Optimal. Leaf size=17 \[ -\frac {c^2 (a x+1)^4}{2 x^2} \]
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Rubi [A] time = 0.07, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 74} \[ -\frac {c^2 (a x+1)^4}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 74
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2}{x^3} \, dx &=c^2 \int \frac {(1-a x) (1+a x)^3}{x^3} \, dx\\ &=-\frac {c^2 (1+a x)^4}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ -\frac {c^2 (a x+1)^4}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 37, normalized size = 2.18 \[ -\frac {a^{4} c^{2} x^{4} + 4 \, a^{3} c^{2} x^{3} + 4 \, a c^{2} x + c^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.56, size = 37, normalized size = 2.18 \[ -\frac {1}{2} \, a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x - \frac {4 \, a c^{2} x + c^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 31, normalized size = 1.82 \[ c^{2} \left (-\frac {x^{2} a^{4}}{2}-2 a^{3} x -\frac {2 a}{x}-\frac {1}{2 x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 37, normalized size = 2.18 \[ -\frac {1}{2} \, a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x - \frac {4 \, a c^{2} x + c^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 29, normalized size = 1.71 \[ -\frac {c^2\,\left (a^4\,x^4+4\,a^3\,x^3+4\,a\,x+1\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.14, size = 39, normalized size = 2.29 \[ - \frac {a^{4} c^{2} x^{2}}{2} - 2 a^{3} c^{2} x - \frac {4 a c^{2} x + c^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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