Optimal. Leaf size=63 \[ \frac {c^3}{5 a^6 x^5}+\frac {c^3}{a^5 x^4}+\frac {5 c^3}{3 a^4 x^3}-\frac {5 c^3}{a^2 x}+c^3 x+\frac {4 c^3 \log (x)}{a} \]
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Rubi [A]
time = 0.08, antiderivative size = 63, 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 = {6292, 6285, 76}
\begin {gather*} \frac {c^3}{5 a^6 x^5}+\frac {c^3}{a^5 x^4}+\frac {5 c^3}{3 a^4 x^3}-\frac {5 c^3}{a^2 x}+\frac {4 c^3 \log (x)}{a}+c^3 x \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 6285
Rule 6292
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{4 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6}\\ &=-\frac {c^3 \int \frac {(1-a x) (1+a x)^5}{x^6} \, dx}{a^6}\\ &=-\frac {c^3 \int \left (-a^6+\frac {1}{x^6}+\frac {4 a}{x^5}+\frac {5 a^2}{x^4}-\frac {5 a^4}{x^2}-\frac {4 a^5}{x}\right ) \, dx}{a^6}\\ &=\frac {c^3}{5 a^6 x^5}+\frac {c^3}{a^5 x^4}+\frac {5 c^3}{3 a^4 x^3}-\frac {5 c^3}{a^2 x}+c^3 x+\frac {4 c^3 \log (x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 63, normalized size = 1.00 \begin {gather*} \frac {c^3}{5 a^6 x^5}+\frac {c^3}{a^5 x^4}+\frac {5 c^3}{3 a^4 x^3}-\frac {5 c^3}{a^2 x}+c^3 x+\frac {4 c^3 \log (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.17, size = 47, normalized size = 0.75
method | result | size |
default | \(\frac {c^{3} \left (a^{6} x +\frac {5 a^{2}}{3 x^{3}}+\frac {a}{x^{4}}-\frac {5 a^{4}}{x}+\frac {1}{5 x^{5}}+4 a^{5} \ln \left (x \right )\right )}{a^{6}}\) | \(47\) |
risch | \(x \,c^{3}+\frac {-5 a^{4} c^{3} x^{4}+\frac {5}{3} a^{2} c^{3} x^{2}+a \,c^{3} x +\frac {1}{5} c^{3}}{a^{6} x^{5}}+\frac {4 c^{3} \ln \left (x \right )}{a}\) | \(58\) |
norman | \(\frac {a^{2} c^{3} x^{3}+a^{7} c^{3} x^{8}-\frac {c^{3}}{5 a}-x \,c^{3}-\frac {22 a \,c^{3} x^{2}}{15}+\frac {20 a^{3} c^{3} x^{4}}{3}-6 a^{5} c^{3} x^{6}}{a^{5} \left (a^{2} x^{2}-1\right ) x^{5}}+\frac {4 c^{3} \ln \left (x \right )}{a}\) | \(96\) |
meijerg | \(\frac {c^{3} \left (\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}-\frac {3 c^{3} \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}-\frac {7 c^{3} \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \arctanh \left (a x \right )}{a}\right )}{\sqrt {-a^{2}}}-\frac {7 c^{3} \left (-\frac {2 \left (-3 a^{2} x^{2}+2\right )}{x \sqrt {-a^{2}}\, \left (-2 a^{2} x^{2}+2\right )}+\frac {3 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{\sqrt {-a^{2}}}+\frac {2 c^{3} \left (\frac {a^{2} x^{2}}{-a^{2} x^{2}+1}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {4 a \,c^{3} x^{2}}{-a^{2} x^{2}+1}-\frac {4 c^{3} \left (-\frac {3 a^{2} x^{2}}{-3 a^{2} x^{2}+3}+2 \ln \left (-a^{2} x^{2}+1\right )-1-4 \ln \left (x \right )-2 \ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )}{a}-\frac {3 c^{3} \left (-\frac {2 \left (-15 a^{4} x^{4}+10 a^{2} x^{2}+2\right )}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}} \left (-2 a^{2} x^{2}+2\right )}+\frac {5 a^{3} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}-\frac {2 c^{3} \left (\frac {4 a^{2} x^{2}}{-4 a^{2} x^{2}+4}-3 \ln \left (-a^{2} x^{2}+1\right )+1+6 \ln \left (x \right )+3 \ln \left (-a^{2}\right )-\frac {1}{2 a^{4} x^{4}}-\frac {2}{a^{2} x^{2}}\right )}{a}+\frac {c^{3} \left (-\frac {2 \left (-105 a^{6} x^{6}+70 a^{4} x^{4}+14 a^{2} x^{2}+6\right )}{5 x^{5} \left (-a^{2}\right )^{\frac {5}{2}} \left (-6 a^{2} x^{2}+6\right )}+\frac {7 \arctanh \left (a x \right ) a^{5}}{\left (-a^{2}\right )^{\frac {5}{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(573\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 59, normalized size = 0.94 \begin {gather*} c^{3} x + \frac {4 \, c^{3} \log \left (x\right )}{a} - \frac {75 \, a^{4} c^{3} x^{4} - 25 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 3 \, c^{3}}{15 \, a^{6} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 67, normalized size = 1.06 \begin {gather*} \frac {15 \, a^{6} c^{3} x^{6} + 60 \, a^{5} c^{3} x^{5} \log \left (x\right ) - 75 \, a^{4} c^{3} x^{4} + 25 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x + 3 \, c^{3}}{15 \, a^{6} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 65, normalized size = 1.03 \begin {gather*} \frac {a^{6} c^{3} x + 4 a^{5} c^{3} \log {\left (x \right )} + \frac {- 75 a^{4} c^{3} x^{4} + 25 a^{2} c^{3} x^{2} + 15 a c^{3} x + 3 c^{3}}{15 x^{5}}}{a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 60, normalized size = 0.95 \begin {gather*} c^{3} x + \frac {4 \, c^{3} \log \left ({\left | x \right |}\right )}{a} - \frac {75 \, a^{4} c^{3} x^{4} - 25 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 3 \, c^{3}}{15 \, a^{6} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 48, normalized size = 0.76 \begin {gather*} \frac {c^3\,\left (a\,x+\frac {5\,a^2\,x^2}{3}-5\,a^4\,x^4+a^6\,x^6+4\,a^5\,x^5\,\ln \left (x\right )+\frac {1}{5}\right )}{a^6\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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