Optimal. Leaf size=89 \[ \frac {19}{a c^3 (1-a x)}-\frac {25}{2 a c^3 (1-a x)^2}+\frac {16}{3 a c^3 (1-a x)^3}-\frac {1}{a c^3 (1-a x)^4}+\frac {7 \log (1-a x)}{a c^3}+\frac {x}{c^3} \]
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Rubi [A] time = 0.17, antiderivative size = 89, 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} \[ \frac {19}{a c^3 (1-a x)}-\frac {25}{2 a c^3 (1-a x)^2}+\frac {16}{3 a c^3 (1-a x)^3}-\frac {1}{a c^3 (1-a x)^4}+\frac {7 \log (1-a x)}{a c^3}+\frac {x}{c^3} \]
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 )^3} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx\\ &=-\frac {a^3 \int \frac {e^{4 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=-\frac {a^3 \int \frac {x^3 (1+a x)^2}{(1-a x)^5} \, dx}{c^3}\\ &=-\frac {a^3 \int \left (-\frac {1}{a^3}-\frac {4}{a^3 (-1+a x)^5}-\frac {16}{a^3 (-1+a x)^4}-\frac {25}{a^3 (-1+a x)^3}-\frac {19}{a^3 (-1+a x)^2}-\frac {7}{a^3 (-1+a x)}\right ) \, dx}{c^3}\\ &=\frac {x}{c^3}-\frac {1}{a c^3 (1-a x)^4}+\frac {16}{3 a c^3 (1-a x)^3}-\frac {25}{2 a c^3 (1-a x)^2}+\frac {19}{a c^3 (1-a x)}+\frac {7 \log (1-a x)}{a c^3}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 71, normalized size = 0.80 \[ \frac {6 a^5 x^5-24 a^4 x^4-78 a^3 x^3+243 a^2 x^2-218 a x+42 (a x-1)^4 \log (1-a x)+65}{6 a c^3 (a x-1)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 126, normalized size = 1.42 \[ \frac {6 \, a^{5} x^{5} - 24 \, a^{4} x^{4} - 78 \, a^{3} x^{3} + 243 \, a^{2} x^{2} - 218 \, a x + 42 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x - 1\right ) + 65}{6 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 109, normalized size = 1.22 \[ \frac {a x - 1}{a c^{3}} - \frac {7 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c^{3}} - \frac {\frac {114 \, a^{7} c^{9}}{a x - 1} + \frac {75 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{2}} + \frac {32 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{4}}}{6 \, a^{8} c^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 81, normalized size = 0.91 \[ \frac {x}{c^{3}}+\frac {7 \ln \left (a x -1\right )}{a \,c^{3}}-\frac {19}{a \,c^{3} \left (a x -1\right )}-\frac {1}{a \,c^{3} \left (a x -1\right )^{4}}-\frac {16}{3 a \,c^{3} \left (a x -1\right )^{3}}-\frac {25}{2 a \,c^{3} \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 93, normalized size = 1.04 \[ -\frac {114 \, a^{3} x^{3} - 267 \, a^{2} x^{2} + 224 \, a x - 65}{6 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} + \frac {x}{c^{3}} + \frac {7 \, \log \left (a x - 1\right )}{a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 90, normalized size = 1.01 \[ \frac {x}{c^3}-\frac {\frac {112\,x}{3}-\frac {89\,a\,x^2}{2}-\frac {65}{6\,a}+19\,a^2\,x^3}{a^4\,c^3\,x^4-4\,a^3\,c^3\,x^3+6\,a^2\,c^3\,x^2-4\,a\,c^3\,x+c^3}+\frac {7\,\ln \left (a\,x-1\right )}{a\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 94, normalized size = 1.06 \[ \frac {- 114 a^{3} x^{3} + 267 a^{2} x^{2} - 224 a x + 65}{6 a^{5} c^{3} x^{4} - 24 a^{4} c^{3} x^{3} + 36 a^{3} c^{3} x^{2} - 24 a^{2} c^{3} x + 6 a c^{3}} + \frac {x}{c^{3}} + \frac {7 \log {\left (a x - 1 \right )}}{a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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