Optimal. Leaf size=191 \[ \frac {\text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {\text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {x \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 \sqrt {a} \sqrt {a+b}} \]
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Rubi [A] time = 0.38, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5630, 3320, 2264, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,-\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {\text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {x \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 \sqrt {a} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3320
Rule 5630
Rubi steps
\begin {align*} \int \frac {x}{a+b \cosh ^2(x)} \, dx &=2 \int \frac {x}{2 a+b+b \cosh (2 x)} \, dx\\ &=4 \int \frac {e^{2 x} x}{b+2 (2 a+b) e^{2 x}+b e^{4 x}} \, dx\\ &=\frac {(2 b) \int \frac {e^{2 x} x}{-4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)+2 b e^{2 x}} \, dx}{\sqrt {a} \sqrt {a+b}}-\frac {(2 b) \int \frac {e^{2 x} x}{4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)+2 b e^{2 x}} \, dx}{\sqrt {a} \sqrt {a+b}}\\ &=\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {\int \log \left (1+\frac {2 b e^{2 x}}{-4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{2 \sqrt {a} \sqrt {a+b}}+\frac {\int \log \left (1+\frac {2 b e^{2 x}}{4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{2 \sqrt {a} \sqrt {a+b}}\\ &=\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{-4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right )}{x} \, dx,x,e^{2 x}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right )}{x} \, dx,x,e^{2 x}\right )}{4 \sqrt {a} \sqrt {a+b}}\\ &=\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}+\frac {\text {Li}_2\left (-\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {\text {Li}_2\left (-\frac {b e^{2 x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}\\ \end {align*}
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Mathematica [C] time = 0.63, size = 536, normalized size = 2.81 \[ -\frac {i \left (\text {Li}_2\left (\frac {\left (2 a+b-2 i \sqrt {-a (a+b)}\right ) \left (a+b-i \sqrt {-a (a+b)} \tanh (x)\right )}{b \left (a+b+i \sqrt {-a (a+b)} \tanh (x)\right )}\right )-\text {Li}_2\left (\frac {\left (2 a+b+2 i \sqrt {-a (a+b)}\right ) \left (a+b-i \sqrt {-a (a+b)} \tanh (x)\right )}{b \left (a+b+i \sqrt {-a (a+b)} \tanh (x)\right )}\right )\right )+4 x \tan ^{-1}\left (\frac {(a+b) \coth (x)}{\sqrt {-a (a+b)}}\right )+2 i \cos ^{-1}\left (-\frac {2 a}{b}-1\right ) \tan ^{-1}\left (\frac {a \tanh (x)}{\sqrt {-a (a+b)}}\right )-\log \left (\frac {2 (a+b) \left (a+i \sqrt {-a (a+b)}\right ) (\tanh (x)-1)}{b \left (i \sqrt {-a (a+b)} \tanh (x)+a+b\right )}\right ) \left (\cos ^{-1}\left (-\frac {2 a}{b}-1\right )-2 \tan ^{-1}\left (\frac {a \tanh (x)}{\sqrt {-a (a+b)}}\right )\right )-\log \left (\frac {2 i (a+b) \left (\sqrt {-a (a+b)}+i a\right ) (\tanh (x)+1)}{b \left (i \sqrt {-a (a+b)} \tanh (x)+a+b\right )}\right ) \left (2 \tan ^{-1}\left (\frac {a \tanh (x)}{\sqrt {-a (a+b)}}\right )+\cos ^{-1}\left (-\frac {2 a}{b}-1\right )\right )+\log \left (\frac {\sqrt {2} e^{-x} \sqrt {-a (a+b)}}{\sqrt {b} \sqrt {2 a+b \cosh (2 x)+b}}\right ) \left (-2 \tan ^{-1}\left (\frac {a \tanh (x)}{\sqrt {-a (a+b)}}\right )+2 \tan ^{-1}\left (\frac {(a+b) \coth (x)}{\sqrt {-a (a+b)}}\right )+\cos ^{-1}\left (-\frac {2 a}{b}-1\right )\right )+\log \left (\frac {\sqrt {2} e^x \sqrt {-a (a+b)}}{\sqrt {b} \sqrt {2 a+b \cosh (2 x)+b}}\right ) \left (2 \tan ^{-1}\left (\frac {a \tanh (x)}{\sqrt {-a (a+b)}}\right )-2 \tan ^{-1}\left (\frac {(a+b) \coth (x)}{\sqrt {-a (a+b)}}\right )+\cos ^{-1}\left (-\frac {2 a}{b}-1\right )\right )}{4 \sqrt {-a (a+b)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 