Optimal. Leaf size=212 \[ -\frac{a f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{a f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^2}-\frac{a (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^2 d}-\frac{a (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^2 d}+\frac{a (e+f x)^2}{2 b^2 f}-\frac{f \cosh (c+d x)}{b d^2}+\frac{(e+f x) \sinh (c+d x)}{b d} \]
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Rubi [A] time = 0.312137, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {5579, 3296, 2638, 5561, 2190, 2279, 2391} \[ -\frac{a f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{a f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^2}-\frac{a (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^2 d}-\frac{a (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^2 d}+\frac{a (e+f x)^2}{2 b^2 f}-\frac{f \cosh (c+d x)}{b d^2}+\frac{(e+f x) \sinh (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 5579
Rule 3296
Rule 2638
Rule 5561
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x) \cosh (c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac{a (e+f x)^2}{2 b^2 f}+\frac{(e+f x) \sinh (c+d x)}{b d}-\frac{a \int \frac{e^{c+d x} (e+f x)}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b}-\frac{a \int \frac{e^{c+d x} (e+f x)}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b}-\frac{f \int \sinh (c+d x) \, dx}{b d}\\ &=\frac{a (e+f x)^2}{2 b^2 f}-\frac{f \cosh (c+d x)}{b d^2}-\frac{a (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{a (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}+\frac{(e+f x) \sinh (c+d x)}{b d}+\frac{(a f) \int \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^2 d}+\frac{(a f) \int \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^2 d}\\ &=\frac{a (e+f x)^2}{2 b^2 f}-\frac{f \cosh (c+d x)}{b d^2}-\frac{a (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{a (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}+\frac{(e+f x) \sinh (c+d x)}{b d}+\frac{(a f) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a-\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^2}+\frac{(a f) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^2}\\ &=\frac{a (e+f x)^2}{2 b^2 f}-\frac{f \cosh (c+d x)}{b d^2}-\frac{a (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{a (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{a f \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{a f \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^2}+\frac{(e+f x) \sinh (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 1.05742, size = 206, normalized size = 0.97 \[ \frac{-a \left (f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))-\frac{1}{2} f (c+d x)^2\right )+b d (e+f x) \sinh (c+d x)-b f \cosh (c+d x)}{b^2 d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 483, normalized size = 2.3 \begin{align*}{\frac{af{x}^{2}}{2\,{b}^{2}}}-{\frac{aex}{{b}^{2}}}+{\frac{ \left ( dfx+de-f \right ){{\rm e}^{dx+c}}}{2\,{d}^{2}b}}-{\frac{ \left ( dfx+de+f \right ){{\rm e}^{-dx-c}}}{2\,{d}^{2}b}}+{\frac{afc\ln \left ( b{{\rm e}^{2\,dx+2\,c}}+2\,a{{\rm e}^{dx+c}}-b \right ) }{{b}^{2}{d}^{2}}}-2\,{\frac{afc\ln \left ({{\rm e}^{dx+c}} \right ) }{{b}^{2}{d}^{2}}}-{\frac{ae\ln \left ( b{{\rm e}^{2\,dx+2\,c}}+2\,a{{\rm e}^{dx+c}}-b \right ) }{{b}^{2}d}}+2\,{\frac{ae\ln \left ({{\rm e}^{dx+c}} \right ) }{{b}^{2}d}}-{\frac{afx}{{b}^{2}d}\ln \left ({ \left ( -b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}-a \right ) \left ( -a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{afc}{{b}^{2}{d}^{2}}\ln \left ({ \left ( -b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}-a \right ) \left ( -a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{afx}{{b}^{2}d}\ln \left ({ \left ( b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}+a \right ) \left ( a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{afc}{{b}^{2}{d}^{2}}\ln \left ({ \left ( b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}+a \right ) \left ( a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{af}{{b}^{2}{d}^{2}}{\it dilog} \left ({ \left ( b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}+a \right ) \left ( a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{af}{{b}^{2}{d}^{2}}{\it dilog} \left ({ \left ( -b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}-a \right ) \left ( -a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }+2\,{\frac{afcx}{{b}^{2}d}}+{\frac{af{c}^{2}}{{b}^{2}{d}^{2}}} \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*} -\frac{1}{2} \, e{\left (\frac{2 \,{\left (d x + c\right )} a}{b^{2} d} - \frac{e^{\left (d x + c\right )}}{b d} + \frac{e^{\left (-d x - c\right )}}{b d} + \frac{2 \, a \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{2} d}\right )} - \frac{1}{4} \, f{\left (\frac{2 \,{\left (a d^{2} x^{2} e^{c} -{\left (b d x e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} +{\left (b d x + b\right )} e^{\left (-d x\right )}\right )} e^{\left (-c\right )}}{b^{2} d^{2}} - \int \frac{8 \,{\left (a^{2} x e^{\left (d x + c\right )} - a b x\right )}}{b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{2} e^{\left (d x + c\right )} - b^{3}}\,{d x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.27905, size = 1760, normalized size = 8.3 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (f x + e\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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