Optimal. Leaf size=42 \[ \frac {a^4 c^2}{x}+\frac {a^3 c^2}{x^2}-\frac {a c^2}{2 x^4}-\frac {c^2}{5 x^5} \]
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Rubi [A] time = 0.08, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 75} \[ \frac {a^3 c^2}{x^2}+\frac {a^4 c^2}{x}-\frac {a c^2}{2 x^4}-\frac {c^2}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 75
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2}{x^6} \, dx &=c^2 \int \frac {(1-a x) (1+a x)^3}{x^6} \, dx\\ &=c^2 \int \left (\frac {1}{x^6}+\frac {2 a}{x^5}-\frac {2 a^3}{x^3}-\frac {a^4}{x^2}\right ) \, dx\\ &=-\frac {c^2}{5 x^5}-\frac {a c^2}{2 x^4}+\frac {a^3 c^2}{x^2}+\frac {a^4 c^2}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.55 \[ \frac {c^2 (a x+1)^4 (3 a x-2)}{10 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 40, normalized size = 0.95 \[ \frac {10 \, a^{4} c^{2} x^{4} + 10 \, a^{3} c^{2} x^{3} - 5 \, a c^{2} x - 2 \, c^{2}}{10 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 40, normalized size = 0.95 \[ \frac {10 \, a^{4} c^{2} x^{4} + 10 \, a^{3} c^{2} x^{3} - 5 \, a c^{2} x - 2 \, c^{2}}{10 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 31, normalized size = 0.74 \[ c^{2} \left (\frac {a^{4}}{x}+\frac {a^{3}}{x^{2}}-\frac {a}{2 x^{4}}-\frac {1}{5 x^{5}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 40, normalized size = 0.95 \[ \frac {10 \, a^{4} c^{2} x^{4} + 10 \, a^{3} c^{2} x^{3} - 5 \, a c^{2} x - 2 \, c^{2}}{10 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 40, normalized size = 0.95 \[ -\frac {-a^4\,c^2\,x^4-a^3\,c^2\,x^3+\frac {a\,c^2\,x}{2}+\frac {c^2}{5}}{x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 42, normalized size = 1.00 \[ - \frac {- 10 a^{4} c^{2} x^{4} - 10 a^{3} c^{2} x^{3} + 5 a c^{2} x + 2 c^{2}}{10 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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