Integrand size = 16, antiderivative size = 132 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=-\frac {7}{8} a c^4 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {17}{15} a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {3}{4} a^3 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{5} a^4 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5+\frac {7 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a} \]
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Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6310, 6313, 1821, 821, 272, 43, 65, 214} \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {7 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a}-\frac {7}{8} a c^4 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {17}{15} a^2 c^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {1}{5} a^4 c^4 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-\frac {3}{4} a^3 c^4 x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \]
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 821
Rule 1821
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = \left (a^4 c^4\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^4 x^4 \, dx \\ & = -\left (\left (a^4 c^4\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3 \sqrt {1-\frac {x^2}{a^2}}}{x^6} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {1}{5} a^4 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5+\frac {1}{5} \left (a^4 c^4\right ) \text {Subst}\left (\int \frac {\left (\frac {15}{a}-\frac {17 x}{a^2}+\frac {5 x^2}{a^3}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x^5} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {3}{4} a^3 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{5} a^4 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5-\frac {1}{20} \left (a^4 c^4\right ) \text {Subst}\left (\int \frac {\left (\frac {68}{a^2}-\frac {35 x}{a^3}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x^4} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {17}{15} a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {3}{4} a^3 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{5} a^4 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5+\frac {1}{4} \left (7 a c^4\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^3} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {17}{15} a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {3}{4} a^3 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{5} a^4 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5+\frac {1}{8} \left (7 a c^4\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a^2}}}{x^2} \, dx,x,\frac {1}{x^2}\right ) \\ & = -\frac {7}{8} a c^4 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {17}{15} a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {3}{4} a^3 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{5} a^4 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5-\frac {\left (7 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{16 a} \\ & = -\frac {7}{8} a c^4 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {17}{15} a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {3}{4} a^3 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{5} a^4 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5+\frac {1}{8} \left (7 a c^4\right ) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ & = -\frac {7}{8} a c^4 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {17}{15} a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {3}{4} a^3 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{5} a^4 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5+\frac {7 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.61 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {c^4 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-136-15 a x+112 a^2 x^2-90 a^3 x^3+24 a^4 x^4\right )+105 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{120 a} \]
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Time = 0.42 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {\left (24 a^{4} x^{4}-90 a^{3} x^{3}+112 a^{2} x^{2}-15 a x -136\right ) \left (a x -1\right ) c^{4}}{120 a \sqrt {\frac {a x -1}{a x +1}}}+\frac {7 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{8 \sqrt {a^{2}}\, \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(128\) |
default | \(\frac {\left (a x -1\right ) c^{4} \left (24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-90 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +16 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-105 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +120 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+105 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{120 a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(183\) |
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Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.95 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {105 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (24 \, a^{5} c^{4} x^{5} - 66 \, a^{4} c^{4} x^{4} + 22 \, a^{3} c^{4} x^{3} + 97 \, a^{2} c^{4} x^{2} - 151 \, a c^{4} x - 136 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{120 \, a} \]
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\[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=c^{4} \left (\int \left (- \frac {4 a x}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {6 a^{2} x^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {4 a^{3} x^{3}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {a^{4} x^{4}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (112) = 224\).
Time = 0.20 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.96 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {1}{120} \, {\left (\frac {105 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {105 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (105 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 790 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 896 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 490 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 105 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - a^{2}}\right )} a \]
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Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.05 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=-\frac {7 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{8 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} - \frac {1}{120} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (\frac {15 \, c^{4}}{\mathrm {sgn}\left (a x + 1\right )} - 2 \, {\left (\frac {56 \, a c^{4}}{\mathrm {sgn}\left (a x + 1\right )} + 3 \, {\left (\frac {4 \, a^{3} c^{4} x}{\mathrm {sgn}\left (a x + 1\right )} - \frac {15 \, a^{2} c^{4}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x\right )} x\right )} x + \frac {136 \, c^{4}}{a \mathrm {sgn}\left (a x + 1\right )}\right )} \]
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Time = 4.33 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.62 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {\frac {49\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{6}-\frac {7\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {224\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{15}+\frac {79\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{6}+\frac {7\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{4}}{a-\frac {5\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {10\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}}+\frac {7\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a} \]
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