Optimal. Leaf size=41 \[ -\frac {c^2}{3 a^4 x^3}-\frac {c^2}{a^3 x^2}-c^2 x-\frac {2 c^2 \log (x)}{a} \]
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Rubi [A]
time = 0.07, antiderivative size = 41, 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, 76}
\begin {gather*} -\frac {c^2}{3 a^4 x^3}-\frac {c^2}{a^3 x^2}-\frac {2 c^2 \log (x)}{a}+c^2 (-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 6285
Rule 6292
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx &=\frac {c^2 \int \frac {e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4}\\ &=\frac {c^2 \int \frac {(1-a x) (1+a x)^3}{x^4} \, dx}{a^4}\\ &=\frac {c^2 \int \left (-a^4+\frac {1}{x^4}+\frac {2 a}{x^3}-\frac {2 a^3}{x}\right ) \, dx}{a^4}\\ &=-\frac {c^2}{3 a^4 x^3}-\frac {c^2}{a^3 x^2}-c^2 x-\frac {2 c^2 \log (x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 41, normalized size = 1.00 \begin {gather*} -\frac {c^2}{3 a^4 x^3}-\frac {c^2}{a^3 x^2}-c^2 x-\frac {2 c^2 \log (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.04, size = 33, normalized size = 0.80
method | result | size |
default | \(\frac {c^{2} \left (-a^{4} x -\frac {1}{3 x^{3}}-\frac {a}{x^{2}}-2 a^{3} \ln \left (x \right )\right )}{a^{4}}\) | \(33\) |
risch | \(-x \,c^{2}+\frac {-a \,c^{2} x -\frac {1}{3} c^{2}}{a^{4} x^{3}}-\frac {2 c^{2} \ln \left (x \right )}{a}\) | \(38\) |
norman | \(\frac {-\frac {c^{2}}{3 a}-x \,c^{2}-a^{3} c^{2} x^{4}}{a^{3} x^{3}}-\frac {2 c^{2} \ln \left (x \right )}{a}\) | \(45\) |
meijerg | \(-\frac {c^{2} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}-\frac {c^{2} \arctanh \left (a x \right )}{a}+\frac {c^{2} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}-\frac {c^{2} \ln \left (-a^{2} x^{2}+1\right )}{a}-\frac {2 c^{2} \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{a}-\frac {c^{2} \left (\ln \left (-a^{2} x^{2}+1\right )-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )}{a}+\frac {c^{2} \left (-\frac {2 a^{2}}{x \left (-a^{2}\right )^{\frac {3}{2}}}-\frac {2}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}}}+\frac {2 a^{3} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(241\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 36, normalized size = 0.88 \begin {gather*} -c^{2} x - \frac {2 \, c^{2} \log \left (x\right )}{a} - \frac {3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 43, normalized size = 1.05 \begin {gather*} -\frac {3 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} \log \left (x\right ) + 3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 41, normalized size = 1.00 \begin {gather*} \frac {- a^{4} c^{2} x - 2 a^{3} c^{2} \log {\left (x \right )} - \frac {3 a c^{2} x + c^{2}}{3 x^{3}}}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 37, normalized size = 0.90 \begin {gather*} -c^{2} x - \frac {2 \, c^{2} \log \left ({\left | x \right |}\right )}{a} - \frac {3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 35, normalized size = 0.85 \begin {gather*} -\frac {c^2\,\left (3\,a\,x+3\,a^4\,x^4+6\,a^3\,x^3\,\ln \left (x\right )+1\right )}{3\,a^4\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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