Integrand size = 14, antiderivative size = 327 \[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x^2 \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}+\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}} \]
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Time = 0.39 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5748, 3401, 2296, 2221, 2611, 2320, 6724} \[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {x^2 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x^2 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}} \]
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Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3401
Rule 5748
Rule 6724
Rubi steps \begin{align*} \text {integral}& = 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 \operatorname {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 \operatorname {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 {\int \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 \operatorname {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 \operatorname {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 {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\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 {\operatorname {PolyLog}\left (2,\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 \operatorname {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 \operatorname {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 {\operatorname {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 {\operatorname {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{align*}
Time = 0.34 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.37 \[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\frac {-x^2 \log \left (1-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-x^2 \log \left (1+\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+x^2 \log \left (1-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+x^2 \log \left (1+\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )-2 x \operatorname {PolyLog}\left (2,-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-2 x \operatorname {PolyLog}\left (2,\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+2 x \operatorname {PolyLog}\left (2,-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+2 x \operatorname {PolyLog}\left (2,\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+2 \operatorname {PolyLog}\left (3,\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-2 \operatorname {PolyLog}\left (3,-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )-2 \operatorname {PolyLog}\left (3,\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )}{2 \sqrt {a (a-b)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(709\) vs. \(2(261)=522\).
Time = 0.13 (sec) , antiderivative size = 710, normalized size of antiderivative = 2.17
method | result | size |
risch | \(-\frac {2 x^{3}}{3 \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) x^{2}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}+\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) x}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}-\frac {\operatorname {polylog}\left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right )}{2 \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {2 x^{3} a}{3 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) a \,x^{2}}{\sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) a x}{\sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {\operatorname {polylog}\left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) a}{2 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {b \,x^{3}}{3 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) b \,x^{2}}{2 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) b x}{2 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {\operatorname {polylog}\left (3, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) b}{4 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {x^{3}}{3 \sqrt {\left (a -b \right ) a}}+\frac {x^{2} \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {\left (a -b \right ) a}-2 a +b}\right )}{2 \sqrt {\left (a -b \right ) a}}+\frac {x \operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {\left (a -b \right ) a}-2 a +b}\right )}{2 \sqrt {\left (a -b \right ) a}}-\frac {\operatorname {polylog}\left (3, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {\left (a -b \right ) a}-2 a +b}\right )}{4 \sqrt {\left (a -b \right ) a}}\) | \(710\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1247 vs. \(2 (252) = 504\).
Time = 0.30 (sec) , antiderivative size = 1247, normalized size of antiderivative = 3.81 \[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\int \frac {x^{2}}{a + b \sinh ^{2}{\left (x \right )}}\, dx \]
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\[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\int { \frac {x^{2}}{b \sinh \left (x\right )^{2} + a} \,d x } \]
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\[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\int { \frac {x^{2}}{b \sinh \left (x\right )^{2} + a} \,d x } \]
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Timed out. \[ \int \frac {x^2}{a+b \sinh ^2(x)} \, dx=\int \frac {x^2}{b\,{\mathrm {sinh}\left (x\right )}^2+a} \,d x \]
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