Optimal. Leaf size=330 \[ -\frac{2 a f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{2 a f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^2}+\frac{2 a f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^3}+\frac{2 a f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^3}-\frac{a (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^2 d}-\frac{a (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^2 d}+\frac{a (e+f x)^3}{3 b^2 f}-\frac{2 f (e+f x) \cosh (c+d x)}{b d^2}+\frac{2 f^2 \sinh (c+d x)}{b d^3}+\frac{(e+f x)^2 \sinh (c+d x)}{b d} \]
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Rubi [A] time = 0.549545, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5579, 3296, 2637, 5561, 2190, 2531, 2282, 6589} \[ -\frac{2 a f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{2 a f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^2}+\frac{2 a f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^3}+\frac{2 a f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b^2 d^3}-\frac{a (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b^2 d}-\frac{a (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b^2 d}+\frac{a (e+f x)^3}{3 b^2 f}-\frac{2 f (e+f x) \cosh (c+d x)}{b d^2}+\frac{2 f^2 \sinh (c+d x)}{b d^3}+\frac{(e+f x)^2 \sinh (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 5579
Rule 3296
Rule 2637
Rule 5561
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \cosh (c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac{a (e+f x)^3}{3 b^2 f}+\frac{(e+f x)^2 \sinh (c+d x)}{b d}-\frac{a \int \frac{e^{c+d x} (e+f x)^2}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b}-\frac{a \int \frac{e^{c+d x} (e+f x)^2}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b}-\frac{(2 f) \int (e+f x) \sinh (c+d x) \, dx}{b d}\\ &=\frac{a (e+f x)^3}{3 b^2 f}-\frac{2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac{a (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{a (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}+\frac{(e+f x)^2 \sinh (c+d x)}{b d}+\frac{(2 a f) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^2 d}+\frac{(2 a f) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^2 d}+\frac{\left (2 f^2\right ) \int \cosh (c+d x) \, dx}{b d^2}\\ &=\frac{a (e+f x)^3}{3 b^2 f}-\frac{2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac{a (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{a (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{2 a f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{2 a f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^2}+\frac{2 f^2 \sinh (c+d x)}{b d^3}+\frac{(e+f x)^2 \sinh (c+d x)}{b d}+\frac{\left (2 a f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^2 d^2}+\frac{\left (2 a f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^2 d^2}\\ &=\frac{a (e+f x)^3}{3 b^2 f}-\frac{2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac{a (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{a (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{2 a f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{2 a f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^2}+\frac{2 f^2 \sinh (c+d x)}{b d^3}+\frac{(e+f x)^2 \sinh (c+d x)}{b d}+\frac{\left (2 a f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{-a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^3}+\frac{\left (2 a f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^3}\\ &=\frac{a (e+f x)^3}{3 b^2 f}-\frac{2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac{a (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{a (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{2 a f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^2}-\frac{2 a f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^2}+\frac{2 a f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 d^3}+\frac{2 a f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 d^3}+\frac{2 f^2 \sinh (c+d x)}{b d^3}+\frac{(e+f x)^2 \sinh (c+d x)}{b d}\\ \end{align*}
