Optimal. Leaf size=278 \[ -\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d}+\frac {a f \cosh (c+d x)}{b^2 d^2}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \sinh (c+d x) \cosh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b d}+\frac {f x}{4 b d} \]
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Rubi [A] time = 0.42, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5579, 5446, 2635, 8, 3296, 2638, 5561, 2190, 2279, 2391} \[ \frac {a^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{b^3 d^2}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d}-\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {a f \cosh (c+d x)}{b^2 d^2}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \sinh (c+d x) \cosh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b d}+\frac {f x}{4 b d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 2635
Rule 2638
Rule 3296
Rule 5446
Rule 5561
Rule 5579
Rubi steps
\begin {align*} \int \frac {(e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {(e+f x) \sinh ^2(c+d x)}{2 b d}-\frac {a \int (e+f x) \cosh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {f \int \sinh ^2(c+d x) \, dx}{2 b d}\\ &=-\frac {a^2 (e+f x)^2}{2 b^3 f}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b d}+\frac {a^2 \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {a^2 \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {(a f) \int \sinh (c+d x) \, dx}{b^2 d}+\frac {f \int 1 \, dx}{4 b d}\\ &=\frac {f x}{4 b d}-\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {a f \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b d}-\frac {\left (a^2 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}-\frac {\left (a^2 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}\\ &=\frac {f x}{4 b d}-\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {a f \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b d}-\frac {\left (a^2 f\right ) \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^3 d^2}-\frac {\left (a^2 f\right ) \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^3 d^2}\\ &=\frac {f x}{4 b d}-\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {a f \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {a (e+f x) \sinh (c+d x)}{b^2 d}-\frac {f \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {(e+f x) \sinh ^2(c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A] time = 0.97, size = 423, normalized size = 1.52 \[ \frac {2 d e \left (\left (4 a^2+b^2\right ) \log (a+b \sinh (c+d x))-4 a b \sinh (c+d x)+2 b^2 \sinh ^2(c+d x)\right )+b^2 f \left (-2 \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )-2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+d x \left (-2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )-2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )+d x\right )\right )+f \left (2 \left (4 a^2+b^2\right ) \left (\text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+\text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )+(c+d x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )-c \log (a+b \sinh (c+d x))-\frac {1}{2} (c+d x)^2\right )-8 a b d x \sinh (c+d x)+8 a b \cosh (c+d x)-b^2 \sinh (2 (c+d x))+2 b^2 d x \cosh (2 (c+d x))\right )-2 b^2 d e \log (a+b \sinh (c+d x))}{8 b^3 d^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 1248, normalized size = 4.49 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (f x + e\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 565, normalized size = 2.03 \[ -\frac {a^{2} f \,x^{2}}{2 b^{3}}+\frac {a^{2} e x}{b^{3}}+\frac {\left (2 d f x +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 d^{2} b}-\frac {a \left (d f x +d e -f \right ) {\mathrm e}^{d x +c}}{2 b^{2} d^{2}}+\frac {a \left (d f x +d e +f \right ) {\mathrm e}^{-d x -c}}{2 b^{2} d^{2}}+\frac {\left (2 d f x +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 d^{2} b}-\frac {a^{2} f c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b^{3}}+\frac {2 a^{2} f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{3}}+\frac {a^{2} e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,b^{3}}-\frac {2 a^{2} e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,b^{3}}+\frac {a^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3}}+\frac {a^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3}}+\frac {a^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3}}+\frac {a^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3}}+\frac {a^{2} f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3}}+\frac {a^{2} f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3}}-\frac {2 f \,a^{2} c x}{d \,b^{3}}-\frac {f \,a^{2} c^{2}}{d^{2} b^{3}} \]
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 \[ \frac {1}{8} \, e {\left (\frac {8 \, {\left (d x + c\right )} a^{2}}{b^{3} d} - \frac {{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{b^{2} d} + \frac {8 \, a^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{3} d} + \frac {4 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )}}{b^{2} d}\right )} + \frac {1}{16} \, f {\left (\frac {{\left (8 \, a^{2} d^{2} x^{2} e^{\left (2 \, c\right )} + {\left (2 \, b^{2} d x e^{\left (4 \, c\right )} - b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 8 \, {\left (a b d x e^{\left (3 \, c\right )} - a b e^{\left (3 \, c\right )}\right )} e^{\left (d x\right )} + 8 \, {\left (a b d x e^{c} + a b e^{c}\right )} e^{\left (-d x\right )} + {\left (2 \, b^{2} d x + b^{2}\right )} e^{\left (-2 \, d x\right )}\right )} e^{\left (-2 \, c\right )}}{b^{3} d^{2}} - 2 \, \int \frac {16 \, {\left (a^{3} x e^{\left (d x + c\right )} - a^{2} b x\right )}}{b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{3} e^{\left (d x + c\right )} - b^{4}}\,{d x}\right )} \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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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