Integrand size = 18, antiderivative size = 17 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {c^2 (1+a x)^3}{3 a} \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6264, 32} \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {c^2 (a x+1)^3}{3 a} \]
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Rule 32
Rule 6264
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int e^{4 \text {arctanh}(a x)} (c-a c x)^2 \, dx \\ & = c^2 \int (1+a x)^2 \, dx \\ & = \frac {c^2 (1+a x)^3}{3 a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=c^2 \left (x+a x^2+\frac {a^2 x^3}{3}\right ) \]
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Time = 0.56 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {c^{2} \left (a x +1\right )^{3}}{3 a}\) | \(16\) |
gosper | \(\frac {x \left (a^{2} x^{2}+3 a x +3\right ) c^{2}}{3}\) | \(20\) |
parallelrisch | \(\frac {1}{3} a^{2} c^{2} x^{3}+a \,c^{2} x^{2}+c^{2} x\) | \(26\) |
risch | \(\frac {a^{2} c^{2} x^{3}}{3}+a \,c^{2} x^{2}+c^{2} x +\frac {c^{2}}{3 a}\) | \(34\) |
norman | \(\frac {-\frac {c^{2}}{a}+\frac {2 a^{2} c^{2} x^{3}}{3}+\frac {a^{3} c^{2} x^{4}}{3}}{a x -1}\) | \(40\) |
meijerg | \(-\frac {c^{2} \left (-\frac {a x \left (-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{15 \left (-a x +1\right )}-4 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{2} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}+\frac {c^{2} x}{-a x +1}\) | \(103\) |
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Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.47 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {a^{2} c^{2} x^{3}}{3} + a c^{2} x^{2} + c^{2} x \]
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Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.47 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (15) = 30\).
Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.35 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {{\left (c^{2} + \frac {6 \, c^{2}}{a x - 1} + \frac {12 \, c^{2}}{{\left (a x - 1\right )}^{2}}\right )} {\left (a x - 1\right )}^{3}}{3 \, a} \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {c^2\,x\,\left (a^2\,x^2+3\,a\,x+3\right )}{3} \]
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