Optimal. Leaf size=52 \[ -\frac {c^3 (a x+1)^7}{7 a}+\frac {2 c^3 (a x+1)^6}{3 a}-\frac {4 c^3 (a x+1)^5}{5 a} \]
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Rubi [A] time = 0.08, antiderivative size = 52, 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} \[ -\frac {c^3 (a x+1)^7}{7 a}+\frac {2 c^3 (a x+1)^6}{3 a}-\frac {4 c^3 (a x+1)^5}{5 a} \]
Antiderivative was successfully verified.
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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 )^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx\\ &=-\left (c^3 \int (1-a x)^2 (1+a x)^4 \, dx\right )\\ &=-\left (c^3 \int \left (4 (1+a x)^4-4 (1+a x)^5+(1+a x)^6\right ) \, dx\right )\\ &=-\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*}
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Mathematica [A] time = 0.03, size = 31, normalized size = 0.60 \[ -\frac {c^3 (a x+1)^5 \left (15 a^2 x^2-40 a x+29\right )}{105 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 70, normalized size = 1.35 \[ -\frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {1}{3} \, a^{5} c^{3} x^{6} + \frac {1}{5} \, a^{4} c^{3} x^{5} + a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} - a c^{3} x^{2} - c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 70, normalized size = 1.35 \[ -\frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {1}{3} \, a^{5} c^{3} x^{6} + \frac {1}{5} \, a^{4} c^{3} x^{5} + a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} - a c^{3} x^{2} - c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 54, normalized size = 1.04 \[ c^{3} \left (-\frac {1}{7} a^{6} x^{7}-\frac {1}{3} x^{6} a^{5}+\frac {1}{5} a^{4} x^{5}+x^{4} a^{3}+\frac {1}{3} x^{3} a^{2}-a \,x^{2}-x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 70, normalized size = 1.35 \[ -\frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {1}{3} \, a^{5} c^{3} x^{6} + \frac {1}{5} \, a^{4} c^{3} x^{5} + a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} - a c^{3} x^{2} - c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 70, normalized size = 1.35 \[ -\frac {a^6\,c^3\,x^7}{7}-\frac {a^5\,c^3\,x^6}{3}+\frac {a^4\,c^3\,x^5}{5}+a^3\,c^3\,x^4+\frac {a^2\,c^3\,x^3}{3}-a\,c^3\,x^2-c^3\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 70, normalized size = 1.35 \[ - \frac {a^{6} c^{3} x^{7}}{7} - \frac {a^{5} c^{3} x^{6}}{3} + \frac {a^{4} c^{3} x^{5}}{5} + a^{3} c^{3} x^{4} + \frac {a^{2} c^{3} x^{3}}{3} - a c^{3} x^{2} - c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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