Optimal. Leaf size=281 \[ \frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {x^2 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{\sqrt {4 a^2+b^2}} \]
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Rubi [A] time = 0.52, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5628, 3322, 2264, 2190, 2531, 2282, 6589} \[ \frac {x \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x \text {PolyLog}\left (2,-\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}\right )}{\sqrt {4 a^2+b^2}}-\frac {\text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {\text {PolyLog}\left (3,-\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {x^2 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{\sqrt {4 a^2+b^2}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 3322
Rule 5628
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2}{a+b \cosh (x) \sinh (x)} \, dx &=\int \frac {x^2}{a+\frac {1}{2} b \sinh (2 x)} \, dx\\ &=2 \int \frac {e^{2 x} x^2}{-\frac {b}{2}+2 a e^{2 x}+\frac {1}{2} b e^{4 x}} \, dx\\ &=\frac {(2 b) \int \frac {e^{2 x} x^2}{2 a-\sqrt {4 a^2+b^2}+b e^{2 x}} \, dx}{\sqrt {4 a^2+b^2}}-\frac {(2 b) \int \frac {e^{2 x} x^2}{2 a+\sqrt {4 a^2+b^2}+b e^{2 x}} \, dx}{\sqrt {4 a^2+b^2}}\\ &=\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {2 \int x \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right ) \, dx}{\sqrt {4 a^2+b^2}}+\frac {2 \int x \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right ) \, dx}{\sqrt {4 a^2+b^2}}\\ &=\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {\int \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right ) \, dx}{\sqrt {4 a^2+b^2}}+\frac {\int \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right ) \, dx}{\sqrt {4 a^2+b^2}}\\ &=\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-2 a+\sqrt {4 a^2+b^2}}\right )}{x} \, dx,x,e^{2 x}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{2 a+\sqrt {4 a^2+b^2}}\right )}{x} \, dx,x,e^{2 x}\right )}{2 \sqrt {4 a^2+b^2}}\\ &=\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 210, normalized size = 0.75 \[ \frac {2 x \text {Li}_2\left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}-2 a}\right )-2 x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )-\text {Li}_3\left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}-2 a}\right )+\text {Li}_3\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )+2 x^2 \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )-2 x^2 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{2 \sqrt {4 a^2+b^2}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.54, size = 1122, normalized size = 3.99 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{2}}{b \cosh \relax (x) \sinh \relax (x) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 530, normalized size = 1.89 \[ -\frac {2 x^{3}}{3 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {x^{2} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{-2 a -\sqrt {4 a^{2}+b^{2}}}+\frac {x \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{-2 a -\sqrt {4 a^{2}+b^{2}}}-\frac {\polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{2 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {4 a \,x^{3}}{3 \sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {2 a \,x^{2} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {2 a x \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {a \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {2 x^{3}}{3 \sqrt {4 a^{2}+b^{2}}}+\frac {x^{2} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{\sqrt {4 a^{2}+b^{2}}}+\frac {x \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{\sqrt {4 a^{2}+b^{2}}}-\frac {\polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{2 \sqrt {4 a^{2}+b^{2}}} \]
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 \[ \int \frac {x^{2}}{b \cosh \relax (x) \sinh \relax (x) + a}\,{d x} \]
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 \[ \int \frac {x^2}{a+b\,\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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