a^{2} + b^{2}} A a \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{a^{2} - b^{2}}\right ) + {\left (B a^{2} - B b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) - {\left (B a^{2} - B b^{2}\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{3} - a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 66, normalized size = 1.02 \[ \frac {2 \, A \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}}} - \frac {B \log \left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} + b\right )}{a} + \frac {B \log \left (e^{\left (2 \, x\right )} + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 125, normalized size = 1.92 \[ -\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -a -b \right ) B}{a -b}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -a -b \right ) B b}{a \left (a -b \right )}+\frac {2 A \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {B \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.14, size = 160, normalized size = 2.46 \[ \frac {B\,\ln \left (16\,B^2\,b^2-16\,B^2\,a^2-16\,B^2\,a^2\,{\mathrm {e}}^{2\,x}+16\,B^2\,b^2\,{\mathrm {e}}^{2\,x}\right )}{a}-\frac {B\,\ln \left (16\,B^2\,b+32\,B^2\,a\,{\mathrm {e}}^x+16\,B^2\,b\,{\mathrm {e}}^{2\,x}\right )}{a}-\frac {2\,\mathrm {atan}\left (\frac {A^2\,b^2\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}+A^2\,a\,b\,\sqrt {b^2-a^2}}{A\,b\,\left (a^2-b^2\right )\,\sqrt {A^2}}\right )\,\sqrt {A^2}}{\sqrt {b^2-a^2}} \]
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 \[ \int \frac {A + B \tanh {\relax (x )}}{a + b \cosh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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