Optimal. Leaf size=386 \[ \frac {x^3 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x^3 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x \text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {3 x \text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}+\frac {3 \text {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}}-\frac {3 \text {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.38, antiderivative size = 386, 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 = {5747, 3403,
2296, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x \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 {3 x \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 {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}}-\frac {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}}+\frac {x^3 \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^3 \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{\sqrt {4 a^2+b^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3403
Rule 5747
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {x^3}{a+b \cosh (x) \sinh (x)} \, dx &=\int \frac {x^3}{a+\frac {1}{2} b \sinh (2 x)} \, dx\\ &=2 \int \frac {e^{2 x} x^3}{-\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^3}{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^3}{2 a+\sqrt {4 a^2+b^2}+b e^{2 x}} \, dx}{\sqrt {4 a^2+b^2}}\\ &=\frac {x^3 \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^3 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {3 \int x^2 \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 {3 \int x^2 \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^3 \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^3 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 \int x \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 {3 \int x \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^3 \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^3 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x \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 {3 x \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 {3 \int \text {Li}_3\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right ) \, dx}{2 \sqrt {4 a^2+b^2}}-\frac {3 \int \text {Li}_3\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right ) \, dx}{2 \sqrt {4 a^2+b^2}}\\ &=\frac {x^3 \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^3 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x \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 {3 x \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 {3 \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-2 a+\sqrt {4 a^2+b^2}}\right )}{x} \, dx,x,e^{2 x}\right )}{4 \sqrt {4 a^2+b^2}}-\frac {3 \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{2 a+\sqrt {4 a^2+b^2}}\right )}{x} \, dx,x,e^{2 x}\right )}{4 \sqrt {4 a^2+b^2}}\\ &=\frac {x^3 \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^3 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {3 x \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 {3 x \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 {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}}-\frac {3 \text {Li}_4\left (-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.27, size = 279, normalized size = 0.72 \begin {gather*} \frac {4 x^3 \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )-4 x^3 \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )+6 x^2 \text {PolyLog}\left (2,\frac {b e^{2 x}}{-2 a+\sqrt {4 a^2+b^2}}\right )-6 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )-6 x \text {PolyLog}\left (3,\frac {b e^{2 x}}{-2 a+\sqrt {4 a^2+b^2}}\right )+6 x \text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )+3 \text {PolyLog}\left (4,\frac {b e^{2 x}}{-2 a+\sqrt {4 a^2+b^2}}\right )-3 \text {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{4 \sqrt {4 a^2+b^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(686\) vs.
\(2(334)=668\).
time = 1.67, size = 687, normalized size = 1.78
method | result | size |
risch | \(\frac {x^{3} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{\sqrt {4 a^{2}+b^{2}}}-\frac {x^{4}}{2 \sqrt {4 a^{2}+b^{2}}}+\frac {3 x^{2} \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{2 \sqrt {4 a^{2}+b^{2}}}-\frac {3 x \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{2 \sqrt {4 a^{2}+b^{2}}}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{4 \sqrt {4 a^{2}+b^{2}}}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) x^{3}}{-2 a -\sqrt {4 a^{2}+b^{2}}}-\frac {x^{4}}{2 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {2 \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) a \,x^{3}}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {a \,x^{4}}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) x^{2}}{2 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) a \,x^{2}}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {3 \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) x}{2 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {3 \polylog \left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) a x}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{4 \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) a}{2 \sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}\) | \(687\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1488 vs.
\(2 (332) = 664\).
time = 0.36, size = 1488, normalized size = 3.85 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{a+b\,\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________