780, normalized size = 4.08 \[ -\frac {b x \sqrt {\frac {a^{2} + a b}{b^{2}}} \log \left (\frac {{\left ({\left (2 \, a + b\right )} \cosh \relax (x) + {\left (2 \, a + b\right )} \sinh \relax (x) - 2 \, {\left (b \cosh \relax (x) + b \sinh \relax (x)\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + b}{b}\right ) + b x \sqrt {\frac {a^{2} + a b}{b^{2}}} \log \left (-\frac {{\left ({\left (2 \, a + b\right )} \cosh \relax (x) + {\left (2 \, a + b\right )} \sinh \relax (x) - 2 \, {\left (b \cosh \relax (x) + b \sinh \relax (x)\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - b}{b}\right ) - b x \sqrt {\frac {a^{2} + a b}{b^{2}}} \log \left (\frac {{\left ({\left (2 \, a + b\right )} \cosh \relax (x) + {\left (2 \, a + b\right )} \sinh \relax (x) + 2 \, {\left (b \cosh \relax (x) + b \sinh \relax (x)\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + b}{b}\right ) - b x \sqrt {\frac {a^{2} + a b}{b^{2}}} \log \left (-\frac {{\left ({\left (2 \, a + b\right )} \cosh \relax (x) + {\left (2 \, a + b\right )} \sinh \relax (x) + 2 \, {\left (b \cosh \relax (x) + b \sinh \relax (x)\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - b}{b}\right ) + b \sqrt {\frac {a^{2} + a b}{b^{2}}} {\rm Li}_2\left (-\frac {{\left ({\left (2 \, a + b\right )} \cosh \relax (x) + {\left (2 \, a + b\right )} \sinh \relax (x) - 2 \, {\left (b \cosh \relax (x) + b \sinh \relax (x)\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + b}{b} + 1\right ) + b \sqrt {\frac {a^{2} + a b}{b^{2}}} {\rm Li}_2\left (\frac {{\left ({\left (2 \, a + b\right )} \cosh \relax (x) + {\left (2 \, a + b\right )} \sinh \relax (x) - 2 \, {\left (b \cosh \relax (x) + b \sinh \relax (x)\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - b}{b} + 1\right ) - b \sqrt {\frac {a^{2} + a b}{b^{2}}} {\rm Li}_2\left (-\frac {{\left ({\left (2 \, a + b\right )} \cosh \relax (x) + {\left (2 \, a + b\right )} \sinh \relax (x) + 2 \, {\left (b \cosh \relax (x) + b \sinh \relax (x)\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + b}{b} + 1\right ) - b \sqrt {\frac {a^{2} + a b}{b^{2}}} {\rm Li}_2\left (\frac {{\left ({\left (2 \, a + b\right )} \cosh \relax (x) + {\left (2 \, a + b\right )} \sinh \relax (x) + 2 \, {\left (b \cosh \relax (x) + b \sinh \relax (x)\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - b}{b} + 1\right )}{2 \, {\left (a^{2} + a b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{b \cosh \relax (x)^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 487, normalized size = 2.55 \[ \frac {x \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{2 \sqrt {a \left (a +b \right )}}-\frac {x^{2}}{2 \sqrt {a \left (a +b \right )}}+\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{4 \sqrt {a \left (a +b \right )}}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) x}{-2 \sqrt {a \left (a +b \right )}-2 a -b}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) a x}{\sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) b x}{2 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {x^{2}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}-\frac {a \,x^{2}}{\sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {b \,x^{2}}{2 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{-4 \sqrt {a \left (a +b \right )}-4 a -2 b}+\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) a}{2 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) b}{4 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{b \cosh \relax (x)^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{b\,{\mathrm {cosh}\relax (x)}^2+a} \,d x \]
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 \[ \int \frac {x}{a + b \cosh ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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