Optimal. Leaf size=100 \[ -\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{3 a^7 x^6}-\frac {4 c^4}{5 a^6 x^5}+\frac {c^4}{a^5 x^4}+\frac {10 c^4}{3 a^4 x^3}+\frac {2 c^4}{a^3 x^2}-\frac {4 c^4}{a^2 x}+\frac {4 c^4 \log (x)}{a}+c^4 x \]
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Rubi [A] time = 0.17, antiderivative size = 100, 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 = {6167, 6157, 6150, 88} \[ \frac {2 c^4}{a^3 x^2}+\frac {10 c^4}{3 a^4 x^3}+\frac {c^4}{a^5 x^4}-\frac {4 c^4}{5 a^6 x^5}-\frac {2 c^4}{3 a^7 x^6}-\frac {c^4}{7 a^8 x^7}-\frac {4 c^4}{a^2 x}+\frac {4 c^4 \log (x)}{a}+c^4 x \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6157
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx\\ &=\frac {c^4 \int \frac {e^{4 \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)^2 (1+a x)^6}{x^8} \, dx}{a^8}\\ &=\frac {c^4 \int \left (a^8+\frac {1}{x^8}+\frac {4 a}{x^7}+\frac {4 a^2}{x^6}-\frac {4 a^3}{x^5}-\frac {10 a^4}{x^4}-\frac {4 a^5}{x^3}+\frac {4 a^6}{x^2}+\frac {4 a^7}{x}\right ) \, dx}{a^8}\\ &=-\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{3 a^7 x^6}-\frac {4 c^4}{5 a^6 x^5}+\frac {c^4}{a^5 x^4}+\frac {10 c^4}{3 a^4 x^3}+\frac {2 c^4}{a^3 x^2}-\frac {4 c^4}{a^2 x}+c^4 x+\frac {4 c^4 \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 100, normalized size = 1.00 \[ -\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{3 a^7 x^6}-\frac {4 c^4}{5 a^6 x^5}+\frac {c^4}{a^5 x^4}+\frac {10 c^4}{3 a^4 x^3}+\frac {2 c^4}{a^3 x^2}-\frac {4 c^4}{a^2 x}+\frac {4 c^4 \log (x)}{a}+c^4 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 100, normalized size = 1.00 \[ \frac {105 \, a^{8} c^{4} x^{8} + 420 \, a^{7} c^{4} x^{7} \log \relax (x) - 420 \, a^{6} c^{4} x^{6} + 210 \, a^{5} c^{4} x^{5} + 350 \, a^{4} c^{4} x^{4} + 105 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x - 15 \, c^{4}}{105 \, a^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 160, normalized size = 1.60 \[ -\frac {4 \, c^{4} \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {4 \, c^{4} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {{\left (105 \, c^{4} + \frac {659 \, c^{4}}{a x - 1} + \frac {1253 \, c^{4}}{{\left (a x - 1\right )}^{2}} - \frac {231 \, c^{4}}{{\left (a x - 1\right )}^{3}} - \frac {3885 \, c^{4}}{{\left (a x - 1\right )}^{4}} - \frac {5250 \, c^{4}}{{\left (a x - 1\right )}^{5}} - \frac {2730 \, c^{4}}{{\left (a x - 1\right )}^{6}} - \frac {420 \, c^{4}}{{\left (a x - 1\right )}^{7}}\right )} {\left (a x - 1\right )}}{105 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 93, normalized size = 0.93 \[ -\frac {c^{4}}{7 a^{8} x^{7}}-\frac {2 c^{4}}{3 a^{7} x^{6}}-\frac {4 c^{4}}{5 a^{6} x^{5}}+\frac {c^{4}}{a^{5} x^{4}}+\frac {10 c^{4}}{3 a^{4} x^{3}}+\frac {2 c^{4}}{x^{2} a^{3}}-\frac {4 c^{4}}{a^{2} x}+c^{4} x +\frac {4 c^{4} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 92, normalized size = 0.92 \[ c^{4} x + \frac {4 \, c^{4} \log \relax (x)}{a} - \frac {420 \, a^{6} c^{4} x^{6} - 210 \, a^{5} c^{4} x^{5} - 350 \, a^{4} c^{4} x^{4} - 105 \, a^{3} c^{4} x^{3} + 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 15 \, c^{4}}{105 \, a^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 72, normalized size = 0.72 \[ \frac {c^4\,\left (a^3\,x^3-\frac {4\,a^2\,x^2}{5}-\frac {2\,a\,x}{3}+\frac {10\,a^4\,x^4}{3}+2\,a^5\,x^5-4\,a^6\,x^6+a^8\,x^8+4\,a^7\,x^7\,\ln \relax (x)-\frac {1}{7}\right )}{a^8\,x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 100, normalized size = 1.00 \[ \frac {a^{8} c^{4} x + 4 a^{7} c^{4} \log {\relax (x )} + \frac {- 420 a^{6} c^{4} x^{6} + 210 a^{5} c^{4} x^{5} + 350 a^{4} c^{4} x^{4} + 105 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} - 70 a c^{4} x - 15 c^{4}}{105 x^{7}}}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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