Optimal. Leaf size=17 \[ \frac {c^2 (a x+1)^5}{5 a} \]
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Rubi [A] time = 0.06, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6167, 6140, 32} \[ \frac {c^2 (a x+1)^5}{5 a} \]
Antiderivative was successfully verified.
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Rule 32
Rule 6140
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx\\ &=c^2 \int (1+a x)^4 \, dx\\ &=\frac {c^2 (1+a x)^5}{5 a}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 37, normalized size = 2.18 \[ c^2 \left (\frac {a^4 x^5}{5}+a^3 x^4+2 a^2 x^3+2 a x^2+x\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 47, normalized size = 2.76 \[ \frac {1}{5} \, a^{4} c^{2} x^{5} + a^{3} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{3} + 2 \, a c^{2} x^{2} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 64, normalized size = 3.76 \[ \frac {{\left (c^{2} + \frac {10 \, c^{2}}{a x - 1} + \frac {40 \, c^{2}}{{\left (a x - 1\right )}^{2}} + \frac {80 \, c^{2}}{{\left (a x - 1\right )}^{3}} + \frac {80 \, c^{2}}{{\left (a x - 1\right )}^{4}}\right )} {\left (a x - 1\right )}^{5}}{5 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 16, normalized size = 0.94 \[ \frac {c^{2} \left (a x +1\right )^{5}}{5 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 47, normalized size = 2.76 \[ \frac {1}{5} \, a^{4} c^{2} x^{5} + a^{3} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{3} + 2 \, a c^{2} x^{2} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 47, normalized size = 2.76 \[ \frac {a^4\,c^2\,x^5}{5}+a^3\,c^2\,x^4+2\,a^2\,c^2\,x^3+2\,a\,c^2\,x^2+c^2\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 48, normalized size = 2.82 \[ \frac {a^{4} c^{2} x^{5}}{5} + a^{3} c^{2} x^{4} + 2 a^{2} c^{2} x^{3} + 2 a c^{2} x^{2} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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