Optimal. Leaf size=91 \[ -\frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}+\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}-\frac {2 c^4}{a^2 x}+\frac {2 c^4 \log (x)}{a}+c^4 (-x) \]
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Rubi [A] time = 0.14, antiderivative size = 91, 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 = {6157, 6150, 88} \[ \frac {3 c^4}{a^3 x^2}-\frac {3 c^4}{2 a^5 x^4}+\frac {2 c^4}{5 a^6 x^5}+\frac {c^4}{3 a^7 x^6}-\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{a^2 x}+\frac {2 c^4 \log (x)}{a}+c^4 (-x) \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6157
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=\frac {c^4 \int \frac {e^{-2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=\frac {c^4 \int \frac {(1-a x)^5 (1+a x)^3}{x^8} \, dx}{a^8}\\ &=\frac {c^4 \int \left (-a^8+\frac {1}{x^8}-\frac {2 a}{x^7}-\frac {2 a^2}{x^6}+\frac {6 a^3}{x^5}-\frac {6 a^5}{x^3}+\frac {2 a^6}{x^2}+\frac {2 a^7}{x}\right ) \, dx}{a^8}\\ &=-\frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}+\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}-\frac {2 c^4}{a^2 x}-c^4 x+\frac {2 c^4 \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 91, normalized size = 1.00 \[ -\frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}+\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}-\frac {2 c^4}{a^2 x}+\frac {2 c^4 \log (x)}{a}+c^4 (-x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 89, normalized size = 0.98 \[ -\frac {210 \, a^{8} c^{4} x^{8} - 420 \, a^{7} c^{4} x^{7} \log \relax (x) + 420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 160, normalized size = 1.76 \[ -\frac {2 \, c^{4} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {2 \, c^{4} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} + \frac {{\left (210 \, c^{4} - \frac {719 \, c^{4}}{a x + 1} - \frac {427 \, c^{4}}{{\left (a x + 1\right )}^{2}} + \frac {5271 \, c^{4}}{{\left (a x + 1\right )}^{3}} - \frac {9485 \, c^{4}}{{\left (a x + 1\right )}^{4}} + \frac {7490 \, c^{4}}{{\left (a x + 1\right )}^{5}} - \frac {2730 \, c^{4}}{{\left (a x + 1\right )}^{6}} + \frac {420 \, c^{4}}{{\left (a x + 1\right )}^{7}}\right )} {\left (a x + 1\right )}}{210 \, a {\left (\frac {1}{a x + 1} - 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 84, normalized size = 0.92 \[ -\frac {c^{4}}{7 a^{8} x^{7}}+\frac {c^{4}}{3 a^{7} x^{6}}+\frac {2 c^{4}}{5 a^{6} x^{5}}-\frac {3 c^{4}}{2 a^{5} x^{4}}+\frac {3 c^{4}}{x^{2} a^{3}}-\frac {2 c^{4}}{a^{2} x}-c^{4} x +\frac {2 c^{4} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 82, normalized size = 0.90 \[ -c^{4} x + \frac {2 \, c^{4} \log \relax (x)}{a} - \frac {420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 66, normalized size = 0.73 \[ \frac {c^4\,\left (\frac {a\,x}{3}+\frac {2\,a^2\,x^2}{5}-\frac {3\,a^3\,x^3}{2}+3\,a^5\,x^5-2\,a^6\,x^6-a^8\,x^8+2\,a^7\,x^7\,\ln \relax (x)-\frac {1}{7}\right )}{a^8\,x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 88, normalized size = 0.97 \[ \frac {- a^{8} c^{4} x + 2 a^{7} c^{4} \log {\relax (x )} - \frac {420 a^{6} c^{4} x^{6} - 630 a^{5} c^{4} x^{5} + 315 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} - 70 a c^{4} x + 30 c^{4}}{210 x^{7}}}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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