Optimal. Leaf size=29 \[ \frac {c x^4}{4}+\frac {2}{5} a c x^5+\frac {1}{6} a^2 c x^6 \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6285, 45}
\begin {gather*} \frac {1}{6} a^2 c x^6+\frac {2}{5} a c x^5+\frac {c x^4}{4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 6285
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx &=c \int x^3 (1+a x)^2 \, dx\\ &=c \int \left (x^3+2 a x^4+a^2 x^5\right ) \, dx\\ &=\frac {c x^4}{4}+\frac {2}{5} a c x^5+\frac {1}{6} a^2 c x^6\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 22, normalized size = 0.76 \begin {gather*} \frac {1}{60} c x^4 \left (15+24 a x+10 a^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 23, normalized size = 0.79
method | result | size |
gosper | \(\frac {c \,x^{4} \left (10 a^{2} x^{2}+24 a x +15\right )}{60}\) | \(21\) |
default | \(c \left (\frac {1}{6} a^{2} x^{6}+\frac {2}{5} a \,x^{5}+\frac {1}{4} x^{4}\right )\) | \(23\) |
norman | \(\frac {1}{4} c \,x^{4}+\frac {2}{5} a c \,x^{5}+\frac {1}{6} a^{2} c \,x^{6}\) | \(24\) |
risch | \(\frac {1}{4} c \,x^{4}+\frac {2}{5} a c \,x^{5}+\frac {1}{6} a^{2} c \,x^{6}\) | \(24\) |
meijerg | \(-\frac {c \left (-\frac {x^{2} a^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{12}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a^{4}}+\frac {c \left (-\frac {2 x \left (-a^{2}\right )^{\frac {7}{2}} \left (21 a^{4} x^{4}+35 a^{2} x^{2}+105\right )}{105 a^{6}}+\frac {2 \left (-a^{2}\right )^{\frac {7}{2}} \arctanh \left (a x \right )}{a^{7}}\right )}{a^{3} \sqrt {-a^{2}}}+\frac {c \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{a^{3} \sqrt {-a^{2}}}+\frac {c \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a^{4}}\) | \(188\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.25, size = 23, normalized size = 0.79 \begin {gather*} \frac {1}{6} \, a^{2} c x^{6} + \frac {2}{5} \, a c x^{5} + \frac {1}{4} \, c x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 23, normalized size = 0.79 \begin {gather*} \frac {1}{6} \, a^{2} c x^{6} + \frac {2}{5} \, a c x^{5} + \frac {1}{4} \, c x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.02, size = 26, normalized size = 0.90 \begin {gather*} \frac {a^{2} c x^{6}}{6} + \frac {2 a c x^{5}}{5} + \frac {c x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.44, size = 23, normalized size = 0.79 \begin {gather*} \frac {1}{6} \, a^{2} c x^{6} + \frac {2}{5} \, a c x^{5} + \frac {1}{4} \, c x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 20, normalized size = 0.69 \begin {gather*} \frac {c\,x^4\,\left (10\,a^2\,x^2+24\,a\,x+15\right )}{60} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________