Optimal. Leaf size=215 \[ \frac{\text{PolyLog}\left (2,-\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{4 \sqrt{a} \sqrt{a-b}}-\frac{\text{PolyLog}\left (2,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{4 \sqrt{a} \sqrt{a-b}}+\frac{x \log \left (\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a-b}+2 a-b}+1\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x \log \left (\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}+1\right )}{2 \sqrt{a} \sqrt{a-b}} \]
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Rubi [A] time = 0.329364, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5629, 3320, 2264, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{4 \sqrt{a} \sqrt{a-b}}-\frac{\text{PolyLog}\left (2,-\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}\right )}{4 \sqrt{a} \sqrt{a-b}}+\frac{x \log \left (\frac{b e^{2 x}}{-2 \sqrt{a} \sqrt{a-b}+2 a-b}+1\right )}{2 \sqrt{a} \sqrt{a-b}}-\frac{x \log \left (\frac{b e^{2 x}}{2 \sqrt{a} \sqrt{a-b}+2 a-b}+1\right )}{2 \sqrt{a} \sqrt{a-b}} \]
Antiderivative was successfully verified.
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Rule 5629
Rule 3320
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x}{a+b \sinh ^2(x)} \, dx &=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{\operatorname{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{\operatorname{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{\text{Li}_2\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}_2\left (-\frac{b e^{2 x}}{2 a+2 \sqrt{a} \sqrt{a-b}-b}\right )}{4 \sqrt{a} \sqrt{a-b}}\\ \end{align*}
Mathematica [C] time = 0.771916, size = 576, normalized size = 2.68 \[ -\frac{i \left (\text{PolyLog}\left (2,\frac{\left (2 i \sqrt{a (b-a)}-2 a+b\right ) \left (\sqrt{a (b-a)} \tanh (x)+i a\right )}{b \sqrt{a (b-a)} \tanh (x)-i a b}\right )-\text{PolyLog}\left (2,\frac{\left (-2 i \sqrt{a (b-a)}-2 a+b\right ) \left (\sqrt{a (b-a)} \tanh (x)+i a\right )}{b \sqrt{a (b-a)} \tanh (x)-i a b}\right )\right )-2 i \cos ^{-1}\left (1-\frac{2 a}{b}\right ) \tan ^{-1}\left (\frac{\sqrt{a b-a^2} \tanh (x)}{a}\right )-\log \left (\frac{2 a \left (\sqrt{a (b-a)}-i a+i b\right ) (\tanh (x)-1)}{b \sqrt{a (b-a)} \tanh (x)-i a b}\right ) \left (2 \tan ^{-1}\left (\frac{\sqrt{a b-a^2} \tanh (x)}{a}\right )+\cos ^{-1}\left (1-\frac{2 a}{b}\right )\right )-\log \left (\frac{2 a \left (\sqrt{a (b-a)}+i a-i b\right ) (\tanh (x)+1)}{b \sqrt{a (b-a)} \tanh (x)-i a b}\right ) \left (\cos ^{-1}\left (1-\frac{2 a}{b}\right )-2 \tan ^{-1}\left (\frac{\sqrt{a b-a^2} \tanh (x)}{a}\right )\right )+\log \left (\frac{\sqrt{2} e^{-x} \sqrt{a (b-a)}}{\sqrt{b} \sqrt{2 a+b \cosh (2 x)-b}}\right ) \left (2 \left (\tan ^{-1}\left (\frac{\sqrt{a b-a^2} \tanh (x)}{a}\right )+\tan ^{-1}\left (\frac{a \coth (x)}{\sqrt{-a (a-b)}}\right )\right )+\cos ^{-1}\left (1-\frac{2 a}{b}\right )\right )+\log \left (\frac{\sqrt{2} e^x \sqrt{a (b-a)}}{\sqrt{b} \sqrt{2 a+b \cosh (2 x)-b}}\right ) \left (\cos ^{-1}\left (1-\frac{2 a}{b}\right )-2 \left (\tan ^{-1}\left (\frac{\sqrt{a b-a^2} \tanh (x)}{a}\right )+\tan ^{-1}\left (\frac{a \coth (x)}{\sqrt{-a (a-b)}}\right )\right )\right )+4 x \tan ^{-1}\left (\frac{a \coth (x)}{\sqrt{-a (a-b)}}\right )}{4 \sqrt{a (b-a)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 505, normalized size = 2.4 \begin{align*}{x\ln \left ( 1-{b{{\rm e}^{2\,x}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}} \right ) \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}}+{ax\ln \left ( 1-{b{{\rm e}^{2\,x}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}}-{\frac{bx}{2}\ln \left ( 1-{b{{\rm e}^{2\,x}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}}-{{x}^{2} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}}-{a{x}^{2}{\frac{1}{\sqrt{a \left ( a-b \right ) }}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}}+{\frac{b{x}^{2}}{2}{\frac{1}{\sqrt{a \left ( a-b \right ) }}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}}+{\frac{1}{2}{\it polylog} \left ( 2,{b{{\rm e}^{2\,x}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}} \right ) \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}}+{\frac{a}{2}{\it polylog} \left ( 2,{b{{\rm e}^{2\,x}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}}-{\frac{b}{4}{\it polylog} \left ( 2,{b{{\rm e}^{2\,x}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}} \left ( -2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}}+{\frac{x}{2}\ln \left ( 1-{b{{\rm e}^{2\,x}} \left ( 2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}}-{\frac{{x}^{2}}{2}{\frac{1}{\sqrt{a \left ( a-b \right ) }}}}+{\frac{1}{4}{\it polylog} \left ( 2,{b{{\rm e}^{2\,x}} \left ( 2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) ^{-1}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{b \sinh \left (x\right )^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.25311, size = 1968, normalized size = 9.15 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{a + b \sinh ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{b \sinh \left (x\right )^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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