Optimal. Leaf size=327 \[ \frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}+\frac {x \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {\text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {\text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}} \]
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Rubi [A]
time = 0.41, 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 = {5748, 3401,
2296, 2221, 2611, 2320, 6724} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3401
Rule 5748
Rule 6724
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 {\text {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 {\text {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 = 0.50, size = 240, normalized size = 0.73 \begin {gather*} \frac {-2 x^2 \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )+2 x^2 \log \left (1-\frac {b e^{2 x}}{-2 a+2 \sqrt {a} \sqrt {a-b}+b}\right )-2 x \text {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )+2 x \text {PolyLog}\left (2,\frac {b e^{2 x}}{-2 a+2 \sqrt {a} \sqrt {a-b}+b}\right )+\text {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )-\text {PolyLog}\left (3,\frac {b e^{2 x}}{-2 a+2 \sqrt {a} \sqrt {a-b}+b}\right )}{4 \sqrt {a} \sqrt {a-b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(709\) vs.
\(2(261)=522\).
time = 0.85, size = 710, normalized size = 2.17
method | result | size |
risch | \(-\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 )}}\) | \(710\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1247 vs.
\(2 (252) = 504\).
time = 0.44, size = 1247, normalized size = 3.81 \begin {gather*} -\frac {b x^{2} \sqrt {\frac {a^{2} - a b}{b^{2}}} \log \left (\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) - 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b}{b}} + b}{b}\right ) + b x^{2} \sqrt {\frac {a^{2} - a b}{b^{2}}} \log \left (-\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) - 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b}{b}} - b}{b}\right ) - b x^{2} \sqrt {\frac {a^{2} - a b}{b^{2}}} \log \left (\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} + b}{b}\right ) - b x^{2} \sqrt {\frac {a^{2} - a b}{b^{2}}} \log \left (-\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} - b}{b}\right ) + 2 \, b x \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm Li}_2\left (-\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) - 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b}{b}} + b}{b} + 1\right ) + 2 \, b x \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm Li}_2\left (\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) - 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b}{b}} - b}{b} + 1\right ) - 2 \, b x \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm Li}_2\left (-\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} + b}{b} + 1\right ) - 2 \, b x \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm Li}_2\left (\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} - b}{b} + 1\right ) - 2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm polylog}\left (3, \frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) - 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b}{b}}}{b}\right ) - 2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm polylog}\left (3, -\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) - 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {-\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b}{b}}}{b}\right ) + 2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm polylog}\left (3, \frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}}}{b}\right ) + 2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} {\rm polylog}\left (3, -\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (x\right ) + {\left (2 \, a - b\right )} \sinh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}}}{b}\right )}{2 \, {\left (a^{2} - a b\right )}} \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{a + b \sinh ^{2}{\left (x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2}{b\,{\mathrm {sinh}\left (x\right )}^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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