Optimal. Leaf size=40 \[ -\frac {c^4}{3 a^4 x^3}+\frac {c^4}{a^3 x^2}-c^4 x+\frac {2 c^4 \log (x)}{a} \]
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Rubi [A]
time = 0.07, antiderivative size = 40, 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 = {6266, 6264, 76}
\begin {gather*} -\frac {c^4}{3 a^4 x^3}+\frac {c^4}{a^3 x^2}+\frac {2 c^4 \log (x)}{a}+c^4 (-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 6264
Rule 6266
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx &=\frac {c^4 \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^4}{x^4} \, dx}{a^4}\\ &=\frac {c^4 \int \frac {(1-a x)^3 (1+a x)}{x^4} \, dx}{a^4}\\ &=\frac {c^4 \int \left (-a^4+\frac {1}{x^4}-\frac {2 a}{x^3}+\frac {2 a^3}{x}\right ) \, dx}{a^4}\\ &=-\frac {c^4}{3 a^4 x^3}+\frac {c^4}{a^3 x^2}-c^4 x+\frac {2 c^4 \log (x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 42, normalized size = 1.05 \begin {gather*} -\frac {c^4}{3 a^4 x^3}+\frac {c^4}{a^3 x^2}-c^4 x+\frac {2 c^4 \log (a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.03, size = 32, normalized size = 0.80
method | result | size |
default | \(\frac {c^{4} \left (-a^{4} x -\frac {1}{3 x^{3}}+\frac {a}{x^{2}}+2 a^{3} \ln \left (x \right )\right )}{a^{4}}\) | \(32\) |
risch | \(-c^{4} x +\frac {a \,c^{4} x -\frac {1}{3} c^{4}}{a^{4} x^{3}}+\frac {2 c^{4} \ln \left (x \right )}{a}\) | \(37\) |
norman | \(\frac {c^{4} x -\frac {c^{4}}{3 a}-c^{4} x^{4} a^{3}}{a^{3} x^{3}}+\frac {2 c^{4} \ln \left (x \right )}{a}\) | \(44\) |
meijerg | \(-\frac {c^{4} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}+\frac {c^{4} \ln \left (-a^{2} x^{2}+1\right )}{a}-\frac {c^{4} \arctanh \left (a x \right )}{a}+\frac {2 c^{4} \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{a}+\frac {c^{4} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}+\frac {c^{4} \left (\ln \left (-a^{2} x^{2}+1\right )-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )}{a}+\frac {c^{4} \left (-\frac {2 a^{2}}{x \left (-a^{2}\right )^{\frac {3}{2}}}-\frac {2}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}}}+\frac {2 a^{3} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(239\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 38, normalized size = 0.95 \begin {gather*} -c^{4} x + \frac {2 \, c^{4} \log \left (x\right )}{a} + \frac {3 \, a c^{4} x - c^{4}}{3 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 43, normalized size = 1.08 \begin {gather*} -\frac {3 \, a^{4} c^{4} x^{4} - 6 \, a^{3} c^{4} x^{3} \log \left (x\right ) - 3 \, a c^{4} x + c^{4}}{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 = 39, normalized size = 0.98 \begin {gather*} \frac {- a^{4} c^{4} x + 2 a^{3} c^{4} \log {\left (x \right )} - \frac {- 3 a c^{4} x + c^{4}}{3 x^{3}}}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 39, normalized size = 0.98 \begin {gather*} -c^{4} x + \frac {2 \, c^{4} \log \left ({\left | x \right |}\right )}{a} + \frac {3 \, a c^{4} x - c^{4}}{3 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 35, normalized size = 0.88 \begin {gather*} \frac {c^4\,\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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