Optimal. Leaf size=73 \[ \frac {c^4 (1-a x)^9}{9 a}-\frac {3 c^4 (1-a x)^8}{4 a}+\frac {12 c^4 (1-a x)^7}{7 a}-\frac {4 c^4 (1-a x)^6}{3 a} \]
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Rubi [A] time = 0.06, antiderivative size = 73, 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} \[ \frac {c^4 (1-a x)^9}{9 a}-\frac {3 c^4 (1-a x)^8}{4 a}+\frac {12 c^4 (1-a x)^7}{7 a}-\frac {4 c^4 (1-a x)^6}{3 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=c^4 \int (1-a x)^5 (1+a x)^3 \, dx\\ &=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\\ &=-\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*}
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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.
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fricas [A] time = 0.53, size = 81, normalized size = 1.11 \[ -\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{4} \, a^{7} c^{4} x^{8} + \frac {2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac {3}{2} \, a^{3} c^{4} x^{4} - \frac {2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.65, size = 78, normalized size = 1.07 \[ -\frac {{\left (28 \, c^{4} - \frac {315 \, c^{4}}{a x + 1} + \frac {1440 \, c^{4}}{{\left (a x + 1\right )}^{2}} - \frac {3360 \, c^{4}}{{\left (a x + 1\right )}^{3}} + \frac {4032 \, c^{4}}{{\left (a x + 1\right )}^{4}} - \frac {2016 \, c^{4}}{{\left (a x + 1\right )}^{5}}\right )} {\left (a x + 1\right )}^{9}}{252 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 61, normalized size = 0.84 \[ c^{4} \left (-\frac {1}{9} x^{9} a^{8}+\frac {1}{4} a^{7} x^{8}+\frac {2}{7} x^{7} a^{6}-x^{6} a^{5}+\frac {3}{2} x^{4} a^{3}-\frac {2}{3} x^{3} a^{2}-a \,x^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 81, normalized size = 1.11 \[ -\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{4} \, a^{7} c^{4} x^{8} + \frac {2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac {3}{2} \, a^{3} c^{4} x^{4} - \frac {2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 81, normalized size = 1.11 \[ -\frac {a^8\,c^4\,x^9}{9}+\frac {a^7\,c^4\,x^8}{4}+\frac {2\,a^6\,c^4\,x^7}{7}-a^5\,c^4\,x^6+\frac {3\,a^3\,c^4\,x^4}{2}-\frac {2\,a^2\,c^4\,x^3}{3}-a\,c^4\,x^2+c^4\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, 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.
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