Optimal. Leaf size=52 \[ \frac {c^4 (a x+1)^9}{9 a}-\frac {c^4 (a x+1)^8}{2 a}+\frac {4 c^4 (a x+1)^7}{7 a} \]
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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} \[ \frac {c^4 (a x+1)^9}{9 a}-\frac {c^4 (a x+1)^8}{2 a}+\frac {4 c^4 (a x+1)^7}{7 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx\\ &=c^4 \int (1-a x)^2 (1+a x)^6 \, dx\\ &=c^4 \int \left (4 (1+a x)^6-4 (1+a x)^7+(1+a x)^8\right ) \, dx\\ &=\frac {4 c^4 (1+a x)^7}{7 a}-\frac {c^4 (1+a x)^8}{2 a}+\frac {c^4 (1+a x)^9}{9 a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 31, normalized size = 0.60 \[ \frac {c^4 (a x+1)^7 \left (14 a^2 x^2-35 a x+23\right )}{126 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 92, normalized size = 1.77 \[ \frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{2} \, a^{7} c^{4} x^{8} + \frac {4}{7} \, a^{6} c^{4} x^{7} - \frac {2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4}{3} \, a^{2} c^{4} x^{3} + 2 \, a c^{4} x^{2} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 90, normalized size = 1.73 \[ \frac {{\left (14 \, c^{4} + \frac {189 \, c^{4}}{a x - 1} + \frac {1080 \, c^{4}}{{\left (a x - 1\right )}^{2}} + \frac {3360 \, c^{4}}{{\left (a x - 1\right )}^{3}} + \frac {6048 \, c^{4}}{{\left (a x - 1\right )}^{4}} + \frac {6048 \, c^{4}}{{\left (a x - 1\right )}^{5}} + \frac {2688 \, c^{4}}{{\left (a x - 1\right )}^{6}}\right )} {\left (a x - 1\right )}^{9}}{126 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 69, normalized size = 1.33 \[ c^{4} \left (\frac {1}{9} x^{9} a^{8}+\frac {1}{2} a^{7} x^{8}+\frac {4}{7} a^{6} x^{7}-\frac {2}{3} x^{6} a^{5}-2 a^{4} x^{5}-x^{4} a^{3}+\frac {4}{3} x^{3} a^{2}+2 a \,x^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 92, normalized size = 1.77 \[ \frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{2} \, a^{7} c^{4} x^{8} + \frac {4}{7} \, a^{6} c^{4} x^{7} - \frac {2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4}{3} \, a^{2} c^{4} x^{3} + 2 \, a c^{4} x^{2} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 92, normalized size = 1.77 \[ \frac {a^8\,c^4\,x^9}{9}+\frac {a^7\,c^4\,x^8}{2}+\frac {4\,a^6\,c^4\,x^7}{7}-\frac {2\,a^5\,c^4\,x^6}{3}-2\,a^4\,c^4\,x^5-a^3\,c^4\,x^4+\frac {4\,a^2\,c^4\,x^3}{3}+2\,a\,c^4\,x^2+c^4\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 100, normalized size = 1.92 \[ \frac {a^{8} c^{4} x^{9}}{9} + \frac {a^{7} c^{4} x^{8}}{2} + \frac {4 a^{6} c^{4} x^{7}}{7} - \frac {2 a^{5} c^{4} x^{6}}{3} - 2 a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac {4 a^{2} c^{4} x^{3}}{3} + 2 a c^{4} x^{2} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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