Optimal. Leaf size=35 \[ \frac {c^3 (a x+1)^6}{3 a}-\frac {c^3 (a x+1)^7}{7 a} \]
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Rubi [A] time = 0.04, antiderivative size = 35, 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} \[ \frac {c^3 (a x+1)^6}{3 a}-\frac {c^3 (a x+1)^7}{7 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int (1-a x) (1+a x)^5 \, dx\\ &=c^3 \int \left (2 (1+a x)^5-(1+a x)^6\right ) \, dx\\ &=\frac {c^3 (1+a x)^6}{3 a}-\frac {c^3 (1+a x)^7}{7 a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.66 \[ -\frac {c^3 (a x+1)^6 (3 a x-4)}{21 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 59, normalized size = 1.69 \[ -\frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac {5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 59, normalized size = 1.69 \[ -\frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac {5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 45, normalized size = 1.29 \[ c^{3} \left (-\frac {1}{7} x^{7} a^{6}-\frac {2}{3} x^{6} a^{5}-a^{4} x^{5}+\frac {5}{3} x^{3} a^{2}+2 a \,x^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 59, normalized size = 1.69 \[ -\frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac {5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 59, normalized size = 1.69 \[ -\frac {a^6\,c^3\,x^7}{7}-\frac {2\,a^5\,c^3\,x^6}{3}-a^4\,c^3\,x^5+\frac {5\,a^2\,c^3\,x^3}{3}+2\,a\,c^3\,x^2+c^3\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 63, normalized size = 1.80 \[ - \frac {a^{6} c^{3} x^{7}}{7} - \frac {2 a^{5} c^{3} x^{6}}{3} - a^{4} c^{3} x^{5} + \frac {5 a^{2} c^{3} x^{3}}{3} + 2 a c^{3} x^{2} + c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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