Mathematica [B] time = 14.2168, size = 1301, normalized size = 3.94 \[ \frac{1}{2} \left (\frac{2 a \left (2 e^{2 c} f^2 x^3+6 e e^{2 c} f x^2-\frac{3 e^{2 c} f^2 \log \left (\frac{e^{2 c+d x} b}{a e^c-\sqrt{\left (a^2+b^2\right ) e^{2 c}}}+1\right ) x^2}{d}+\frac{3 f^2 \log \left (\frac{e^{2 c+d x} b}{a e^c-\sqrt{\left (a^2+b^2\right ) e^{2 c}}}+1\right ) x^2}{d}-\frac{3 e^{2 c} f^2 \log \left (\frac{e^{2 c+d x} b}{e^c a+\sqrt{\left (a^2+b^2\right ) e^{2 c}}}+1\right ) x^2}{d}+\frac{3 f^2 \log \left (\frac{e^{2 c+d x} b}{e^c a+\sqrt{\left (a^2+b^2\right ) e^{2 c}}}+1\right ) x^2}{d}+6 e^2 e^{2 c} x-\frac{6 e e^{2 c} f \log \left (\frac{e^{2 c+d x} b}{a e^c-\sqrt{\left (a^2+b^2\right ) e^{2 c}}}+1\right ) x}{d}+\frac{6 e f \log \left (\frac{e^{2 c+d x} b}{a e^c-\sqrt{\left (a^2+b^2\right ) e^{2 c}}}+1\right ) x}{d}-\frac{6 e e^{2 c} f \log \left (\frac{e^{2 c+d x} b}{e^c a+\sqrt{\left (a^2+b^2\right ) e^{2 c}}}+1\right ) x}{d}+\frac{6 e f \log \left (\frac{e^{2 c+d x} b}{e^c a+\sqrt{\left (a^2+b^2\right ) e^{2 c}}}+1\right ) x}{d}+\frac{6 a \sqrt{-\left (a^2+b^2\right )^2} e^2 e^{2 c} \tan ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{-a^2-b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac{6 a \sqrt{a^2+b^2} e^2 \tan ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-\left (a^2+b^2\right )^2} d}+\frac{6 a \sqrt{-\left (a^2+b^2\right )^2} e^2 e^{2 c} \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2} d}-\frac{6 a \sqrt{-\left (a^2+b^2\right )^2} e^2 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2} d}-\frac{3 e^2 e^{2 c} \log \left (2 e^{c+d x} a+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}+\frac{3 e^2 \log \left (2 e^{c+d x} a+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}-\frac{6 \left (-1+e^{2 c}\right ) f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{2 c+d x}}{a e^c-\sqrt{\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^2}-\frac{6 \left (-1+e^{2 c}\right ) f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{2 c+d x}}{e^c a+\sqrt{\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^2}+\frac{6 e^{2 c} f^2 \text{PolyLog}\left (3,-\frac{b e^{2 c+d x}}{a e^c-\sqrt{\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}-\frac{6 f^2 \text{PolyLog}\left (3,-\frac{b e^{2 c+d x}}{a e^c-\sqrt{\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}+\frac{6 e^{2 c} f^2 \text{PolyLog}\left (3,-\frac{b e^{2 c+d x}}{e^c a+\sqrt{\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}-\frac{6 f^2 \text{PolyLog}\left (3,-\frac{b e^{2 c+d x}}{e^c a+\sqrt{\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}\right )}{3 b^2 \left (-1+e^{2 c}\right )}-\frac{a x \left (3 e^2+3 f x e+f^2 x^2\right ) \cosh (c) \text{csch}\left (\frac{c}{2}\right ) \text{sech}\left (\frac{c}{2}\right )}{3 b^2}+\frac{2 \cosh (d x) \left (e^2 \sinh (c) d^2+f^2 x^2 \sinh (c) d^2+2 e f x \sinh (c) d^2-2 e f \cosh (c) d-2 f^2 x \cosh (c) d+2 f^2 \sinh (c)\right )}{b d^3}+\frac{2 \left (e^2 \cosh (c) d^2+f^2 x^2 \cosh (c) d^2+2 e f x \cosh (c) d^2-2 e f \sinh (c) d-2 f^2 x \sinh (c) d+2 f^2 \cosh (c)\right ) \sinh (d x)}{b d^3}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.126, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2}\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{a+b\sinh \left ( dx+c \right ) }}\, dx \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^{2}{\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{{\left (2 \, a d^{3} f^{2} x^{3} e^{c} + 6 \, a d^{3} e f x^{2} e^{c} - 3 \,{\left (b d^{2} f^{2} x^{2} e^{\left (2 \, c\right )} + 2 \,{\left (d^{2} e f - d f^{2}\right )} b x e^{\left (2 \, c\right )} - 2 \,{\left (d e f - f^{2}\right )} b e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + 3 \,{\left (b d^{2} f^{2} x^{2} + 2 \,{\left (d^{2} e f + d f^{2}\right )} b x + 2 \,{\left (d e f + f^{2}\right )} b\right )} e^{\left (-d x\right )}\right )} e^{\left (-c\right )}}{6 \, b^{2} d^{3}} + \int -\frac{2 \,{\left (a b f^{2} x^{2} + 2 \, a b e f x -{\left (a^{2} f^{2} x^{2} e^{c} + 2 \, a^{2} e f x e^{c}\right )} e^{\left (d x\right )}\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.47519, size = 3051, normalized size = 9.25 \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 )}^{2} \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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