Integrand size = 18, antiderivative size = 66 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {4 c (c-a c x)^{-1+p}}{a (1-p)}+\frac {4 (c-a c x)^p}{a p}-\frac {(c-a c x)^{1+p}}{a c (1+p)} \]
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Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6302, 6265, 21, 45} \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {4 c (c-a c x)^{p-1}}{a (1-p)}+\frac {4 (c-a c x)^p}{a p}-\frac {(c-a c x)^{p+1}}{a c (p+1)} \]
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Rule 21
Rule 45
Rule 6265
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int e^{4 \text {arctanh}(a x)} (c-a c x)^p \, dx \\ & = \int \frac {(1+a x)^2 (c-a c x)^p}{(1-a x)^2} \, dx \\ & = c^2 \int (1+a x)^2 (c-a c x)^{-2+p} \, dx \\ & = c^2 \int \left (4 (c-a c x)^{-2+p}-\frac {4 (c-a c x)^{-1+p}}{c}+\frac {(c-a c x)^p}{c^2}\right ) \, dx \\ & = \frac {4 c (c-a c x)^{-1+p}}{a (1-p)}+\frac {4 (c-a c x)^p}{a p}-\frac {(c-a c x)^{1+p}}{a c (1+p)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {(c-a c x)^p \left (\frac {4+3 p}{p (1+p)}+\frac {a x}{1+p}+\frac {4}{(-1+p) (-1+a x)}\right )}{a} \]
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Time = 0.52 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.12
method | result | size |
gosper | \(\frac {\left (a^{2} p^{2} x^{2}-a^{2} x^{2} p +2 a \,p^{2} x +2 p a x -4 a x +p^{2}+3 p +4\right ) \left (-a c x +c \right )^{p}}{\left (a x -1\right ) a p \left (p^{2}-1\right )}\) | \(74\) |
risch | \(\frac {\left (a^{2} p^{2} x^{2}-a^{2} x^{2} p +2 a \,p^{2} x +2 p a x -4 a x +p^{2}+3 p +4\right ) \left (-a c x +c \right )^{p}}{a p \left (p +1\right ) \left (-1+p \right ) \left (a x -1\right )}\) | \(77\) |
norman | \(\frac {\frac {a \,x^{2} {\mathrm e}^{p \ln \left (-a c x +c \right )}}{p +1}+\frac {\left (p^{2}+3 p +4\right ) {\mathrm e}^{p \ln \left (-a c x +c \right )}}{a p \left (p^{2}-1\right )}+\frac {2 \left (2+p \right ) x \,{\mathrm e}^{p \ln \left (-a c x +c \right )}}{p \left (p +1\right )}}{a x -1}\) | \(89\) |
parallelrisch | \(\frac {x^{2} \left (-a c x +c \right )^{p} a^{2} p^{2}-x^{2} \left (-a c x +c \right )^{p} a^{2} p +2 x \left (-a c x +c \right )^{p} a \,p^{2}+2 x \left (-a c x +c \right )^{p} a p -4 \left (-a c x +c \right )^{p} x a +\left (-a c x +c \right )^{p} p^{2}+3 \left (-a c x +c \right )^{p} p +4 \left (-a c x +c \right )^{p}}{\left (a x -1\right ) a p \left (p^{2}-1\right )}\) | \(139\) |
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none
Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.23 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^p \, dx=-\frac {{\left ({\left (a^{2} p^{2} - a^{2} p\right )} x^{2} + p^{2} + 2 \, {\left (a p^{2} + a p - 2 \, a\right )} x + 3 \, p + 4\right )} {\left (-a c x + c\right )}^{p}}{a p^{3} - a p - {\left (a^{2} p^{3} - a^{2} p\right )} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (48) = 96\).
Time = 0.58 (sec) , antiderivative size = 530, normalized size of antiderivative = 8.03 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\begin {cases} c^{p} x & \text {for}\: a = 0 \\- \frac {a^{2} x^{2} \log {\left (x - \frac {1}{a} \right )}}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac {2 a x \log {\left (x - \frac {1}{a} \right )}}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac {4 a x}{a^{3} c x^{2} - 2 a^{2} c x + a c} - \frac {\log {\left (x - \frac {1}{a} \right )}}{a^{3} c x^{2} - 2 a^{2} c x + a c} - \frac {2}{a^{3} c x^{2} - 2 a^{2} c x + a c} & \text {for}\: p = -1 \\\frac {a^{2} x^{2}}{a^{2} x - a} + \frac {4 a x \log {\left (x - \frac {1}{a} \right )}}{a^{2} x - a} - \frac {4 \log {\left (x - \frac {1}{a} \right )}}{a^{2} x - a} - \frac {5}{a^{2} x - a} & \text {for}\: p = 0 \\- \frac {a c x^{2}}{2} - 3 c x - \frac {4 c \log {\left (x - \frac {1}{a} \right )}}{a} & \text {for}\: p = 1 \\\frac {a^{2} p^{2} x^{2} \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} - \frac {a^{2} p x^{2} \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac {2 a p^{2} x \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac {2 a p x \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} - \frac {4 a x \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac {p^{2} \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac {3 p \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac {4 \left (- a c x + c\right )^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (64) = 128\).
Time = 0.23 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.32 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {{\left ({\left (p^{2} - p\right )} a^{2} c^{p} x^{2} + 2 \, a c^{p} {\left (p - 1\right )} x + 2 \, c^{p}\right )} {\left (-a x + 1\right )}^{p} a^{2}}{{\left (p^{3} - p\right )} a^{4} x - {\left (p^{3} - p\right )} a^{3}} + \frac {2 \, {\left (a c^{p} {\left (p - 1\right )} x + c^{p}\right )} {\left (-a x + 1\right )}^{p} a}{{\left (p^{2} - p\right )} a^{3} x - {\left (p^{2} - p\right )} a^{2}} + \frac {{\left (-a x + 1\right )}^{p} c^{p}}{a^{2} {\left (p - 1\right )} x - a {\left (p - 1\right )}} \]
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\[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (a x + 1\right )}^{2} {\left (-a c x + c\right )}^{p}}{{\left (a x - 1\right )}^{2}} \,d x } \]
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Time = 4.38 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {4\,{\left (c-a\,c\,x\right )}^p}{a\,\left (a\,x-1\right )\,\left (p-1\right )}+\frac {{\left (c-a\,c\,x\right )}^p\,\left (3\,p+a\,p\,x+4\right )}{a\,p\,\left (p+1\right )} \]
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