Optimal. Leaf size=111 \[ \frac {111}{16 a c^3 (1-a x)}-\frac {49}{16 a c^3 (1-a x)^2}+\frac {11}{12 a c^3 (1-a x)^3}-\frac {1}{8 a c^3 (1-a x)^4}+\frac {129 \log (1-a x)}{32 a c^3}-\frac {\log (a x+1)}{32 a c^3}+\frac {x}{c^3} \]
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Rubi [A] time = 0.20, antiderivative size = 111, 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, 6157, 6150, 88} \[ \frac {111}{16 a c^3 (1-a x)}-\frac {49}{16 a c^3 (1-a x)^2}+\frac {11}{12 a c^3 (1-a x)^3}-\frac {1}{8 a c^3 (1-a x)^4}+\frac {129 \log (1-a x)}{32 a c^3}-\frac {\log (a x+1)}{32 a c^3}+\frac {x}{c^3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6157
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx\\ &=-\frac {a^6 \int \frac {e^{4 \tanh ^{-1}(a x)} x^6}{\left (1-a^2 x^2\right )^3} \, dx}{c^3}\\ &=-\frac {a^6 \int \frac {x^6}{(1-a x)^5 (1+a x)} \, dx}{c^3}\\ &=-\frac {a^6 \int \left (-\frac {1}{a^6}-\frac {1}{2 a^6 (-1+a x)^5}-\frac {11}{4 a^6 (-1+a x)^4}-\frac {49}{8 a^6 (-1+a x)^3}-\frac {111}{16 a^6 (-1+a x)^2}-\frac {129}{32 a^6 (-1+a x)}+\frac {1}{32 a^6 (1+a x)}\right ) \, dx}{c^3}\\ &=\frac {x}{c^3}-\frac {1}{8 a c^3 (1-a x)^4}+\frac {11}{12 a c^3 (1-a x)^3}-\frac {49}{16 a c^3 (1-a x)^2}+\frac {111}{16 a c^3 (1-a x)}+\frac {129 \log (1-a x)}{32 a c^3}-\frac {\log (1+a x)}{32 a c^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 89, normalized size = 0.80 \[ \frac {2 \left (48 a^5 x^5-192 a^4 x^4-45 a^3 x^3+660 a^2 x^2-701 a x+224\right )+387 (a x-1)^4 \log (1-a x)-3 (a x-1)^4 \log (a x+1)}{96 a c^3 (a x-1)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 163, normalized size = 1.47 \[ \frac {96 \, a^{5} x^{5} - 384 \, a^{4} x^{4} - 90 \, a^{3} x^{3} + 1320 \, a^{2} x^{2} - 1402 \, a x - 3 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x + 1\right ) + 387 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x - 1\right ) + 448}{96 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 130, normalized size = 1.17 \[ \frac {a x - 1}{a c^{3}} - \frac {4 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c^{3}} - \frac {\log \left ({\left | -\frac {2}{a x - 1} - 1 \right |}\right )}{32 \, a c^{3}} - \frac {\frac {333 \, a^{11} c^{9}}{a x - 1} + \frac {147 \, a^{11} c^{9}}{{\left (a x - 1\right )}^{2}} + \frac {44 \, a^{11} c^{9}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, a^{11} c^{9}}{{\left (a x - 1\right )}^{4}}}{48 \, a^{12} c^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 95, normalized size = 0.86 \[ \frac {x}{c^{3}}-\frac {1}{8 a \,c^{3} \left (a x -1\right )^{4}}-\frac {11}{12 c^{3} a \left (a x -1\right )^{3}}-\frac {49}{16 c^{3} a \left (a x -1\right )^{2}}-\frac {111}{16 a \,c^{3} \left (a x -1\right )}+\frac {129 \ln \left (a x -1\right )}{32 c^{3} a}-\frac {\ln \left (a x +1\right )}{32 a \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 107, normalized size = 0.96 \[ -\frac {333 \, a^{3} x^{3} - 852 \, a^{2} x^{2} + 749 \, a x - 224}{48 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} + \frac {x}{c^{3}} - \frac {\log \left (a x + 1\right )}{32 \, a c^{3}} + \frac {129 \, \log \left (a x - 1\right )}{32 \, a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 104, normalized size = 0.94 \[ \frac {x}{c^3}-\frac {\frac {749\,x}{48}-\frac {71\,a\,x^2}{4}-\frac {14}{3\,a}+\frac {111\,a^2\,x^3}{16}}{a^4\,c^3\,x^4-4\,a^3\,c^3\,x^3+6\,a^2\,c^3\,x^2-4\,a\,c^3\,x+c^3}+\frac {129\,\ln \left (a\,x-1\right )}{32\,a\,c^3}-\frac {\ln \left (a\,x+1\right )}{32\,a\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 114, normalized size = 1.03 \[ a^{6} \left (\frac {- 333 a^{3} x^{3} + 852 a^{2} x^{2} - 749 a x + 224}{48 a^{11} c^{3} x^{4} - 192 a^{10} c^{3} x^{3} + 288 a^{9} c^{3} x^{2} - 192 a^{8} c^{3} x + 48 a^{7} c^{3}} + \frac {x}{a^{6} c^{3}} + \frac {\frac {129 \log {\left (x - \frac {1}{a} \right )}}{32} - \frac {\log {\left (x + \frac {1}{a} \right )}}{32}}{a^{7} c^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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