Optimal. Leaf size=69 \[ -\frac {c^4 (a x+1)^9}{9 a}+\frac {3 c^4 (a x+1)^8}{4 a}-\frac {12 c^4 (a x+1)^7}{7 a}+\frac {4 c^4 (a x+1)^6}{3 a} \]
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Rubi [A] time = 0.06, antiderivative size = 69, 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} \[ -\frac {c^4 (a x+1)^9}{9 a}+\frac {3 c^4 (a x+1)^8}{4 a}-\frac {12 c^4 (a x+1)^7}{7 a}+\frac {4 c^4 (a x+1)^6}{3 a} \]
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)^3 (1+a x)^5 \, 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 = 0.57 \[ -\frac {c^4 (a x+1)^6 \left (28 a^3 x^3-105 a^2 x^2+138 a x-65\right )}{252 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 79, normalized size = 1.14 \[ -\frac {1}{9} \, a^{8} c^{4} x^{9} - \frac {1}{4} \, a^{7} c^{4} x^{8} + \frac {2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac {3}{2} \, a^{3} c^{4} x^{4} - \frac {2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 79, normalized size = 1.14 \[ -\frac {1}{9} \, a^{8} c^{4} x^{9} - \frac {1}{4} \, a^{7} c^{4} x^{8} + \frac {2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac {3}{2} \, a^{3} c^{4} x^{4} - \frac {2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 59, normalized size = 0.86 \[ c^{4} \left (-\frac {1}{9} x^{9} a^{8}-\frac {1}{4} a^{7} x^{8}+\frac {2}{7} x^{7} a^{6}+x^{6} a^{5}-\frac {3}{2} x^{4} a^{3}-\frac {2}{3} x^{3} a^{2}+a \,x^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 79, normalized size = 1.14 \[ -\frac {1}{9} \, a^{8} c^{4} x^{9} - \frac {1}{4} \, a^{7} c^{4} x^{8} + \frac {2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac {3}{2} \, a^{3} c^{4} x^{4} - \frac {2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 79, normalized size = 1.14 \[ -\frac {a^8\,c^4\,x^9}{9}-\frac {a^7\,c^4\,x^8}{4}+\frac {2\,a^6\,c^4\,x^7}{7}+a^5\,c^4\,x^6-\frac {3\,a^3\,c^4\,x^4}{2}-\frac {2\,a^2\,c^4\,x^3}{3}+a\,c^4\,x^2+c^4\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 87, normalized size = 1.26 \[ - \frac {a^{8} c^{4} x^{9}}{9} - \frac {a^{7} c^{4} x^{8}}{4} + \frac {2 a^{6} c^{4} x^{7}}{7} + a^{5} c^{4} x^{6} - \frac {3 a^{3} c^{4} x^{4}}{2} - \frac {2 a^{2} c^{4} x^{3}}{3} + a c^{4} x^{2} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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