Optimal. Leaf size=78 \[ \frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}-c^3 x+\frac {2 c^3 \log (x)}{a} \]
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Rubi [A]
time = 0.08, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6292, 6285, 90}
\begin {gather*} \frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}+\frac {2 c^3 \log (x)}{a}+c^3 (-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
Rule 6292
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{-2 \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)^4 (1+a x)^2}{x^6} \, dx}{a^6}\\ &=-\frac {c^3 \int \left (a^6+\frac {1}{x^6}-\frac {2 a}{x^5}-\frac {a^2}{x^4}+\frac {4 a^3}{x^3}-\frac {a^4}{x^2}-\frac {2 a^5}{x}\right ) \, dx}{a^6}\\ &=\frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}-c^3 x+\frac {2 c^3 \log (x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 78, normalized size = 1.00 \begin {gather*} \frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}-c^3 x+\frac {2 c^3 \log (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.92, size = 57, normalized size = 0.73
method | result | size |
default | \(\frac {c^{3} \left (-a^{6} x -\frac {a^{2}}{3 x^{3}}-\frac {a}{2 x^{4}}+\frac {2 a^{3}}{x^{2}}-\frac {a^{4}}{x}+\frac {1}{5 x^{5}}+2 a^{5} \ln \left (x \right )\right )}{a^{6}}\) | \(57\) |
risch | \(-x \,c^{3}+\frac {-a^{4} c^{3} x^{4}+2 a^{3} c^{3} x^{3}-\frac {1}{3} a^{2} c^{3} x^{2}-\frac {1}{2} a \,c^{3} x +\frac {1}{5} c^{3}}{a^{6} x^{5}}+\frac {2 c^{3} \ln \left (x \right )}{a}\) | \(71\) |
norman | \(\frac {a^{3} c^{3} x^{4}+\frac {c^{3}}{5 a}-\frac {3 x \,c^{3}}{10}-\frac {5 a \,c^{3} x^{2}}{6}+\frac {5 a^{2} c^{3} x^{3}}{3}-c^{3} a^{6} x^{7}}{a^{5} x^{5} \left (a x +1\right )}+\frac {2 c^{3} \ln \left (x \right )}{a}\) | \(82\) |
meijerg | \(-\frac {c^{3} \left (\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )\right )}{a}+\frac {4 c^{3} x}{a x +1}-\frac {6 c^{3} \left (\frac {3 a x}{3 a x +3}+2 \ln \left (a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (a \right )-\frac {1}{a x}\right )}{a}+\frac {4 c^{3} \left (\frac {5 a x}{5 a x +5}+4 \ln \left (a x +1\right )-1-4 \ln \left (x \right )-4 \ln \left (a \right )-\frac {1}{3 a^{3} x^{3}}+\frac {1}{a^{2} x^{2}}-\frac {3}{a x}\right )}{a}-\frac {c^{3} \left (\frac {7 a x}{7 a x +7}+6 \ln \left (a x +1\right )-1-6 \ln \left (x \right )-6 \ln \left (a \right )-\frac {1}{5 a^{5} x^{5}}+\frac {1}{2 a^{4} x^{4}}-\frac {1}{a^{3} x^{3}}+\frac {2}{a^{2} x^{2}}-\frac {5}{a x}\right )}{a}\) | \(234\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 71, normalized size = 0.91 \begin {gather*} -c^{3} x + \frac {2 \, c^{3} \log \left (x\right )}{a} - \frac {30 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 78, normalized size = 1.00 \begin {gather*} -\frac {30 \, a^{6} c^{3} x^{6} - 60 \, a^{5} c^{3} x^{5} \log \left (x\right ) + 30 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 76, normalized size = 0.97 \begin {gather*} \frac {- a^{6} c^{3} x + 2 a^{5} c^{3} \log {\left (x \right )} - \frac {30 a^{4} c^{3} x^{4} - 60 a^{3} c^{3} x^{3} + 10 a^{2} c^{3} x^{2} + 15 a c^{3} x - 6 c^{3}}{30 x^{5}}}{a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 136, normalized size = 1.74 \begin {gather*} -\frac {2 \, c^{3} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {2 \, c^{3} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} + \frac {{\left (30 \, c^{3} - \frac {71 \, c^{3}}{a x + 1} - \frac {65 \, c^{3}}{{\left (a x + 1\right )}^{2}} + \frac {310 \, c^{3}}{{\left (a x + 1\right )}^{3}} - \frac {270 \, c^{3}}{{\left (a x + 1\right )}^{4}} + \frac {60 \, c^{3}}{{\left (a x + 1\right )}^{5}}\right )} {\left (a x + 1\right )}}{30 \, a {\left (\frac {1}{a x + 1} - 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.85, size = 57, normalized size = 0.73 \begin {gather*} -\frac {c^3\,\left (\frac {a\,x}{2}+\frac {a^2\,x^2}{3}-2\,a^3\,x^3+a^4\,x^4+a^6\,x^6-2\,a^5\,x^5\,\ln \left (x\right )-\frac {1}{5}\right )}{a^6\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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