Optimal. Leaf size=64 \[ \frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}+c^5 x-\frac {c^5 \log (x)}{a} \]
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Rubi [A]
time = 0.07, antiderivative size = 64, 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, 90}
\begin {gather*} \frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}-\frac {c^5 \log (x)}{a}+c^5 x \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6264
Rule 6266
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx &=-\frac {c^5 \int \frac {e^{4 \tanh ^{-1}(a x)} (1-a x)^5}{x^5} \, dx}{a^5}\\ &=-\frac {c^5 \int \frac {(1-a x)^3 (1+a x)^2}{x^5} \, dx}{a^5}\\ &=-\frac {c^5 \int \left (-a^5+\frac {1}{x^5}-\frac {a}{x^4}-\frac {2 a^2}{x^3}+\frac {2 a^3}{x^2}+\frac {a^4}{x}\right ) \, dx}{a^5}\\ &=\frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}+c^5 x-\frac {c^5 \log (x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 66, normalized size = 1.03 \begin {gather*} \frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}+c^5 x-\frac {c^5 \log (a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.25, size = 48, normalized size = 0.75
method | result | size |
default | \(\frac {c^{5} \left (a^{5} x -\frac {a}{3 x^{3}}+\frac {1}{4 x^{4}}-\frac {a^{2}}{x^{2}}+\frac {2 a^{3}}{x}-a^{4} \ln \left (x \right )\right )}{a^{5}}\) | \(48\) |
risch | \(x \,c^{5}+\frac {2 a^{3} c^{5} x^{3}-a^{2} c^{5} x^{2}-\frac {1}{3} a \,c^{5} x +\frac {1}{4} c^{5}}{a^{5} x^{4}}-\frac {c^{5} \ln \left (x \right )}{a}\) | \(59\) |
norman | \(\frac {-c^{5} a^{5} x^{6}+a^{6} c^{5} x^{7}+c^{5} a^{4} x^{5}-\frac {c^{5}}{4 a}+\frac {x \,c^{5}}{3}+\frac {5 a \,c^{5} x^{2}}{4}-\frac {7 a^{2} c^{5} x^{3}}{3}}{\left (a^{2} x^{2}-1\right ) a^{4} x^{4}}-\frac {c^{5} \ln \left (x \right )}{a}\) | \(96\) |
meijerg | \(\frac {2 a \,c^{5} x^{2}}{-a^{2} x^{2}+1}+\frac {c^{5} \left (\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}-\frac {c^{5} \left (\frac {a^{2} x^{2}}{-a^{2} x^{2}+1}+\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a}+\frac {2 c^{5} \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}+\frac {3 c^{5} \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \arctanh \left (a x \right )}{a}\right )}{\sqrt {-a^{2}}}-\frac {3 c^{5} \left (\frac {2 a^{2} x^{2}}{-2 a^{2} x^{2}+2}-\ln \left (-a^{2} x^{2}+1\right )+1+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{a}-\frac {2 c^{5} \left (-\frac {3 a^{2} x^{2}}{-3 a^{2} x^{2}+3}+2 \ln \left (-a^{2} x^{2}+1\right )-1-4 \ln \left (x \right )-2 \ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )}{a}+\frac {c^{5} \left (-\frac {2 \left (-15 a^{4} x^{4}+10 a^{2} x^{2}+2\right )}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}} \left (-2 a^{2} x^{2}+2\right )}+\frac {5 a^{3} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}+\frac {2 c^{5} \left (-\frac {2 \left (-3 a^{2} x^{2}+2\right )}{x \sqrt {-a^{2}}\, \left (-2 a^{2} x^{2}+2\right )}+\frac {3 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{\sqrt {-a^{2}}}-\frac {c^{5} \left (\frac {4 a^{2} x^{2}}{-4 a^{2} x^{2}+4}-3 \ln \left (-a^{2} x^{2}+1\right )+1+6 \ln \left (x \right )+3 \ln \left (-a^{2}\right )-\frac {1}{2 a^{4} x^{4}}-\frac {2}{a^{2} x^{2}}\right )}{2 a}\) | \(547\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 59, normalized size = 0.92 \begin {gather*} c^{5} x - \frac {c^{5} \log \left (x\right )}{a} + \frac {24 \, a^{3} c^{5} x^{3} - 12 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + 3 \, c^{5}}{12 \, a^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 67, normalized size = 1.05 \begin {gather*} \frac {12 \, a^{5} c^{5} x^{5} - 12 \, a^{4} c^{5} x^{4} \log \left (x\right ) + 24 \, a^{3} c^{5} x^{3} - 12 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + 3 \, c^{5}}{12 \, a^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 63, normalized size = 0.98 \begin {gather*} \frac {a^{5} c^{5} x - a^{4} c^{5} \log {\left (x \right )} + \frac {24 a^{3} c^{5} x^{3} - 12 a^{2} c^{5} x^{2} - 4 a c^{5} x + 3 c^{5}}{12 x^{4}}}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 60, normalized size = 0.94 \begin {gather*} c^{5} x - \frac {c^{5} \log \left ({\left | x \right |}\right )}{a} + \frac {24 \, a^{3} c^{5} x^{3} - 12 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + 3 \, c^{5}}{12 \, a^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 51, normalized size = 0.80 \begin {gather*} -\frac {c^5\,\left (4\,a\,x+12\,a^2\,x^2-24\,a^3\,x^3-12\,a^5\,x^5+12\,a^4\,x^4\,\ln \left (x\right )-3\right )}{12\,a^5\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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