Optimal. Leaf size=215 \[ \frac {\text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {\text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {x \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}} \]
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Rubi [A] time = 0.33, antiderivative size = 215, 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 = {5629, 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 5629
Rubi steps
\begin {align*} \int \frac {x}{a+b \sinh ^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-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-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-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-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-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}+\frac {\text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {\text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}\\ \end {align*}
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Mathematica [C] time = 0.82, size = 576, normalized size = 2.68 \[ -\frac {-2 i \cos ^{-1}\left (1-\frac {2 a}{b}\right ) \tan ^{-1}\left (\frac {\sqrt {a b-a^2} \tanh (x)}{a}\right )-\log \left (\frac {2 a \left (\sqrt {a (b-a)}-i a+i b\right ) (\tanh (x)-1)}{b \sqrt {a (b-a)} \tanh (x)-i a b}\right ) \left (2 \tan ^{-1}\left (\frac {\sqrt {a b-a^2} \tanh (x)}{a}\right )+\cos ^{-1}\left (1-\frac {2 a}{b}\right )\right )-\log \left (\frac {2 a \left (\sqrt {a (b-a)}+i a-i b\right ) (\tanh (x)+1)}{b \sqrt {a (b-a)} \tanh (x)-i a b}\right ) \left (\cos ^{-1}\left (1-\frac {2 a}{b}\right )-2 \tan ^{-1}\left (\frac {\sqrt {a b-a^2} \tanh (x)}{a}\right )\right )+\log \left (\frac {\sqrt {2} e^{-x} \sqrt {a (b-a)}}{\sqrt {b} \sqrt {2 a+b \cosh (2 x)-b}}\right ) \left (2 \left (\tan ^{-1}\left (\frac {\sqrt {a b-a^2} \tanh (x)}{a}\right )+\tan ^{-1}\left (\frac {a \coth (x)}{\sqrt {-a (a-b)}}\right )\right )+\cos ^{-1}\left (1-\frac {2 a}{b}\right )\right )+\log \left (\frac {\sqrt {2} e^x \sqrt {a (b-a)}}{\sqrt {b} \sqrt {2 a+b \cosh (2 x)-b}}\right ) \left (\cos ^{-1}\left (1-\frac {2 a}{b}\right )-2 \left (\tan ^{-1}\left (\frac {\sqrt {a b-a^2} \tanh (x)}{a}\right )+\tan ^{-1}\left (\frac {a \coth (x)}{\sqrt {-a (a-b)}}\right )\right )\right )+i \left (\text {Li}_2\left (\frac {\left (-2 a+b+2 i \sqrt {a (b-a)}\right ) \left (i a+\sqrt {a (b-a)} \tanh (x)\right )}{b \sqrt {a (b-a)} \tanh (x)-i a b}\right )-\text {Li}_2\left (\frac {\left (-2 a+b-2 i \sqrt {a (b-a)}\right ) \left (i a+\sqrt {a (b-a)} \tanh (x)\right )}{b \sqrt {a (b-a)} \tanh (x)-i a b}\right )\right )+4 x \tan ^{-1}\left (\frac {a \coth (x)}{\sqrt {-a (a-b)}}\right )}{4 \sqrt {a (b-a)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.28, size = 837, normalized size = 3.89 \[ -\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 \[ \int \frac {x}{b \sinh \relax (x)^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 505, normalized size = 2.35 \[ \frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) x}{-2 \sqrt {a \left (a -b \right )}-2 a +b}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) a x}{\sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) b x}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {x^{2}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}-\frac {a \,x^{2}}{\sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {b \,x^{2}}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{-4 \sqrt {a \left (a -b \right )}-4 a +2 b}+\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) a}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right ) b}{4 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {x \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{2 \sqrt {a \left (a -b \right )}}-\frac {x^{2}}{2 \sqrt {a \left (a -b \right )}}+\frac {\polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{4 \sqrt {a \left (a -b \right )}} \]
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 \[ \int \frac {x}{b \sinh \relax (x)^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{b\,{\mathrm {sinh}\relax (x)}^2+a} \,d x \]
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 \[ \int \frac {x}{a + b \sinh ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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