Optimal. Leaf size=63 \[ \frac {c^3}{5 a^6 x^5}+\frac {c^3}{a^5 x^4}+\frac {5 c^3}{3 a^4 x^3}-\frac {5 c^3}{a^2 x}+\frac {4 c^3 \log (x)}{a}+c^3 x \]
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Rubi [A] time = 0.15, antiderivative size = 63, 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, 75} \[ \frac {5 c^3}{3 a^4 x^3}+\frac {c^3}{a^5 x^4}+\frac {c^3}{5 a^6 x^5}-\frac {5 c^3}{a^2 x}+\frac {4 c^3 \log (x)}{a}+c^3 x \]
Antiderivative was successfully verified.
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Rule 75
Rule 6150
Rule 6157
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx\\ &=-\frac {c^3 \int \frac {e^{4 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6}\\ &=-\frac {c^3 \int \frac {(1-a x) (1+a x)^5}{x^6} \, dx}{a^6}\\ &=-\frac {c^3 \int \left (-a^6+\frac {1}{x^6}+\frac {4 a}{x^5}+\frac {5 a^2}{x^4}-\frac {5 a^4}{x^2}-\frac {4 a^5}{x}\right ) \, dx}{a^6}\\ &=\frac {c^3}{5 a^6 x^5}+\frac {c^3}{a^5 x^4}+\frac {5 c^3}{3 a^4 x^3}-\frac {5 c^3}{a^2 x}+c^3 x+\frac {4 c^3 \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 1.00 \[ \frac {c^3}{5 a^6 x^5}+\frac {c^3}{a^5 x^4}+\frac {5 c^3}{3 a^4 x^3}-\frac {5 c^3}{a^2 x}+\frac {4 c^3 \log (x)}{a}+c^3 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 67, normalized size = 1.06 \[ \frac {15 \, a^{6} c^{3} x^{6} + 60 \, a^{5} c^{3} x^{5} \log \relax (x) - 75 \, a^{4} c^{3} x^{4} + 25 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x + 3 \, c^{3}}{15 \, a^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 136, normalized size = 2.16 \[ -\frac {4 \, c^{3} \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {4 \, c^{3} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {{\left (15 \, c^{3} + \frac {107 \, c^{3}}{a x - 1} + \frac {235 \, c^{3}}{{\left (a x - 1\right )}^{2}} + \frac {170 \, c^{3}}{{\left (a x - 1\right )}^{3}} - \frac {30 \, c^{3}}{{\left (a x - 1\right )}^{4}} - \frac {60 \, c^{3}}{{\left (a x - 1\right )}^{5}}\right )} {\left (a x - 1\right )}}{15 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 60, normalized size = 0.95 \[ \frac {c^{3}}{5 a^{6} x^{5}}+\frac {c^{3}}{a^{5} x^{4}}+\frac {5 c^{3}}{3 a^{4} x^{3}}-\frac {5 c^{3}}{a^{2} x}+c^{3} x +\frac {4 c^{3} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 59, normalized size = 0.94 \[ c^{3} x + \frac {4 \, c^{3} \log \relax (x)}{a} - \frac {75 \, a^{4} c^{3} x^{4} - 25 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 3 \, c^{3}}{15 \, a^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 48, normalized size = 0.76 \[ \frac {c^3\,\left (a\,x+\frac {5\,a^2\,x^2}{3}-5\,a^4\,x^4+a^6\,x^6+4\,a^5\,x^5\,\ln \relax (x)+\frac {1}{5}\right )}{a^6\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 65, normalized size = 1.03 \[ \frac {a^{6} c^{3} x + 4 a^{5} c^{3} \log {\relax (x )} + \frac {- 75 a^{4} c^{3} x^{4} + 25 a^{2} c^{3} x^{2} + 15 a c^{3} x + 3 c^{3}}{15 x^{5}}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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