Optimal. Leaf size=73 \[ -\frac {c^4 (1-a x)^9}{9 a}+\frac {3 c^4 (1-a x)^8}{4 a}-\frac {12 c^4 (1-a x)^7}{7 a}+\frac {4 c^4 (1-a x)^6}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6167, 6140, 43} \[ -\frac {c^4 (1-a x)^9}{9 a}+\frac {3 c^4 (1-a x)^8}{4 a}-\frac {12 c^4 (1-a x)^7}{7 a}+\frac {4 c^4 (1-a x)^6}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6140
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx\\ &=-\left (c^4 \int (1-a x)^5 (1+a x)^3 \, dx\right )\\ &=-\left (c^4 \int \left (8 (1-a x)^5-12 (1-a x)^6+6 (1-a x)^7-(1-a x)^8\right ) \, dx\right )\\ &=\frac {4 c^4 (1-a x)^6}{3 a}-\frac {12 c^4 (1-a x)^7}{7 a}+\frac {3 c^4 (1-a x)^8}{4 a}-\frac {c^4 (1-a x)^9}{9 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 39, normalized size = 0.53 \[ \frac {c^4 (a x-1)^6 \left (28 a^3 x^3+105 a^2 x^2+138 a x+65\right )}{252 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 80, normalized size = 1.10 \[ \frac {1}{9} \, a^{8} c^{4} x^{9} - \frac {1}{4} \, a^{7} c^{4} x^{8} - \frac {2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac {3}{2} \, a^{3} c^{4} x^{4} + \frac {2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} - c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 80, normalized size = 1.10 \[ \frac {1}{9} \, a^{8} c^{4} x^{9} - \frac {1}{4} \, a^{7} c^{4} x^{8} - \frac {2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac {3}{2} \, a^{3} c^{4} x^{4} + \frac {2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} - c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 61, normalized size = 0.84 \[ c^{4} \left (\frac {1}{9} x^{9} a^{8}-\frac {1}{4} a^{7} x^{8}-\frac {2}{7} a^{6} x^{7}+x^{6} a^{5}-\frac {3}{2} x^{4} a^{3}+\frac {2}{3} x^{3} a^{2}+a \,x^{2}-x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 80, normalized size = 1.10 \[ \frac {1}{9} \, a^{8} c^{4} x^{9} - \frac {1}{4} \, a^{7} c^{4} x^{8} - \frac {2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac {3}{2} \, a^{3} c^{4} x^{4} + \frac {2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} - c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 80, normalized size = 1.10 \[ \frac {a^8\,c^4\,x^9}{9}-\frac {a^7\,c^4\,x^8}{4}-\frac {2\,a^6\,c^4\,x^7}{7}+a^5\,c^4\,x^6-\frac {3\,a^3\,c^4\,x^4}{2}+\frac {2\,a^2\,c^4\,x^3}{3}+a\,c^4\,x^2-c^4\,x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.31, size = 87, normalized size = 1.19 \[ \frac {a^{8} c^{4} x^{9}}{9} - \frac {a^{7} c^{4} x^{8}}{4} - \frac {2 a^{6} c^{4} x^{7}}{7} + a^{5} c^{4} x^{6} - \frac {3 a^{3} c^{4} x^{4}}{2} + \frac {2 a^{2} c^{4} x^{3}}{3} + a c^{4} x^{2} - c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________