Optimal. Leaf size=105 \[ \frac {26}{a c^4 (1-a x)}-\frac {22}{a c^4 (1-a x)^2}+\frac {41}{3 a c^4 (1-a x)^3}-\frac {5}{a c^4 (1-a x)^4}+\frac {4}{5 a c^4 (1-a x)^5}+\frac {8 \log (1-a x)}{a c^4}+\frac {x}{c^4} \]
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Rubi [A] time = 0.18, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6167, 6131, 6129, 88} \[ \frac {26}{a c^4 (1-a x)}-\frac {22}{a c^4 (1-a x)^2}+\frac {41}{3 a c^4 (1-a x)^3}-\frac {5}{a c^4 (1-a x)^4}+\frac {4}{5 a c^4 (1-a x)^5}+\frac {8 \log (1-a x)}{a c^4}+\frac {x}{c^4} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx\\ &=\frac {a^4 \int \frac {e^{4 \tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=\frac {a^4 \int \frac {x^4 (1+a x)^2}{(1-a x)^6} \, dx}{c^4}\\ &=\frac {a^4 \int \left (\frac {1}{a^4}+\frac {4}{a^4 (-1+a x)^6}+\frac {20}{a^4 (-1+a x)^5}+\frac {41}{a^4 (-1+a x)^4}+\frac {44}{a^4 (-1+a x)^3}+\frac {26}{a^4 (-1+a x)^2}+\frac {8}{a^4 (-1+a x)}\right ) \, dx}{c^4}\\ &=\frac {x}{c^4}+\frac {4}{5 a c^4 (1-a x)^5}-\frac {5}{a c^4 (1-a x)^4}+\frac {41}{3 a c^4 (1-a x)^3}-\frac {22}{a c^4 (1-a x)^2}+\frac {26}{a c^4 (1-a x)}+\frac {8 \log (1-a x)}{a c^4}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 79, normalized size = 0.75 \[ \frac {15 a^6 x^6-75 a^5 x^5-240 a^4 x^4+1080 a^3 x^3-1480 a^2 x^2+890 a x+120 (a x-1)^5 \log (1-a x)-202}{15 a c^4 (a x-1)^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 154, normalized size = 1.47 \[ \frac {15 \, a^{6} x^{6} - 75 \, a^{5} x^{5} - 240 \, a^{4} x^{4} + 1080 \, a^{3} x^{3} - 1480 \, a^{2} x^{2} + 890 \, a x + 120 \, {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (a x - 1\right ) - 202}{15 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 124, normalized size = 1.18 \[ \frac {a x - 1}{a c^{4}} - \frac {8 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c^{4}} - \frac {\frac {390 \, a^{9} c^{16}}{a x - 1} + \frac {330 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{2}} + \frac {205 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{3}} + \frac {75 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{4}} + \frac {12 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{5}}}{15 \, a^{10} c^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 96, normalized size = 0.91 \[ \frac {x}{c^{4}}+\frac {8 \ln \left (a x -1\right )}{a \,c^{4}}-\frac {26}{a \,c^{4} \left (a x -1\right )}-\frac {5}{a \,c^{4} \left (a x -1\right )^{4}}-\frac {4}{5 a \,c^{4} \left (a x -1\right )^{5}}-\frac {41}{3 a \,c^{4} \left (a x -1\right )^{3}}-\frac {22}{a \,c^{4} \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 113, normalized size = 1.08 \[ -\frac {390 \, a^{4} x^{4} - 1230 \, a^{3} x^{3} + 1555 \, a^{2} x^{2} - 905 \, a x + 202}{15 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} + \frac {x}{c^{4}} + \frac {8 \, \log \left (a x - 1\right )}{a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 109, normalized size = 1.04 \[ \frac {x}{c^4}+\frac {\frac {311\,a\,x^2}{3}-\frac {181\,x}{3}+\frac {202}{15\,a}-82\,a^2\,x^3+26\,a^3\,x^4}{-a^5\,c^4\,x^5+5\,a^4\,c^4\,x^4-10\,a^3\,c^4\,x^3+10\,a^2\,c^4\,x^2-5\,a\,c^4\,x+c^4}+\frac {8\,\ln \left (a\,x-1\right )}{a\,c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 114, normalized size = 1.09 \[ \frac {- 390 a^{4} x^{4} + 1230 a^{3} x^{3} - 1555 a^{2} x^{2} + 905 a x - 202}{15 a^{6} c^{4} x^{5} - 75 a^{5} c^{4} x^{4} + 150 a^{4} c^{4} x^{3} - 150 a^{3} c^{4} x^{2} + 75 a^{2} c^{4} x - 15 a c^{4}} + \frac {x}{c^{4}} + \frac {8 \log {\left (a x - 1 \right )}}{a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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