Optimal. Leaf size=30 \[ -\frac {c^4}{3 a^4 x^3}+\frac {2 c^4}{a^2 x}+c^4 x \]
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Rubi [A]
time = 0.08, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6302, 6266,
6264, 74, 276} \begin {gather*} -\frac {c^4}{3 a^4 x^3}+\frac {2 c^4}{a^2 x}+c^4 x \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 276
Rule 6264
Rule 6266
Rule 6302
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx\\ &=\frac {c^4 \int \frac {e^{4 \tanh ^{-1}(a x)} (1-a x)^4}{x^4} \, dx}{a^4}\\ &=\frac {c^4 \int \frac {(1-a x)^2 (1+a x)^2}{x^4} \, dx}{a^4}\\ &=\frac {c^4 \int \frac {\left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4}\\ &=\frac {c^4 \int \left (a^4+\frac {1}{x^4}-\frac {2 a^2}{x^2}\right ) \, dx}{a^4}\\ &=-\frac {c^4}{3 a^4 x^3}+\frac {2 c^4}{a^2 x}+c^4 x\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 30, normalized size = 1.00 \begin {gather*} -\frac {c^4}{3 a^4 x^3}+\frac {2 c^4}{a^2 x}+c^4 x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 27, normalized size = 0.90
method | result | size |
default | \(\frac {c^{4} \left (a^{4} x +\frac {2 a^{2}}{x}-\frac {1}{3 x^{3}}\right )}{a^{4}}\) | \(27\) |
gosper | \(\frac {c^{4} \left (3 a^{4} x^{4}+6 a^{2} x^{2}-1\right )}{3 x^{3} a^{4}}\) | \(30\) |
risch | \(c^{4} x +\frac {2 a^{2} c^{4} x^{2}-\frac {1}{3} c^{4}}{a^{4} x^{3}}\) | \(31\) |
norman | \(\frac {a^{3} c^{4} x^{4}+a^{4} c^{4} x^{5}+\frac {c^{4}}{3 a}-\frac {c^{4} x}{3}-2 c^{4} a \,x^{2}}{\left (a x -1\right ) a^{3} x^{3}}\) | \(59\) |
meijerg | \(-\frac {c^{4} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{4} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}-\frac {c^{4} x}{-a x +1}+\frac {4 c^{4} \left (\frac {2 a x}{-2 a x +2}-\ln \left (-a x +1\right )+1+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}+\frac {c^{4} \left (-\frac {3 a x}{-3 a x +3}+2 \ln \left (-a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (-a \right )+\frac {1}{a x}\right )}{a}-\frac {2 c^{4} \left (\frac {4 a x}{-4 a x +4}-3 \ln \left (-a x +1\right )+1+3 \ln \left (x \right )+3 \ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {2}{a x}\right )}{a}-\frac {c^{4} \left (-\frac {5 a x}{-5 a x +5}+4 \ln \left (-a x +1\right )-1-4 \ln \left (x \right )-4 \ln \left (-a \right )+\frac {1}{3 x^{3} a^{3}}+\frac {1}{a^{2} x^{2}}+\frac {3}{a x}\right )}{a}\) | \(284\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 31, normalized size = 1.03 \begin {gather*} c^{4} x + \frac {6 \, a^{2} c^{4} x^{2} - 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 = 0.33, size = 36, normalized size = 1.20 \begin {gather*} \frac {3 \, a^{4} c^{4} x^{4} + 6 \, a^{2} c^{4} x^{2} - 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 = 0.07, size = 31, normalized size = 1.03 \begin {gather*} \frac {a^{4} c^{4} x + \frac {6 a^{2} c^{4} x^{2} - c^{4}}{3 x^{3}}}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs.
\(2 (28) = 56\).
time = 0.40, size = 59, normalized size = 1.97 \begin {gather*} \frac {{\left (a x - 1\right )} c^{4}}{a} - \frac {5 \, c^{4} + \frac {9 \, c^{4}}{a x - 1} + \frac {3 \, c^{4}}{{\left (a x - 1\right )}^{2}}}{3 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 27, normalized size = 0.90 \begin {gather*} \frac {c^4\,\left (a^4\,x^4+2\,a^2\,x^2-\frac {1}{3}\right )}{a^4\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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