Optimal. Leaf size=15 \[ -\frac{c (a x+1)^3}{3 x^3} \]
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Rubi [A] time = 0.0448639, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {6150, 37} \[ -\frac{c (a x+1)^3}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )}{x^4} \, dx &=c \int \frac{(1+a x)^2}{x^4} \, dx\\ &=-\frac{c (1+a x)^3}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0087305, size = 15, normalized size = 1. \[ -\frac{c (a x+1)^3}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 23, normalized size = 1.5 \begin{align*} c \left ( -{\frac{{a}^{2}}{x}}-{\frac{a}{{x}^{2}}}-{\frac{1}{3\,{x}^{3}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.940828, size = 28, normalized size = 1.87 \begin{align*} -\frac{3 \, a^{2} c x^{2} + 3 \, a c x + c}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66385, size = 51, normalized size = 3.4 \begin{align*} -\frac{3 \, a^{2} c x^{2} + 3 \, a c x + c}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.330752, size = 24, normalized size = 1.6 \begin{align*} - \frac{3 a^{2} c x^{2} + 3 a c x + c}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13783, size = 28, normalized size = 1.87 \begin{align*} -\frac{3 \, a^{2} c x^{2} + 3 \, a c x + c}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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