Optimal. Leaf size=391 \[ \frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 x \text {Li}_3\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 x \text {Li}_3\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 \sqrt {a} \sqrt {a+b}}+\frac {x^3 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x^3 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 \sqrt {a} \sqrt {a+b}} \]
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Rubi [A] time = 0.60, antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5630, 3320, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 x \text {PolyLog}\left (3,-\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 x \text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \text {PolyLog}\left (4,-\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \text {PolyLog}\left (4,-\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{8 \sqrt {a} \sqrt {a+b}}+\frac {x^3 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x^3 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 \sqrt {a} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 3320
Rule 5630
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^3}{a+b \cosh ^2(x)} \, dx &=2 \int \frac {x^3}{2 a+b+b \cosh (2 x)} \, dx\\ &=4 \int \frac {e^{2 x} x^3}{b+2 (2 a+b) e^{2 x}+b e^{4 x}} \, dx\\ &=\frac {(2 b) \int \frac {e^{2 x} x^3}{-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^3}{4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)+2 b e^{2 x}} \, dx}{\sqrt {a} \sqrt {a+b}}\\ &=\frac {x^3 \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^3 \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 {3 \int x^2 \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 {3 \int x^2 \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^3 \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^3 \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 {3 x^2 \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 {3 x^2 \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 {3 \int x \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 {3 \int x \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^3 \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^3 \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 {3 x^2 \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 {3 x^2 \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 {3 x \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 {3 x \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 {3 \int \text {Li}_3\left (-\frac {2 b e^{2 x}}{-4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 \int \text {Li}_3\left (-\frac {2 b e^{2 x}}{4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{4 \sqrt {a} \sqrt {a+b}}\\ &=\frac {x^3 \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^3 \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 {3 x^2 \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 {3 x^2 \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 {3 x \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 {3 x \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 {3 \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{x} \, dx,x,e^{2 x}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{x} \, dx,x,e^{2 x}\right )}{8 \sqrt {a} \sqrt {a+b}}\\ &=\frac {x^3 \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^3 \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 {3 x^2 \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 {3 x^2 \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 {3 x \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 {3 x \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 {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 295, normalized size = 0.75 \[ \frac {6 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )-6 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )-6 x \text {Li}_3\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )+6 x \text {Li}_3\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )+3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )-3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )+4 x^3 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )-4 x^3 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{8 \sqrt {a} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.57, size = 1542, normalized size = 3.94 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{3}}{b \cosh \relax (x)^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 889, normalized size = 2.27 \[ \frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) x^{3}}{-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^{3}}{\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^{3}}{2 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {x^{4}}{2 \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {a \,x^{4}}{2 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {b \,x^{4}}{4 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) x^{2}}{2 \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) a \,x^{2}}{2 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) b \,x^{2}}{4 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {3 \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) x}{2 \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {3 \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) a x}{2 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {3 \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) b x}{4 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{4 \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) a}{4 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) b}{8 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {x^{3} \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^{4}}{4 \sqrt {a \left (a +b \right )}}+\frac {3 x^{2} \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 {3 x \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 )}}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{8 \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 \[ \int \frac {x^{3}}{b \cosh \relax (x)^{2} + a}\,{d x} \]
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 \[ \int \frac {x^3}{b\,{\mathrm {cosh}\relax (x)}^2+a} \,d x \]
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 \[ \int \frac {x^{3}}{a + b \cosh ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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