Optimal. Leaf size=17 \[ \frac {c^2 (1+a x)^3}{3 a} \]
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Rubi [A]
time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6264, 32}
\begin {gather*} \frac {c^2 (a x+1)^3}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 6264
Rule 6302
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx\\ &=c^2 \int (1+a x)^2 \, dx\\ &=\frac {c^2 (1+a x)^3}{3 a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 27, normalized size = 1.59 \begin {gather*} c^2 x+a c^2 x^2+\frac {1}{3} a^2 c^2 x^3 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 16, normalized size = 0.94
method | result | size |
default | \(\frac {c^{2} \left (a x +1\right )^{3}}{3 a}\) | \(16\) |
gosper | \(\frac {x \left (a^{2} x^{2}+3 a x +3\right ) c^{2}}{3}\) | \(20\) |
risch | \(\frac {a^{2} c^{2} x^{3}}{3}+a \,c^{2} x^{2}+c^{2} x +\frac {c^{2}}{3 a}\) | \(34\) |
norman | \(\frac {-\frac {c^{2}}{a}+\frac {2 a^{2} c^{2} x^{3}}{3}+\frac {a^{3} c^{2} x^{4}}{3}}{a x -1}\) | \(40\) |
meijerg | \(-\frac {c^{2} \left (-\frac {a x \left (-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{15 \left (-a x +1\right )}-4 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{2} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}+\frac {c^{2} x}{-a x +1}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 25, normalized size = 1.47 \begin {gather*} \frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 25, normalized size = 1.47 \begin {gather*} \frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 24, normalized size = 1.41 \begin {gather*} \frac {a^{2} c^{2} x^{3}}{3} + a c^{2} x^{2} + c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs.
\(2 (15) = 30\).
time = 0.40, size = 40, normalized size = 2.35 \begin {gather*} \frac {{\left (c^{2} + \frac {6 \, c^{2}}{a x - 1} + \frac {12 \, c^{2}}{{\left (a x - 1\right )}^{2}}\right )} {\left (a x - 1\right )}^{3}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 19, normalized size = 1.12 \begin {gather*} \frac {c^2\,x\,\left (a^2\,x^2+3\,a\,x+3\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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