Optimal. Leaf size=212 \[ -\frac {a f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {a f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {a (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^2 d}-\frac {a (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^2 d}+\frac {a (e+f x)^2}{2 b^2 f}-\frac {f \cosh (c+d x)}{b d^2}+\frac {(e+f x) \sinh (c+d x)}{b d} \]
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Rubi [A] time = 0.31, antiderivative size = 212, normalized size of antiderivative = 1.00, 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 2190
Rule 2279
Rule 2391
Rule 2638
Rule 3296
Rule 5561
Rule 5579
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*}
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Mathematica [A] time = 1.07, size = 206, normalized size = 0.97 \[ \frac {-a \left (f \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))-\frac {1}{2} f (c+d x)^2\right )+b d (e+f x) \sinh (c+d x)-b f \cosh (c+d x)}{b^2 d^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 692, normalized size = 3.26 \[ -\frac {b d f x + b d e - {\left (b d f x + b d e - b f\right )} \cosh \left (d x + c\right )^{2} - {\left (b d f x + b d e - b f\right )} \sinh \left (d x + c\right )^{2} + b f - {\left (a d^{2} f x^{2} + 2 \, a d^{2} e x + 4 \, a c d e - 2 \, a c^{2} f\right )} \cosh \left (d x + c\right ) + 2 \, {\left (a f \cosh \left (d x + c\right ) + a f \sinh \left (d x + c\right )\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 ) + 2 \, {\left (a f \cosh \left (d x + c\right ) + a f \sinh \left (d x + c\right )\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 ) + 2 \, {\left ({\left (a d e - a c f\right )} \cosh \left (d x + c\right ) + {\left (a d e - a c f\right )} \sinh \left (d x + c\right )\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 ) + 2 \, {\left ({\left (a d e - a c f\right )} \cosh \left (d x + c\right ) + {\left (a d e - a c f\right )} \sinh \left (d x + c\right )\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 ) + 2 \, {\left ({\left (a d f x + a c f\right )} \cosh \left (d x + c\right ) + {\left (a d f x + a c f\right )} \sinh \left (d x + c\right )\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 ) + 2 \, {\left ({\left (a d f x + a c f\right )} \cosh \left (d x + c\right ) + {\left (a d f x + a c f\right )} \sinh \left (d x + c\right )\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 ) - {\left (a d^{2} f x^{2} + 2 \, a d^{2} e x + 4 \, a c d e - 2 \, a c^{2} f + 2 \, {\left (b d f x + b d e - b f\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (b^{2} d^{2} \cosh \left (d x + c\right ) + b^{2} d^{2} \sinh \left (d x + c\right )\right )}} \]
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 \[ \int \frac {{\left (f x + e\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 483, normalized size = 2.28 \[ \frac {a f \,x^{2}}{2 b^{2}}-\frac {a e x}{b^{2}}+\frac {\left (d f x +d e -f \right ) {\mathrm e}^{d x +c}}{2 d^{2} b}-\frac {\left (d f x +d e +f \right ) {\mathrm e}^{-d x -c}}{2 d^{2} b}+\frac {a f c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b^{2}}-\frac {2 a f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{2}}-\frac {a e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,b^{2}}+\frac {2 a e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,b^{2}}-\frac {a f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{2}}-\frac {a f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{2}}-\frac {a f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{2}}-\frac {a f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{2}}-\frac {a f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{2}}-\frac {a f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{2}}+\frac {2 a f c x}{d \,b^{2}}+\frac {a f \,c^{2}}{d^{2} b^{2}} \]
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 \[ -\frac {1}{2} \, e {\left (\frac {2 \, {\left (d x + c\right )} a}{b^{2} d} - \frac {e^{\left (d x + c\right )}}{b d} + \frac {e^{\left (-d x - c\right )}}{b d} + \frac {2 \, a \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{2} d}\right )} - \frac {1}{4} \, f {\left (\frac {2 \, {\left (a d^{2} x^{2} e^{c} - {\left (b d x e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + {\left (b d x + b\right )} e^{\left (-d x\right )}\right )} e^{\left (-c\right )}}{b^{2} d^{2}} - \int \frac {8 \, {\left (a^{2} x e^{\left (d x + c\right )} - a b x\right )}}{b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{2} e^{\left (d x + c\right )} - b^{3}}\,{d x}\right )} \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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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