Optimal. Leaf size=327 \[ \frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {x^2 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x^2 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}} \]
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Rubi [A] time = 0.54, antiderivative size = 327, 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 = {5629, 3320, 2264, 2190, 2531, 2282, 6589} \[ \frac {x \text {PolyLog}\left (2,-\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {\text {PolyLog}\left (3,-\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {\text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {x^2 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x^2 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 3320
Rule 5629
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2}{a+b \sinh ^2(x)} \, dx &=2 \int \frac {x^2}{2 a-b+b \cosh (2 x)} \, dx\\ &=4 \int \frac {e^{2 x} x^2}{b+2 (2 a-b) e^{2 x}+b e^{4 x}} \, dx\\ &=\frac {(2 b) \int \frac {e^{2 x} x^2}{-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^2}{4 \sqrt {a} \sqrt {a-b}+2 (2 a-b)+2 b e^{2 x}} \, dx}{\sqrt {a} \sqrt {a-b}}\\ &=\frac {x^2 \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^2 \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 x \log \left (1+\frac {2 b e^{2 x}}{-4 \sqrt {a} \sqrt {a-b}+2 (2 a-b)}\right ) \, dx}{\sqrt {a} \sqrt {a-b}}+\frac {\int x \log \left (1+\frac {2 b e^{2 x}}{4 \sqrt {a} \sqrt {a-b}+2 (2 a-b)}\right ) \, dx}{\sqrt {a} \sqrt {a-b}}\\ &=\frac {x^2 \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^2 \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 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {\int \text {Li}_2\left (-\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 \text {Li}_2\left (-\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^2 \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^2 \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 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \text {Li}_2\left (-\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 {\text {Li}_2\left (-\frac {b x}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{x} \, dx,x,e^{2 x}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-2 a+2 \sqrt {a} \sqrt {a-b}+b}\right )}{x} \, dx,x,e^{2 x}\right )}{4 \sqrt {a} \sqrt {a-b}}\\ &=\frac {x^2 \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^2 \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 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {\text {Li}_3\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}_3\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 [A] time = 0.75, size = 240, normalized size = 0.73 \[ \frac {-2 x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )+2 x \text {Li}_2\left (\frac {b e^{2 x}}{-2 a+2 \sqrt {a-b} \sqrt {a}+b}\right )+\text {Li}_3\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )-\text {Li}_3\left (\frac {b e^{2 x}}{-2 a+2 \sqrt {a-b} \sqrt {a}+b}\right )-2 x^2 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )+2 x^2 \log \left (1-\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}-2 a+b}\right )}{4 \sqrt {a} \sqrt {a-b}} \]
Antiderivative was successfully verified.
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fricas [C] time = 1.92, size = 1247, normalized size = 3.81 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{2}}{b \sinh \relax (x)^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 710, normalized size = 2.17 \[ -\frac {2 x^{3}}{3 \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {x^{2} \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 )}-2 a +b}+\frac {x \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{-2 \sqrt {a \left (a -b \right )}-2 a +b}-\frac {\polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{2 \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {2 a \,x^{3}}{3 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {a \,x^{2} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{\sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {a x \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{\sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {a \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {b \,x^{3}}{3 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {b \,x^{2} \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 )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {b x \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{2 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}+\frac {b \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a -b \right )}-2 a +b}\right )}{4 \sqrt {a \left (a -b \right )}\, \left (-2 \sqrt {a \left (a -b \right )}-2 a +b \right )}-\frac {x^{3}}{3 \sqrt {a \left (a -b \right )}}+\frac {x^{2} \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 \polylog \left (2, \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 {\polylog \left (3, \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 \[ \int \frac {x^{2}}{b \sinh \relax (x)^{2} + a}\,{d x} \]
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 \[ \int \frac {x^2}{b\,{\mathrm {sinh}\relax (x)}^2+a} \,d x \]
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 \[ \int \frac {x^{2}}{a + b \sinh ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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