Optimal. Leaf size=55 \[ -\frac {c^3 (1-a x)^7}{7 a}+\frac {2 c^3 (1-a x)^6}{3 a}-\frac {4 c^3 (1-a x)^5}{5 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6140, 43} \[ -\frac {c^3 (1-a x)^7}{7 a}+\frac {2 c^3 (1-a x)^6}{3 a}-\frac {4 c^3 (1-a x)^5}{5 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6140
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int (1-a x)^4 (1+a x)^2 \, dx\\ &=c^3 \int \left (4 (1-a x)^4-4 (1-a x)^5+(1-a x)^6\right ) \, dx\\ &=-\frac {4 c^3 (1-a x)^5}{5 a}+\frac {2 c^3 (1-a x)^6}{3 a}-\frac {c^3 (1-a x)^7}{7 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 31, normalized size = 0.56 \[ \frac {c^3 (a x-1)^5 \left (15 a^2 x^2+40 a x+29\right )}{105 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 69, normalized size = 1.25 \[ \frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {1}{3} \, a^{5} c^{3} x^{6} - \frac {1}{5} \, a^{4} c^{3} x^{5} + a^{3} c^{3} x^{4} - \frac {1}{3} \, a^{2} c^{3} x^{3} - a c^{3} x^{2} + c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 66, normalized size = 1.20 \[ \frac {{\left (15 \, c^{3} - \frac {140 \, c^{3}}{a x + 1} + \frac {504 \, c^{3}}{{\left (a x + 1\right )}^{2}} - \frac {840 \, c^{3}}{{\left (a x + 1\right )}^{3}} + \frac {560 \, c^{3}}{{\left (a x + 1\right )}^{4}}\right )} {\left (a x + 1\right )}^{7}}{105 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 52, normalized size = 0.95 \[ c^{3} \left (\frac {1}{7} x^{7} a^{6}-\frac {1}{3} x^{6} a^{5}-\frac {1}{5} a^{4} x^{5}+x^{4} a^{3}-\frac {1}{3} x^{3} a^{2}-a \,x^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 69, normalized size = 1.25 \[ \frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {1}{3} \, a^{5} c^{3} x^{6} - \frac {1}{5} \, a^{4} c^{3} x^{5} + a^{3} c^{3} x^{4} - \frac {1}{3} \, a^{2} c^{3} x^{3} - a c^{3} x^{2} + c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 69, normalized size = 1.25 \[ \frac {a^6\,c^3\,x^7}{7}-\frac {a^5\,c^3\,x^6}{3}-\frac {a^4\,c^3\,x^5}{5}+a^3\,c^3\,x^4-\frac {a^2\,c^3\,x^3}{3}-a\,c^3\,x^2+c^3\,x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 70, normalized size = 1.27 \[ \frac {a^{6} c^{3} x^{7}}{7} - \frac {a^{5} c^{3} x^{6}}{3} - \frac {a^{4} c^{3} x^{5}}{5} + a^{3} c^{3} x^{4} - \frac {a^{2} c^{3} x^{3}}{3} - a c^{3} x^{2} + c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________