Optimal. Leaf size=62 \[ \frac {2 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a b}+\frac {x}{a}-\frac {\tan ^{-1}(\sinh (x))}{b} \]
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Rubi [A] time = 0.17, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3894, 4051, 3770, 3919, 3831, 2659, 205} \[ \frac {2 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a b}+\frac {x}{a}-\frac {\tan ^{-1}(\sinh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2659
Rule 3770
Rule 3831
Rule 3894
Rule 3919
Rule 4051
Rubi steps
\begin {align*} \int \frac {\tanh ^2(x)}{a+b \text {sech}(x)} \, dx &=-\int \frac {-1+\text {sech}^2(x)}{a+b \text {sech}(x)} \, dx\\ &=-\frac {\int \text {sech}(x) \, dx}{b}-\frac {\int \frac {-b-a \text {sech}(x)}{a+b \text {sech}(x)} \, dx}{b}\\ &=\frac {x}{a}-\frac {\tan ^{-1}(\sinh (x))}{b}+\left (\frac {a}{b}-\frac {b}{a}\right ) \int \frac {\text {sech}(x)}{a+b \text {sech}(x)} \, dx\\ &=\frac {x}{a}-\frac {\tan ^{-1}(\sinh (x))}{b}+\frac {\left (\frac {a}{b}-\frac {b}{a}\right ) \int \frac {1}{1+\frac {a \cosh (x)}{b}} \, dx}{b}\\ &=\frac {x}{a}-\frac {\tan ^{-1}(\sinh (x))}{b}+\frac {\left (2 \left (\frac {a}{b}-\frac {b}{a}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a}{b}-\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b}\\ &=\frac {x}{a}-\frac {\tan ^{-1}(\sinh (x))}{b}+\frac {2 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a b}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 62, normalized size = 1.00 \[ \frac {-2 \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {(b-a) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )-2 a \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )+b x}{a b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 193, normalized size = 3.11 \[ \left [\frac {b x - 2 \, a \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + \sqrt {-a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \relax (x)^{2} + a^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) - a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x) + b\right )}}{a \cosh \relax (x)^{2} + a \sinh \relax (x)^{2} + 2 \, b \cosh \relax (x) + 2 \, {\left (a \cosh \relax (x) + b\right )} \sinh \relax (x) + a}\right )}{a b}, \frac {b x - 2 \, a \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) - 2 \, \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cosh \relax (x) + a \sinh \relax (x) + b}{\sqrt {a^{2} - b^{2}}}\right )}{a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 52, normalized size = 0.84 \[ \frac {x}{a} - \frac {2 \, \arctan \left (e^{x}\right )}{b} + \frac {2 \, \sqrt {a^{2} - b^{2}} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 113, normalized size = 1.82 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {2 a \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {2 b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{b} \]
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 = 3.92, size = 273, normalized size = 4.40 \[ \frac {\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b}+\frac {\ln \left (2\,a\,b^3-2\,a^3\,b+a^3\,\sqrt {b^2-a^2}+a^4\,{\mathrm {e}}^x+4\,b^4\,{\mathrm {e}}^x-2\,a\,b^2\,\sqrt {b^2-a^2}-4\,b^3\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}-5\,a^2\,b^2\,{\mathrm {e}}^x+3\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}\right )\,\sqrt {b^2-a^2}-\ln \left (2\,a\,b^3-2\,a^3\,b-a^3\,\sqrt {b^2-a^2}+a^4\,{\mathrm {e}}^x+4\,b^4\,{\mathrm {e}}^x+2\,a\,b^2\,\sqrt {b^2-a^2}+4\,b^3\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}-5\,a^2\,b^2\,{\mathrm {e}}^x-3\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}\right )\,\sqrt {b^2-a^2}+b\,x}{a\,b} \]
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 \[ \int \frac {\tanh ^{2}{\relax (x )}}{a + b \operatorname {sech}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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