Integrand size = 18, antiderivative size = 53 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {c^5 (1-a x)^4}{a}+\frac {4 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a} \]
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Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6264, 45} \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {c^5 (1-a x)^6}{6 a}+\frac {4 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^4}{a} \]
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Rule 45
Rule 6264
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int e^{4 \text {arctanh}(a x)} (c-a c x)^5 \, dx \\ & = c^5 \int (1-a x)^3 (1+a x)^2 \, dx \\ & = c^5 \int \left (4 (1-a x)^3-4 (1-a x)^4+(1-a x)^5\right ) \, dx \\ & = -\frac {c^5 (1-a x)^4}{a}+\frac {4 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.58 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {c^5 (-1+a x)^4 \left (11+14 a x+5 a^2 x^2\right )}{30 a} \]
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Time = 0.58 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(-\frac {x \left (5 a^{5} x^{5}-6 a^{4} x^{4}-15 a^{3} x^{3}+20 a^{2} x^{2}+15 a x -30\right ) c^{5}}{30}\) | \(45\) |
default | \(c^{5} \left (-\frac {1}{6} a^{5} x^{6}+\frac {1}{5} a^{4} x^{5}+\frac {1}{2} a^{3} x^{4}-\frac {2}{3} a^{2} x^{3}-\frac {1}{2} a \,x^{2}+x \right )\) | \(45\) |
risch | \(-\frac {1}{6} a^{5} c^{5} x^{6}+\frac {1}{5} a^{4} c^{5} x^{5}+\frac {1}{2} a^{3} c^{5} x^{4}-\frac {2}{3} c^{5} a^{2} x^{3}-\frac {1}{2} a \,c^{5} x^{2}+c^{5} x\) | \(60\) |
parallelrisch | \(-\frac {1}{6} a^{5} c^{5} x^{6}+\frac {1}{5} a^{4} c^{5} x^{5}+\frac {1}{2} a^{3} c^{5} x^{4}-\frac {2}{3} c^{5} a^{2} x^{3}-\frac {1}{2} a \,c^{5} x^{2}+c^{5} x\) | \(60\) |
norman | \(\frac {-c^{5} x +\frac {3}{2} a \,c^{5} x^{2}-\frac {7}{6} a^{3} c^{5} x^{4}+\frac {3}{10} a^{4} c^{5} x^{5}+\frac {11}{30} a^{5} c^{5} x^{6}-\frac {1}{6} a^{6} c^{5} x^{7}+\frac {1}{6} c^{5} a^{2} x^{3}}{a x -1}\) | \(80\) |
meijerg | \(-\frac {c^{5} \left (\frac {a x \left (-20 a^{6} x^{6}-28 a^{5} x^{5}-42 a^{4} x^{4}-70 a^{3} x^{3}-140 a^{2} x^{2}-420 a x +840\right )}{-120 a x +120}+7 \ln \left (-a x +1\right )\right )}{a}-\frac {3 c^{5} \left (-\frac {a x \left (-14 a^{5} x^{5}-21 a^{4} x^{4}-35 a^{3} x^{3}-70 a^{2} x^{2}-210 a x +420\right )}{70 \left (-a x +1\right )}-6 \ln \left (-a x +1\right )\right )}{a}-\frac {c^{5} \left (\frac {a x \left (-3 a^{4} x^{4}-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{-12 a x +12}+5 \ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (-\frac {a x \left (-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{15 \left (-a x +1\right )}-4 \ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (\frac {a x \left (-2 a^{2} x^{2}-6 a x +12\right )}{-4 a x +4}+3 \ln \left (-a x +1\right )\right )}{a}-\frac {c^{5} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {3 c^{5} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} x}{-a x +1}\) | \(378\) |
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Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {1}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {1}{2} \, a c^{5} x^{2} + c^{5} x \]
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Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.19 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=- \frac {a^{5} c^{5} x^{6}}{6} + \frac {a^{4} c^{5} x^{5}}{5} + \frac {a^{3} c^{5} x^{4}}{2} - \frac {2 a^{2} c^{5} x^{3}}{3} - \frac {a c^{5} x^{2}}{2} + c^{5} x \]
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Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {1}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {1}{2} \, a c^{5} x^{2} + c^{5} x \]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {{\left (5 \, c^{5} + \frac {24 \, c^{5}}{a x - 1} + \frac {30 \, c^{5}}{{\left (a x - 1\right )}^{2}}\right )} {\left (a x - 1\right )}^{6}}{30 \, a} \]
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Time = 4.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {a^5\,c^5\,x^6}{6}+\frac {a^4\,c^5\,x^5}{5}+\frac {a^3\,c^5\,x^4}{2}-\frac {2\,a^2\,c^5\,x^3}{3}-\frac {a\,c^5\,x^2}{2}+c^5\,x \]
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