Optimal. Leaf size=322 \[ -\frac {f \left (a^2+b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {f \left (a^2+b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a b^2 d}+\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a b^2 f}+\frac {f \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}+\frac {(e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {(e+f x)^2}{2 a f}-\frac {f \cosh (c+d x)}{b d^2}+\frac {(e+f x) \sinh (c+d x)}{b d} \]
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Rubi [A] time = 0.59, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 13, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {5585, 5450, 5446, 2635, 8, 3716, 2190, 2279, 2391, 5565, 3296, 2638, 5561} \[ -\frac {f \left (a^2+b^2\right ) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {f \left (a^2+b^2\right ) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a b^2 d^2}+\frac {f \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a b^2 d}+\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a b^2 f}+\frac {(e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {(e+f x)^2}{2 a f}-\frac {f \cosh (c+d x)}{b d^2}+\frac {(e+f x) \sinh (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 2635
Rule 2638
Rule 3296
Rule 3716
Rule 5446
Rule 5450
Rule 5561
Rule 5565
Rule 5585
Rubi steps
\begin {align*} \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \cosh ^2(c+d x) \coth (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {\int (e+f x) \coth (c+d x) \, dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \, dx}{b}-\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a b}\\ &=-\frac {(e+f x)^2}{2 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a b^2 f}+\frac {(e+f x) \sinh (c+d x)}{b d}-\frac {2 \int \frac {e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a}-\frac {\left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a b}-\frac {\left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a b}-\frac {f \int \sinh (c+d x) \, dx}{b d}\\ &=-\frac {(e+f x)^2}{2 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a b^2 f}-\frac {f \cosh (c+d x)}{b d^2}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d}+\frac {(e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {(e+f x) \sinh (c+d x)}{b d}-\frac {f \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d}+\frac {\left (\left (a^2+b^2\right ) f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a b^2 d}+\frac {\left (\left (a^2+b^2\right ) f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a b^2 d}\\ &=-\frac {(e+f x)^2}{2 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a b^2 f}-\frac {f \cosh (c+d x)}{b d^2}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d}+\frac {(e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {(e+f x) \sinh (c+d x)}{b d}-\frac {f \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {\left (\left (a^2+b^2\right ) 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 )}{a b^2 d^2}+\frac {\left (\left (a^2+b^2\right ) 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 )}{a b^2 d^2}\\ &=-\frac {(e+f x)^2}{2 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a b^2 f}-\frac {f \cosh (c+d x)}{b d^2}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d}+\frac {(e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {\left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {\left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^2}+\frac {f \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}+\frac {(e+f x) \sinh (c+d x)}{b d}\\ \end {align*}
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Mathematica [A] time = 1.63, size = 296, normalized size = 0.92 \[ \frac {\left (a^2+b^2\right ) \left (-f \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )-f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )-f (c+d x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )-d e \log (a+b \sinh (c+d x))+c f \log (a+b \sinh (c+d x))+\frac {1}{2} f (c+d x)^2\right )+a b d (e+f x) \sinh (c+d x)-a b f \cosh (c+d x)+b^2 d e \log (\sinh (c+d x))+\frac {1}{2} b^2 f \left ((c+d x) \left (2 \log \left (1-e^{-2 (c+d x)}\right )+c+d x\right )-\text {Li}_2\left (e^{-2 (c+d x)}\right )\right )-b^2 c f \log (\sinh (c+d x))}{a b^2 d^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 1108, normalized size = 3.44 \[ \text {result too large to display} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 932, normalized size = 2.89 \[ \frac {e \ln \left ({\mathrm e}^{d x +c}+1\right )}{a d}+\frac {e \ln \left ({\mathrm e}^{d x +c}-1\right )}{a d}-\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 \,x^{2}}{2 b^{2}}-\frac {e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d a}+\frac {f \dilog \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a}-\frac {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} a}-\frac {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} a}-\frac {f \dilog \left ({\mathrm e}^{d x +c}\right )}{d^{2} a}+\frac {2 a f c x}{d \,b^{2}}+\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 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 \,c^{2}}{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 \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 {\ln \left ({\mathrm e}^{d x +c}+1\right ) f x}{a d}-\frac {f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}-\frac {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 a}-\frac {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} a}-\frac {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 a}-\frac {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} a}+\frac {f c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} a} \]
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 \[ -\frac {1}{2} \, e {\left (\frac {2 \, {\left (d x + c\right )} a}{b^{2} d} - \frac {e^{\left (d x + c\right )}}{b d} + \frac {e^{\left (-d x - c\right )}}{b d} - \frac {2 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} - \frac {2 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} + \frac {2 \, {\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a b^{2} d}\right )} - \frac {1}{4} \, f {\left (\frac {2 \, {\left (a d^{2} x^{2} e^{c} - {\left (b d x e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + {\left (b d x + b\right )} e^{\left (-d x\right )}\right )} e^{\left (-c\right )}}{b^{2} d^{2}} - \int \frac {8 \, {\left ({\left (a^{3} e^{c} + a b^{2} e^{c}\right )} x e^{\left (d x\right )} - {\left (a^{2} b + b^{3}\right )} x\right )}}{a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} b^{2} e^{\left (d x + c\right )} - a b^{3}}\,{d x} + 4 \, \int \frac {x}{a e^{\left (d x + c\right )} + a}\,{d x} - 4 \, \int \frac {x}{a e^{\left (d x + c\right )} - a}\,{d x}\right )} \]
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 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {coth}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right ) \cosh ^{2}{\left (c + d x \right )} \coth {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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