Optimal. Leaf size=324 \[ \frac {f \left (a^2+b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {f \left (a^2+b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {\left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 b d}+\frac {\left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {b f \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a^2 d^2}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {b (e+f x)^2}{2 a^2 f}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d} \]
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Rubi [A] time = 0.74, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 15, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {5585, 5450, 3296, 2638, 5452, 3770, 5446, 2635, 8, 3716, 2190, 2279, 2391, 5565, 5561} \[ \frac {f \left (a^2+b^2\right ) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {f \left (a^2+b^2\right ) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a^2 b d^2}-\frac {b f \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 b d}+\frac {\left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {b (e+f x)^2}{2 a^2 f}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d} \]
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 3770
Rule 5446
Rule 5450
Rule 5452
Rule 5561
Rule 5565
Rule 5585
Rubi steps
\begin {align*} \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \cosh (c+d x) \coth ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {\int (e+f x) \cosh (c+d x) \, dx}{a}+\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x) \, dx}{a}-\frac {b \int (e+f x) \cosh ^2(c+d x) \coth (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}\\ &=-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {(e+f x) \sinh (c+d x)}{a d}-\frac {\int (e+f x) \cosh (c+d x) \, dx}{a}-\frac {b \int (e+f x) \coth (c+d x) \, dx}{a^2}+\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {f \int \text {csch}(c+d x) \, dx}{a d}-\frac {f \int \sinh (c+d x) \, dx}{a d}\\ &=\frac {b (e+f x)^2}{2 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {f \cosh (c+d x)}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {(2 b) \int \frac {e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a^2}+\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^2}+\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^2}+\frac {f \int \sinh (c+d x) \, dx}{a d}\\ &=\frac {b (e+f x)^2}{2 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a 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^2 b 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^2 b d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {(b f) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^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^2 b 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^2 b d}\\ &=\frac {b (e+f x)^2}{2 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a 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^2 b 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^2 b d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {(b f) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^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^2 b 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^2 b d^2}\\ &=\frac {b (e+f x)^2}{2 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a 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^2 b 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^2 b d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 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^2 b 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^2 b d^2}-\frac {b f \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a^2 d^2}\\ \end {align*}
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Mathematica [A] time = 2.38, size = 313, normalized size = 0.97 \[ \frac {\frac {2 \left (a^2+b^2\right ) \left (f \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))-\frac {1}{2} f (c+d x)^2\right )}{b}+a d (e+f x) \tanh \left (\frac {1}{2} (c+d x)\right )-a d (e+f x) \coth \left (\frac {1}{2} (c+d x)\right )+2 a f \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-2 b d e \log (\sinh (c+d x))+b f \left (\text {Li}_2\left (e^{-2 (c+d x)}\right )-(c+d x) \left (2 \log \left (1-e^{-2 (c+d x)}\right )+c+d x\right )\right )+2 b c f \log (\sinh (c+d x))}{2 a^2 d^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 1735, normalized size = 5.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 0.32, size = 938, normalized size = 2.90 \[ -\frac {f \,x^{2}}{2 b}+\frac {b f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a^{2}}-\frac {b f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d \,a^{2}}+\frac {e x}{b}+\frac {b 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^{2}}-\frac {b f c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} a^{2}}+\frac {b 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^{2}}+\frac {b 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^{2}}+\frac {b 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^{2}}-\frac {f \,c^{2}}{d^{2} b}+\frac {e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d b}-\frac {2 e \ln \left ({\mathrm e}^{d x +c}\right )}{d b}+\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} b}+\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} b}-\frac {2 \left (f x +e \right ) {\mathrm e}^{d x +c}}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {2 f c x}{d b}-\frac {f c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b}+\frac {2 f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b}+\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 b}+\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} b}+\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 b}+\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} b}-\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a}+\frac {f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}+\frac {b e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,a^{2}}+\frac {b 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^{2}}+\frac {b 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^{2}}-\frac {b e \ln \left ({\mathrm e}^{d x +c}+1\right )}{d \,a^{2}}-\frac {b e \ln \left ({\mathrm e}^{d x +c}-1\right )}{d \,a^{2}}-\frac {b f \dilog \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a^{2}}+\frac {b f \dilog \left ({\mathrm e}^{d x +c}\right )}{d^{2} a^{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 \[ \frac {1}{2} \, {\left (2 \, b d \int \frac {x}{a^{2} d e^{\left (d x + c\right )} + a^{2} d}\,{d x} - 2 \, b d \int \frac {x}{a^{2} d e^{\left (d x + c\right )} - a^{2} d}\,{d x} + 2 \, a {\left (\frac {d x + c}{a^{2} d^{2}} - \frac {\log \left (e^{\left (d x + c\right )} + 1\right )}{a^{2} d^{2}}\right )} - 2 \, a {\left (\frac {d x + c}{a^{2} d^{2}} - \frac {\log \left (e^{\left (d x + c\right )} - 1\right )}{a^{2} d^{2}}\right )} + \frac {a d x^{2} e^{\left (2 \, d x + 2 \, c\right )} - a d x^{2} - 4 \, b x e^{\left (d x + c\right )}}{a b d e^{\left (2 \, d x + 2 \, c\right )} - a b d} - \int \frac {4 \, {\left ({\left (a^{3} e^{c} + a b^{2} e^{c}\right )} x e^{\left (d x\right )} - {\left (a^{2} b + b^{3}\right )} x\right )}}{a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b e^{\left (d x + c\right )} - a^{2} b^{2}}\,{d x}\right )} f + e {\left (\frac {d x + c}{b d} + \frac {2 \, e^{\left (-d x - c\right )}}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} d} - \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{2} b d}\right )} \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {coth}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right ) \cosh {\left (c + d x \right )} \coth ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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