Optimal. Leaf size=330 \[ \frac {2 a f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {2 a f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}-\frac {2 a f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {2 a f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {a (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^2 d}-\frac {a (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^2 d}+\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 f^2 \sinh (c+d x)}{b d^3}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}+\frac {(e+f x)^2 \sinh (c+d x)}{b d} \]
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Rubi [A] time = 0.55, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2190
Rule 2282
Rule 2531
Rule 2637
Rule 3296
Rule 5561
Rule 5579
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*}
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Mathematica [B] time = 11.65, 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 {Li}_2\left (-\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 {Li}_2\left (-\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 {Li}_3\left (-\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 {Li}_3\left (-\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 {Li}_3\left (-\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 {Li}_3\left (-\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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fricas [C] time = 0.47, size = 1265, normalized size = 3.83 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, e^{2} {\left (\frac {2 \, {\left (d x + c\right )} a}{b^{2} d} - \frac {e^{\left (d x + c\right )}}{b d} + \frac {e^{\left (-d x - c\right )}}{b d} + \frac {2 \, a \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{2} d}\right )} - \frac {{\left (2 \, a d^{3} f^{2} x^{3} e^{c} + 6 \, a d^{3} e f x^{2} e^{c} - 3 \, {\left (b d^{2} f^{2} x^{2} e^{\left (2 \, c\right )} + 2 \, {\left (d^{2} e f - d f^{2}\right )} b x e^{\left (2 \, c\right )} - 2 \, {\left (d e f - f^{2}\right )} b e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + 3 \, {\left (b d^{2} f^{2} x^{2} + 2 \, {\left (d^{2} e f + d f^{2}\right )} b x + 2 \, {\left (d e f + f^{2}\right )} b\right )} e^{\left (-d x\right )}\right )} e^{\left (-c\right )}}{6 \, b^{2} d^{3}} + \int -\frac {2 \, {\left (a b f^{2} x^{2} + 2 \, a b e f x - {\left (a^{2} f^{2} x^{2} e^{c} + 2 \, a^{2} e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{2} e^{\left (d x + c\right )} - b^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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