Optimal. Leaf size=291 \[ \frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {\text {Li}_3\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {x^2 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x^2 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 \sqrt {a} \sqrt {a+b}} \]
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Rubi [A] time = 0.57, antiderivative size = 291, 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 = {5630, 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 5630
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2}{a+b \cosh ^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+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x^2 \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 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+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x^2 \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 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+b+2 \sqrt {a} \sqrt {a+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+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x^2 \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 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x \text {Li}_2\left (-\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 {\text {Li}_2\left (-\frac {b x}{2 a+b-2 \sqrt {a} \sqrt {a+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+b+2 \sqrt {a} \sqrt {a+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+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x^2 \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 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {\text {Li}_3\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}_3\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 [A] time = 0.77, size = 221, normalized size = 0.76 \[ \frac {2 x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )-2 x \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )-\text {Li}_3\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )+\text {Li}_3\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )+2 x^2 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )-2 x^2 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{4 \sqrt {a} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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fricas [C] time = 1.38, size = 1162, 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 \[ \int \frac {x^{2}}{b \cosh \relax (x)^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 686, normalized size = 2.36 \[ -\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 \[ \int \frac {x^{2}}{b \cosh \relax (x)^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{b\,{\mathrm {cosh}\relax (x)}^2+a} \,d x \]
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 \[ \int \frac {x^{2}}{a + b \cosh ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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