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 = 0.06, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {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
Rubi steps
\begin {align*} \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 = 0.77, size = 27, normalized size = 0.90
method | result | size |
default | \(\frac {c^{4} \left (a^{4} x -\frac {1}{3 x^{3}}+\frac {2 a^{2}}{x}\right )}{a^{4}}\) | \(27\) |
gosper | \(\frac {\left (3 a^{4} x^{4}+6 a^{2} x^{2}-1\right ) c^{4}}{3 x^{3} a^{4}}\) | \(30\) |
risch | \(c^{4} x +\frac {2 a^{2} c^{4} x^{2}-\frac {1}{3} c^{4}}{a^{4} x^{3}}\) | \(31\) |
norman | \(\frac {c^{4} a^{5} x^{6}+c^{4} x^{4} a^{3}+\frac {c^{4}}{3 a}-\frac {7 c^{4} a \,x^{2}}{3}}{\left (a^{2} x^{2}-1\right ) a^{3} x^{3}}\) | \(57\) |
meijerg | \(\frac {c^{4} \left (\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}+\frac {2 c^{4} \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}+\frac {3 c^{4} \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \arctanh \left (a x \right )}{a}\right )}{\sqrt {-a^{2}}}+\frac {c^{4} \left (-\frac {2 \left (-15 a^{4} x^{4}+10 a^{2} x^{2}+2\right )}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}} \left (-2 a^{2} x^{2}+2\right )}+\frac {5 a^{3} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}+\frac {2 c^{4} \left (-\frac {2 \left (-3 a^{2} x^{2}+2\right )}{x \sqrt {-a^{2}}\, \left (-2 a^{2} x^{2}+2\right )}+\frac {3 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{\sqrt {-a^{2}}}\) | \(301\) |
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 = 0.34, 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 = 0.08, 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 [A]
time = 0.38, 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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Mupad [B]
time = 0.81, 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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