Optimal. Leaf size=87 \[ \frac {1}{10} a^6 c^3 x^{10}+\frac {2}{9} a^5 c^3 x^9-\frac {1}{8} a^4 c^3 x^8-\frac {4}{7} a^3 c^3 x^7-\frac {1}{6} a^2 c^3 x^6+\frac {2}{5} a c^3 x^5+\frac {c^3 x^4}{4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 88} \[ \frac {1}{10} a^6 c^3 x^{10}+\frac {2}{9} a^5 c^3 x^9-\frac {1}{8} a^4 c^3 x^8-\frac {4}{7} a^3 c^3 x^7-\frac {1}{6} a^2 c^3 x^6+\frac {2}{5} a c^3 x^5+\frac {c^3 x^4}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 6150
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x^3 (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (x^3+2 a x^4-a^2 x^5-4 a^3 x^6-a^4 x^7+2 a^5 x^8+a^6 x^9\right ) \, dx\\ &=\frac {c^3 x^4}{4}+\frac {2}{5} a c^3 x^5-\frac {1}{6} a^2 c^3 x^6-\frac {4}{7} a^3 c^3 x^7-\frac {1}{8} a^4 c^3 x^8+\frac {2}{9} a^5 c^3 x^9+\frac {1}{10} a^6 c^3 x^{10}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 70, normalized size = 0.80 \[ c^3 \left (\frac {a^6 x^{10}}{10}+\frac {2 a^5 x^9}{9}-\frac {a^4 x^8}{8}-\frac {4 a^3 x^7}{7}-\frac {a^2 x^6}{6}+\frac {2 a x^5}{5}+\frac {x^4}{4}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 73, normalized size = 0.84 \[ \frac {1}{10} \, a^{6} c^{3} x^{10} + \frac {2}{9} \, a^{5} c^{3} x^{9} - \frac {1}{8} \, a^{4} c^{3} x^{8} - \frac {4}{7} \, a^{3} c^{3} x^{7} - \frac {1}{6} \, a^{2} c^{3} x^{6} + \frac {2}{5} \, a c^{3} x^{5} + \frac {1}{4} \, c^{3} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 73, normalized size = 0.84 \[ \frac {1}{10} \, a^{6} c^{3} x^{10} + \frac {2}{9} \, a^{5} c^{3} x^{9} - \frac {1}{8} \, a^{4} c^{3} x^{8} - \frac {4}{7} \, a^{3} c^{3} x^{7} - \frac {1}{6} \, a^{2} c^{3} x^{6} + \frac {2}{5} \, a c^{3} x^{5} + \frac {1}{4} \, c^{3} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 57, normalized size = 0.66 \[ c^{3} \left (\frac {1}{10} a^{6} x^{10}+\frac {2}{9} a^{5} x^{9}-\frac {1}{8} a^{4} x^{8}-\frac {4}{7} a^{3} x^{7}-\frac {1}{6} a^{2} x^{6}+\frac {2}{5} a \,x^{5}+\frac {1}{4} x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 73, normalized size = 0.84 \[ \frac {1}{10} \, a^{6} c^{3} x^{10} + \frac {2}{9} \, a^{5} c^{3} x^{9} - \frac {1}{8} \, a^{4} c^{3} x^{8} - \frac {4}{7} \, a^{3} c^{3} x^{7} - \frac {1}{6} \, a^{2} c^{3} x^{6} + \frac {2}{5} \, a c^{3} x^{5} + \frac {1}{4} \, c^{3} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 73, normalized size = 0.84 \[ \frac {a^6\,c^3\,x^{10}}{10}+\frac {2\,a^5\,c^3\,x^9}{9}-\frac {a^4\,c^3\,x^8}{8}-\frac {4\,a^3\,c^3\,x^7}{7}-\frac {a^2\,c^3\,x^6}{6}+\frac {2\,a\,c^3\,x^5}{5}+\frac {c^3\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.10, size = 82, normalized size = 0.94 \[ \frac {a^{6} c^{3} x^{10}}{10} + \frac {2 a^{5} c^{3} x^{9}}{9} - \frac {a^{4} c^{3} x^{8}}{8} - \frac {4 a^{3} c^{3} x^{7}}{7} - \frac {a^{2} c^{3} x^{6}}{6} + \frac {2 a c^{3} x^{5}}{5} + \frac {c^{3} x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________