Integrand size = 12, antiderivative size = 215 \[ \int \frac {x}{a+b \sinh ^2(x)} \, dx=\frac {x \log \left (1+\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{2 \sqrt {a} \sqrt {a-b}}+\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}} \]
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Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5748, 3401, 2296, 2221, 2317, 2438} \[ \int \frac {x}{a+b \sinh ^2(x)} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a-b} \sqrt {a}-b}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {x \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}}-\frac {x \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a-b}+2 a-b}+1\right )}{2 \sqrt {a} \sqrt {a-b}} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3401
Rule 5748
Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {x}{2 a-b+b \cosh (2 x)} \, dx \\ & = 4 \int \frac {e^{2 x} x}{b+2 (2 a-b) e^{2 x}+b e^{4 x}} \, dx \\ & = \frac {(2 b) \int \frac {e^{2 x} x}{-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}{4 \sqrt {a} \sqrt {a-b}+2 (2 a-b)+2 b e^{2 x}} \, dx}{\sqrt {a} \sqrt {a-b}} \\ & = \frac {x \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 \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 \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 {\int \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 \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 \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 {\text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{-4 \sqrt {a} \sqrt {a-b}+2 (2 a-b)}\right )}{x} \, dx,x,e^{2 x}\right )}{4 \sqrt {a} \sqrt {a-b}}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{4 \sqrt {a} \sqrt {a-b}+2 (2 a-b)}\right )}{x} \, dx,x,e^{2 x}\right )}{4 \sqrt {a} \sqrt {a-b}} \\ & = \frac {x \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 \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 {\operatorname {PolyLog}\left (2,-\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 (2,-\frac {b e^{2 x}}{2 a+2 \sqrt {a} \sqrt {a-b}-b}\right )}{4 \sqrt {a} \sqrt {a-b}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.37 \[ \int \frac {x}{a+b \sinh ^2(x)} \, dx=\frac {-x \log \left (1-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-x \log \left (1+\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+x \log \left (1-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+x \log \left (1+\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )-\operatorname {PolyLog}\left (2,-\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )-\operatorname {PolyLog}\left (2,\frac {e^x}{\sqrt {\frac {-2 a-2 \sqrt {a (a-b)}+b}{b}}}\right )+\operatorname {PolyLog}\left (2,-\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )+\operatorname {PolyLog}\left (2,\frac {e^x}{\sqrt {\frac {-2 a+2 \sqrt {a (a-b)}+b}{b}}}\right )}{2 \sqrt {a (a-b)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(504\) vs. \(2(171)=342\).
Time = 0.11 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.35
method | result | size |
risch | \(\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) x}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}+\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) a x}{\sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) b x}{2 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {x^{2}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}-\frac {x^{2} a}{\sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {b \,x^{2}}{2 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right )}{-4 \sqrt {\left (a -b \right ) a}-4 a +2 b}+\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) a}{2 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}-\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 \sqrt {\left (a -b \right ) a}-2 a +b}\right ) b}{4 \sqrt {\left (a -b \right ) a}\, \left (-2 \sqrt {\left (a -b \right ) a}-2 a +b \right )}+\frac {x \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {\left (a -b \right ) a}-2 a +b}\right )}{2 \sqrt {\left (a -b \right ) a}}-\frac {x^{2}}{2 \sqrt {\left (a -b \right ) a}}+\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{2 \sqrt {\left (a -b \right ) a}-2 a +b}\right )}{4 \sqrt {\left (a -b \right ) a}}\) | \(505\) |
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Leaf count of result is larger than twice the leaf count of optimal. 837 vs. \(2 (165) = 330\).
Time = 0.32 (sec) , antiderivative size = 837, normalized size of antiderivative = 3.89 \[ \int \frac {x}{a+b \sinh ^2(x)} \, dx=-\frac {b x \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 \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 \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 \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 \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 ) + b \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 ) - b \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 ) - b \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 \, {\left (a^{2} - a b\right )}} \]
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\[ \int \frac {x}{a+b \sinh ^2(x)} \, dx=\int \frac {x}{a + b \sinh ^{2}{\left (x \right )}}\, dx \]
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\[ \int \frac {x}{a+b \sinh ^2(x)} \, dx=\int { \frac {x}{b \sinh \left (x\right )^{2} + a} \,d x } \]
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\[ \int \frac {x}{a+b \sinh ^2(x)} \, dx=\int { \frac {x}{b \sinh \left (x\right )^{2} + a} \,d x } \]
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Timed out. \[ \int \frac {x}{a+b \sinh ^2(x)} \, dx=\int \frac {x}{b\,{\mathrm {sinh}\left (x\right )}^2+a} \,d x \]
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