Optimal. Leaf size=37 \[ \frac {2 c^3 (1-a x)^3}{3 a}-\frac {c^3 (1-a x)^4}{4 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6264, 45}
\begin {gather*} \frac {2 c^3 (1-a x)^3}{3 a}-\frac {c^3 (1-a x)^4}{4 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 6264
Rule 6302
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx\\ &=-\left (c^3 \int (1-a x)^2 (1+a x) \, dx\right )\\ &=-\left (c^3 \int \left (2 (1-a x)^2-(1-a x)^3\right ) \, dx\right )\\ &=\frac {2 c^3 (1-a x)^3}{3 a}-\frac {c^3 (1-a x)^4}{4 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 30, normalized size = 0.81 \begin {gather*} -\frac {1}{12} c^3 x \left (12-6 a x-4 a^2 x^2+3 a^3 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.13, size = 31, normalized size = 0.84
method | result | size |
gosper | \(-\frac {x \left (3 a^{3} x^{3}-4 a^{2} x^{2}-6 a x +12\right ) c^{3}}{12}\) | \(29\) |
default | \(c^{3} \left (-\frac {1}{4} a^{3} x^{4}+\frac {1}{3} a^{2} x^{3}+\frac {1}{2} a \,x^{2}-x \right )\) | \(31\) |
norman | \(-c^{3} x +\frac {1}{3} a^{2} c^{3} x^{3}-\frac {1}{4} a^{3} c^{3} x^{4}+\frac {1}{2} c^{3} a \,x^{2}\) | \(39\) |
risch | \(-c^{3} x +\frac {1}{3} a^{2} c^{3} x^{3}-\frac {1}{4} a^{3} c^{3} x^{4}+\frac {1}{2} c^{3} a \,x^{2}\) | \(39\) |
meijerg | \(-\frac {c^{3} \left (\frac {a x \left (15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}+\ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{3} \left (-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{3} \left (-a x -\ln \left (-a x +1\right )\right )}{a}+\frac {c^{3} \ln \left (-a x +1\right )}{a}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 38, normalized size = 1.03 \begin {gather*} -\frac {1}{4} \, a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} - c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.33, size = 38, normalized size = 1.03 \begin {gather*} -\frac {1}{4} \, a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} - c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.02, size = 37, normalized size = 1.00 \begin {gather*} - \frac {a^{3} c^{3} x^{4}}{4} + \frac {a^{2} c^{3} x^{3}}{3} + \frac {a c^{3} x^{2}}{2} - c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 38, normalized size = 1.03 \begin {gather*} -\frac {1}{4} \, a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} - c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 38, normalized size = 1.03 \begin {gather*} -\frac {a^3\,c^3\,x^4}{4}+\frac {a^2\,c^3\,x^3}{3}+\frac {a\,c^3\,x^2}{2}-c^3\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________