Optimal. Leaf size=40 \[ 2 a c^2 x-\frac {2}{3} a^3 c^2 x^3-\frac {1}{4} a^4 c^2 x^4+c^2 \log (x) \]
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Rubi [A]
time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6285, 76}
\begin {gather*} -\frac {1}{4} a^4 c^2 x^4-\frac {2}{3} a^3 c^2 x^3+2 a c^2 x+c^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2}{x} \, dx &=c^2 \int \frac {(1-a x) (1+a x)^3}{x} \, dx\\ &=c^2 \int \left (2 a+\frac {1}{x}-2 a^3 x^2-a^4 x^3\right ) \, dx\\ &=2 a c^2 x-\frac {2}{3} a^3 c^2 x^3-\frac {1}{4} a^4 c^2 x^4+c^2 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.82 \begin {gather*} -\frac {1}{12} c^2 \left (3-24 a x+8 a^3 x^3+3 a^4 x^4-12 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 28, normalized size = 0.70
method | result | size |
default | \(c^{2} \left (-\frac {a^{4} x^{4}}{4}-\frac {2 a^{3} x^{3}}{3}+2 a x +\ln \left (x \right )\right )\) | \(28\) |
norman | \(2 a \,c^{2} x -\frac {2 a^{3} c^{2} x^{3}}{3}-\frac {a^{4} c^{2} x^{4}}{4}+c^{2} \ln \left (x \right )\) | \(37\) |
risch | \(2 a \,c^{2} x -\frac {2 a^{3} c^{2} x^{3}}{3}-\frac {a^{4} c^{2} x^{4}}{4}+c^{2} \ln \left (x \right )\) | \(37\) |
meijerg | \(-\frac {c^{2} \left (\frac {x^{2} a^{2} \left (3 a^{2} x^{2}+6\right )}{6}+\ln \left (-a^{2} x^{2}+1\right )\right )}{2}-\frac {c^{2} \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2}+\frac {c^{2} \ln \left (-a^{2} x^{2}+1\right )}{2}+\frac {a \,c^{2} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{\sqrt {-a^{2}}}+\frac {2 a \,c^{2} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}+2 c^{2} \arctanh \left (a x \right )+\frac {c^{2} \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{2}\) | \(213\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 36, normalized size = 0.90 \begin {gather*} -\frac {1}{4} \, a^{4} c^{2} x^{4} - \frac {2}{3} \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x + c^{2} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 36, normalized size = 0.90 \begin {gather*} -\frac {1}{4} \, a^{4} c^{2} x^{4} - \frac {2}{3} \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x + c^{2} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 39, normalized size = 0.98 \begin {gather*} - \frac {a^{4} c^{2} x^{4}}{4} - \frac {2 a^{3} c^{2} x^{3}}{3} + 2 a c^{2} x + c^{2} \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 37, normalized size = 0.92 \begin {gather*} -\frac {1}{4} \, a^{4} c^{2} x^{4} - \frac {2}{3} \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x + c^{2} \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 36, normalized size = 0.90 \begin {gather*} c^2\,\ln \left (x\right )-\frac {2\,a^3\,c^2\,x^3}{3}-\frac {a^4\,c^2\,x^4}{4}+2\,a\,c^2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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