Optimal. Leaf size=264 \[ \frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^2}+\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^2}-\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^3}-\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^3}+\frac{(e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b d}+\frac{(e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b d}-\frac{(e+f x)^3}{3 b f} \]
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Rubi [A] time = 0.411303, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {5561, 2190, 2531, 2282, 6589} \[ \frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^2}+\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^2}-\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^3}-\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^3}+\frac{(e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b d}+\frac{(e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b d}-\frac{(e+f x)^3}{3 b f} \]
Antiderivative was successfully verified.
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Rule 5561
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac{(e+f x)^3}{3 b f}+\int \frac{e^{c+d x} (e+f x)^2}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx+\int \frac{e^{c+d x} (e+f x)^2}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx\\ &=-\frac{(e+f x)^3}{3 b f}+\frac{(e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d}+\frac{(e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b d}-\frac{(2 f) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b d}-\frac{(2 f) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b d}\\ &=-\frac{(e+f x)^3}{3 b f}+\frac{(e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d}+\frac{(e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b d}+\frac{2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^2}+\frac{2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b d^2}-\frac{\left (2 f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b d^2}-\frac{\left (2 f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b d^2}\\ &=-\frac{(e+f x)^3}{3 b f}+\frac{(e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d}+\frac{(e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b d}+\frac{2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^2}+\frac{2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b d^2}-\frac{\left (2 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 d^3}-\frac{\left (2 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 d^3}\\ &=-\frac{(e+f x)^3}{3 b f}+\frac{(e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d}+\frac{(e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b d}+\frac{2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^2}+\frac{2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b d^2}-\frac{2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^3}-\frac{2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b d^3}\\ \end{align*}
Mathematica [A] time = 0.145766, size = 244, normalized size = 0.92 \[ \frac{\frac{6 f \left (d (e+f x) \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-f \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )\right )}{d^3}+\frac{6 f \left (d (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )-f \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )\right )}{d^3}+\frac{3 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{d}+\frac{3 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{d}-\frac{(e+f x)^3}{f}}{3 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.204, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2}\cosh \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{e^{2} \log \left (b \sinh \left (d x + c\right ) + a\right )}{b d} + \frac{f^{2} x^{3} + 3 \, e f x^{2}}{3 \, b} - \int -\frac{2 \,{\left (b f^{2} x^{2} + 2 \, b e f x -{\left (a f^{2} x^{2} e^{c} + 2 \, a e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} - b^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.31073, size = 1504, normalized size = 5.7 \begin{align*} -\frac{d^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2} + 3 \, d^{3} e^{2} x + 6 \, f^{2}{\rm polylog}\left (3, \frac{a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) +{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt{\frac{a^{2} + b^{2}}{b^{2}}}}{b}\right ) + 6 \, f^{2}{\rm polylog}\left (3, \frac{a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) -{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt{\frac{a^{2} + b^{2}}{b^{2}}}}{b}\right ) - 6 \,{\left (d f^{2} x + d e f\right )}{\rm Li}_2\left (\frac{a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) +{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) - 6 \,{\left (d f^{2} x + d e f\right )}{\rm Li}_2\left (\frac{a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) -{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) - 3 \,{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - 3 \,{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - 3 \,{\left (d^{2} f^{2} x^{2} + 2 \, d^{2} e f x + 2 \, c d e f - c^{2} f^{2}\right )} \log \left (-\frac{a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) +{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) - 3 \,{\left (d^{2} f^{2} x^{2} + 2 \, d^{2} e f x + 2 \, c d e f - c^{2} f^{2}\right )} \log \left (-\frac{a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) -{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt{\frac{a^{2} + b^{2}}{b^{2}}} - b}{b}\right )}{3 \, b d^{3}} \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 )}{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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