Optimal. Leaf size=85 \[ -\frac {2 b^3 \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^3 \sqrt {a-b} \sqrt {a+b}}-\frac {b \sinh (x)}{a^2}+\frac {x \left (a^2+2 b^2\right )}{2 a^3}+\frac {\sinh (x) \cosh (x)}{2 a} \]
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Rubi [A] time = 0.26, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3853, 4104, 3919, 3831, 2659, 205} \[ \frac {x \left (a^2+2 b^2\right )}{2 a^3}-\frac {2 b^3 \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^3 \sqrt {a-b} \sqrt {a+b}}-\frac {b \sinh (x)}{a^2}+\frac {\sinh (x) \cosh (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2659
Rule 3831
Rule 3853
Rule 3919
Rule 4104
Rubi steps
\begin {align*} \int \frac {\cosh ^2(x)}{a+b \text {sech}(x)} \, dx &=\frac {\cosh (x) \sinh (x)}{2 a}+\frac {\int \frac {\cosh (x) \left (-2 b+a \text {sech}(x)+b \text {sech}^2(x)\right )}{a+b \text {sech}(x)} \, dx}{2 a}\\ &=-\frac {b \sinh (x)}{a^2}+\frac {\cosh (x) \sinh (x)}{2 a}-\frac {\int \frac {-a^2-2 b^2-a b \text {sech}(x)}{a+b \text {sech}(x)} \, dx}{2 a^2}\\ &=\frac {\left (a^2+2 b^2\right ) x}{2 a^3}-\frac {b \sinh (x)}{a^2}+\frac {\cosh (x) \sinh (x)}{2 a}-\frac {b^3 \int \frac {\text {sech}(x)}{a+b \text {sech}(x)} \, dx}{a^3}\\ &=\frac {\left (a^2+2 b^2\right ) x}{2 a^3}-\frac {b \sinh (x)}{a^2}+\frac {\cosh (x) \sinh (x)}{2 a}-\frac {b^2 \int \frac {1}{1+\frac {a \cosh (x)}{b}} \, dx}{a^3}\\ &=\frac {\left (a^2+2 b^2\right ) x}{2 a^3}-\frac {b \sinh (x)}{a^2}+\frac {\cosh (x) \sinh (x)}{2 a}-\frac {\left (2 b^2\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 )}{a^3}\\ &=\frac {\left (a^2+2 b^2\right ) x}{2 a^3}-\frac {2 b^3 \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^3 \sqrt {a-b} \sqrt {a+b}}-\frac {b \sinh (x)}{a^2}+\frac {\cosh (x) \sinh (x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 78, normalized size = 0.92 \[ \frac {\frac {8 b^3 \tan ^{-1}\left (\frac {(b-a) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+2 a^2 x+a^2 \sinh (2 x)-4 a b \sinh (x)+4 b^2 x}{4 a^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 860, normalized size = 10.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 92, normalized size = 1.08 \[ -\frac {2 \, b^{3} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{3}} + \frac {a e^{\left (2 \, x\right )} - 4 \, b e^{x}}{8 \, a^{2}} + \frac {{\left (a^{2} + 2 \, b^{2}\right )} x}{2 \, a^{3}} + \frac {{\left (4 \, a b e^{x} - a^{2}\right )} e^{\left (-2 \, x\right )}}{8 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 174, normalized size = 2.05 \[ \frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b}{a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) b^{2}}{a^{3}}-\frac {2 b^{3} \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{3} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b}{a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) b^{2}}{a^{3}} \]
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 = 1.58, size = 167, normalized size = 1.96 \[ \frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {b\,{\mathrm {e}}^x}{2\,a^2}+\frac {b\,{\mathrm {e}}^{-x}}{2\,a^2}+\frac {x\,\left (a^2+2\,b^2\right )}{2\,a^3}+\frac {b^3\,\ln \left (\frac {2\,b^3\,{\mathrm {e}}^x}{a^4}-\frac {2\,b^3\,\left (a+b\,{\mathrm {e}}^x\right )}{a^4\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a^3\,\sqrt {a+b}\,\sqrt {b-a}}-\frac {b^3\,\ln \left (\frac {2\,b^3\,{\mathrm {e}}^x}{a^4}+\frac {2\,b^3\,\left (a+b\,{\mathrm {e}}^x\right )}{a^4\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a^3\,\sqrt {a+b}\,\sqrt {b-a}} \]
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 \[ \int \frac {\cosh ^{2}{\relax (x )}}{a + b \operatorname {sech}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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