Optimal. Leaf size=54 \[ -\frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {a x}{b^2}+\frac {\cosh (x)}{b} \]
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Rubi [A] time = 0.11, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2695, 2735, 2660, 618, 206} \[ -\frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {a x}{b^2}+\frac {\cosh (x)}{b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 2695
Rule 2735
Rubi steps
\begin {align*} \int \frac {\cosh ^2(x)}{a+b \sinh (x)} \, dx &=\frac {\cosh (x)}{b}+\frac {i \int \frac {-i b+i a \sinh (x)}{a+b \sinh (x)} \, dx}{b}\\ &=-\frac {a x}{b^2}+\frac {\cosh (x)}{b}+\frac {\left (a^2+b^2\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{b^2}\\ &=-\frac {a x}{b^2}+\frac {\cosh (x)}{b}+\frac {\left (2 \left (a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^2}\\ &=-\frac {a x}{b^2}+\frac {\cosh (x)}{b}-\frac {\left (4 \left (a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{b^2}\\ &=-\frac {a x}{b^2}-\frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2}+\frac {\cosh (x)}{b}\\ \end {align*}
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Mathematica [C] time = 0.77, size = 396, normalized size = 7.33 \[ \frac {\cosh (x) \left (\sqrt {a+i b} \sqrt {-\frac {b (\sinh (x)-i)}{a+i b}} \left (\sqrt {a-i b} \sqrt {1+i \sinh (x)} \sqrt {-\frac {b (\sinh (x)+i)}{a-i b}}-2 (-1)^{3/4} \sqrt {b} \sin ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a-i b} \sqrt {-\frac {b (\sinh (x)+i)}{a-i b}}}{\sqrt {b}}\right )\right )-2 \sqrt {a-i b} \sqrt {a+i b} \sqrt {1+i \sinh (x)} \tanh ^{-1}\left (\frac {\sqrt {-\frac {b (\sinh (x)+i)}{a-i b}}}{\sqrt {-\frac {b (\sinh (x)-i)}{a+i b}}}\right )+2 (a-i b) \sqrt {1+i \sinh (x)} \tanh ^{-1}\left (\frac {\sqrt {a-i b} \sqrt {-\frac {b (\sinh (x)+i)}{a-i b}}}{\sqrt {a+i b} \sqrt {-\frac {b (\sinh (x)-i)}{a+i b}}}\right )\right )}{b \sqrt {a-i b} \sqrt {a+i b} \sqrt {1+i \sinh (x)} \sqrt {-\frac {b (\sinh (x)-i)}{a+i b}} \sqrt {-\frac {b (\sinh (x)+i)}{a-i b}}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 171, normalized size = 3.17 \[ -\frac {2 \, a x \cosh \relax (x) - b \cosh \relax (x)^{2} - b \sinh \relax (x)^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) - b}\right ) + 2 \, {\left (a x - b \cosh \relax (x)\right )} \sinh \relax (x) - b}{2 \, {\left (b^{2} \cosh \relax (x) + b^{2} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 83, normalized size = 1.54 \[ -\frac {a x}{b^{2}} + \frac {e^{\left (-x\right )}}{2 \, b} + \frac {e^{x}}{2 \, b} + \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 126, normalized size = 2.33 \[ \frac {2 a^{2} \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{2} \sqrt {a^{2}+b^{2}}}+\frac {2 \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{2}}+\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 81, normalized size = 1.50 \[ -\frac {a x}{b^{2}} + \frac {e^{\left (-x\right )}}{2 \, b} + \frac {e^{x}}{2 \, b} + \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 87, normalized size = 1.61 \[ \frac {{\mathrm {e}}^{-x}}{2\,b}+\frac {{\mathrm {e}}^x}{2\,b}-\frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {-b^4}}{b^2\,\sqrt {a^2+b^2}}+\frac {{\mathrm {e}}^x\,\sqrt {-b^4}}{b\,\sqrt {a^2+b^2}}\right )\,\sqrt {a^2+b^2}}{\sqrt {-b^4}}-\frac {a\,x}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 138.91, size = 377, normalized size = 6.98 \[ \begin {cases} \tilde {\infty } \left (\frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {2}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {2}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1}}{b} & \text {for}\: a = 0 \\\frac {- \frac {x \sinh ^{2}{\relax (x )}}{2} + \frac {x \cosh ^{2}{\relax (x )}}{2} + \frac {\sinh {\relax (x )} \cosh {\relax (x )}}{2}}{a} & \text {for}\: b = 0 \\- \frac {a x \tanh ^{2}{\left (\frac {x}{2} \right )}}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} + \frac {a x}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} - \frac {2 b}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} - \frac {\sqrt {a^{2} + b^{2}} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} + \frac {\sqrt {a^{2} + b^{2}} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} + \frac {\sqrt {a^{2} + b^{2}} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} - \frac {\sqrt {a^{2} + b^{2}} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} - b^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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