Optimal. Leaf size=322 \[ -\frac {(e+f x)^2}{2 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a b^2 f}-\frac {f \cosh (c+d x)}{b d^2}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d}+\frac {(e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {\left (a^2+b^2\right ) f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {\left (a^2+b^2\right ) f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^2}+\frac {f \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {(e+f x) \sinh (c+d x)}{b d} \]
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Rubi [A]
time = 0.42, 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 = {5704, 5558, 5554, 2715, 8, 3797, 2221, 2317, 2438, 5684, 3377, 2718, 5680}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 2715
Rule 2718
Rule 3377
Rule 3797
Rule 5554
Rule 5558
Rule 5680
Rule 5684
Rule 5704
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 \text {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 ) \text {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 ) \text {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 = 1.20, size = 296, normalized size = 0.92 \begin {gather*} \frac {-a b f \cosh (c+d x)+b^2 d e \log (\sinh (c+d x))-b^2 c f \log (\sinh (c+d x))+\frac {1}{2} b^2 f \left ((c+d x) \left (c+d x+2 \log \left (1-e^{-2 (c+d x)}\right )\right )-\text {PolyLog}\left (2,e^{-2 (c+d x)}\right )\right )+\left (a^2+b^2\right ) \left (\frac {1}{2} f (c+d x)^2-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-d e \log (a+b \sinh (c+d x))+c f \log (a+b \sinh (c+d x))-f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )+a b d (e+f x) \sinh (c+d x)}{a b^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(931\) vs.
\(2(304)=608\).
time = 10.71, size = 932, normalized size = 2.89
method | result | size |
risch | \(-\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 \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{2}}+\frac {2 e a \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 ) 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 \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \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 \,x^{2}}{2 b^{2}}-\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 {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 {f a \,c^{2}}{d^{2} b^{2}}+\frac {\left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 d^{2} b}+\frac {2 f a c x}{d \,b^{2}}+\frac {f \dilog \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a}-\frac {f \dilog \left ({\mathrm e}^{d x +c}\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 {a e x}{b^{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 ) 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 \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 {\left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 d^{2} b}+\frac {e \ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}-\frac {e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d a}+\frac {e \ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}+\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d a}-\frac {f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{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}-\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}}\) | \(932\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1266 vs.
\(2 (307) = 614\).
time = 0.44, size = 1266, normalized size = 3.93 \begin {gather*} -\frac {a b d f x + a b d \cosh \left (1\right ) + a b d \sinh \left (1\right ) + a b f - {\left (a b d f x + a b d \cosh \left (1\right ) + a b d \sinh \left (1\right ) - a b f\right )} \cosh \left (d x + c\right )^{2} - {\left (a b d f x + a b d \cosh \left (1\right ) + a b d \sinh \left (1\right ) - a b f\right )} \sinh \left (d x + c\right )^{2} - {\left (a^{2} d^{2} f x^{2} - 2 \, a^{2} c^{2} f + 2 \, {\left (a^{2} d^{2} x + 2 \, a^{2} c d\right )} \cosh \left (1\right ) + 2 \, {\left (a^{2} d^{2} x + 2 \, a^{2} c d\right )} \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) + 2 \, {\left ({\left (a^{2} + b^{2}\right )} f \cosh \left (d x + c\right ) + {\left (a^{2} + b^{2}\right )} 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^{2} + b^{2}\right )} f \cosh \left (d x + c\right ) + {\left (a^{2} + b^{2}\right )} 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 (b^{2} f \cosh \left (d x + c\right ) + b^{2} f \sinh \left (d x + c\right )\right )} {\rm Li}_2\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - 2 \, {\left (b^{2} f \cosh \left (d x + c\right ) + b^{2} f \sinh \left (d x + c\right )\right )} {\rm Li}_2\left (-\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )\right ) - 2 \, {\left ({\left ({\left (a^{2} + b^{2}\right )} c f - {\left (a^{2} + b^{2}\right )} d \cosh \left (1\right ) - {\left (a^{2} + b^{2}\right )} d \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{2} + b^{2}\right )} c f - {\left (a^{2} + b^{2}\right )} d \cosh \left (1\right ) - {\left (a^{2} + b^{2}\right )} d \sinh \left (1\right )\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 ({\left (a^{2} + b^{2}\right )} c f - {\left (a^{2} + b^{2}\right )} d \cosh \left (1\right ) - {\left (a^{2} + b^{2}\right )} d \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{2} + b^{2}\right )} c f - {\left (a^{2} + b^{2}\right )} d \cosh \left (1\right ) - {\left (a^{2} + b^{2}\right )} d \sinh \left (1\right )\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 ({\left (a^{2} + b^{2}\right )} d f x + {\left (a^{2} + b^{2}\right )} c f\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{2} + b^{2}\right )} d f x + {\left (a^{2} + b^{2}\right )} 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 ({\left (a^{2} + b^{2}\right )} d f x + {\left (a^{2} + b^{2}\right )} c f\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{2} + b^{2}\right )} d f x + {\left (a^{2} + b^{2}\right )} 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 (b^{2} d f x + b^{2} d \cosh \left (1\right ) + b^{2} d \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) + {\left (b^{2} d f x + b^{2} d \cosh \left (1\right ) + b^{2} d \sinh \left (1\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 2 \, {\left ({\left (b^{2} c f - b^{2} d \cosh \left (1\right ) - b^{2} d \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) + {\left (b^{2} c f - b^{2} d \cosh \left (1\right ) - b^{2} d \sinh \left (1\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) - 2 \, {\left ({\left (b^{2} d f x + b^{2} c f\right )} \cosh \left (d x + c\right ) + {\left (b^{2} d f x + b^{2} c f\right )} \sinh \left (d x + c\right )\right )} \log \left (-\cosh \left (d x + c\right ) - \sinh \left (d x + c\right ) + 1\right ) - {\left (a^{2} d^{2} f x^{2} - 2 \, a^{2} c^{2} f + 2 \, {\left (a^{2} d^{2} x + 2 \, a^{2} c d\right )} \cosh \left (1\right ) + 2 \, {\left (a b d f x + a b d \cosh \left (1\right ) + a b d \sinh \left (1\right ) - a b f\right )} \cosh \left (d x + c\right ) + 2 \, {\left (a^{2} d^{2} x + 2 \, a^{2} c d\right )} \sinh \left (1\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (a b^{2} d^{2} \cosh \left (d x + c\right ) + a b^{2} d^{2} \sinh \left (d x + c\right )\right )}} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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