Optimal. Leaf size=31 \[ \frac {4 a x+1}{60 a^3 c^9 (1-a x)^6 (a x+1)^{10}} \]
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Rubi [A] time = 0.09, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 81} \[ \frac {4 a x+1}{60 a^3 c^9 (1-a x)^6 (a x+1)^{10}} \]
Antiderivative was successfully verified.
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Rule 81
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{-4 \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^9} \, dx &=\frac {\int \frac {x^2}{(1-a x)^7 (1+a x)^{11}} \, dx}{c^9}\\ &=\frac {1+4 a x}{60 a^3 c^9 (1-a x)^6 (1+a x)^{10}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 30, normalized size = 0.97 \[ \frac {4 a x+1}{60 a^3 c^9 (a x-1)^6 (a x+1)^{10}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 169, normalized size = 5.45 \[ \frac {4 \, a x + 1}{60 \, {\left (a^{19} c^{9} x^{16} + 4 \, a^{18} c^{9} x^{15} - 20 \, a^{16} c^{9} x^{13} - 20 \, a^{15} c^{9} x^{12} + 36 \, a^{14} c^{9} x^{11} + 64 \, a^{13} c^{9} x^{10} - 20 \, a^{12} c^{9} x^{9} - 90 \, a^{11} c^{9} x^{8} - 20 \, a^{10} c^{9} x^{7} + 64 \, a^{9} c^{9} x^{6} + 36 \, a^{8} c^{9} x^{5} - 20 \, a^{7} c^{9} x^{4} - 20 \, a^{6} c^{9} x^{3} + 4 \, a^{4} c^{9} x + a^{3} c^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 139, normalized size = 4.48 \[ -\frac {2145 \, a^{5} x^{5} - 12540 \, a^{4} x^{4} + 30030 \, a^{3} x^{3} - 37080 \, a^{2} x^{2} + 23841 \, a x - 6476}{983040 \, {\left (a x - 1\right )}^{6} a^{3} c^{9}} + \frac {2145 \, a^{9} x^{9} + 21780 \, a^{8} x^{8} + 99660 \, a^{7} x^{7} + 270480 \, a^{6} x^{6} + 481446 \, a^{5} x^{5} + 584920 \, a^{4} x^{4} + 486220 \, a^{3} x^{3} + 265680 \, a^{2} x^{2} + 84065 \, a x + 9908}{983040 \, {\left (a x + 1\right )}^{10} a^{3} c^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 186, normalized size = 6.00 \[ \frac {\frac {1}{12288 a^{3} \left (a x -1\right )^{6}}-\frac {7}{20480 a^{3} \left (a x -1\right )^{5}}-\frac {11}{8192 a^{3} \left (a x -1\right )^{3}}+\frac {121}{65536 a^{3} \left (a x -1\right )^{2}}-\frac {143}{65536 a^{3} \left (a x -1\right )}+\frac {13}{16384 a^{3} \left (a x -1\right )^{4}}-\frac {1}{1280 a^{3} \left (a x +1\right )^{10}}-\frac {1}{768 a^{3} \left (a x +1\right )^{9}}+\frac {7}{6144 a^{3} \left (a x +1\right )^{6}}+\frac {21}{10240 a^{3} \left (a x +1\right )^{5}}+\frac {21}{8192 a^{3} \left (a x +1\right )^{4}}+\frac {11}{4096 a^{3} \left (a x +1\right )^{3}}+\frac {165}{65536 a^{3} \left (a x +1\right )^{2}}+\frac {143}{65536 a^{3} \left (a x +1\right )}-\frac {1}{1024 a^{3} \left (a x +1\right )^{8}}}{c^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 169, normalized size = 5.45 \[ \frac {4 \, a x + 1}{60 \, {\left (a^{19} c^{9} x^{16} + 4 \, a^{18} c^{9} x^{15} - 20 \, a^{16} c^{9} x^{13} - 20 \, a^{15} c^{9} x^{12} + 36 \, a^{14} c^{9} x^{11} + 64 \, a^{13} c^{9} x^{10} - 20 \, a^{12} c^{9} x^{9} - 90 \, a^{11} c^{9} x^{8} - 20 \, a^{10} c^{9} x^{7} + 64 \, a^{9} c^{9} x^{6} + 36 \, a^{8} c^{9} x^{5} - 20 \, a^{7} c^{9} x^{4} - 20 \, a^{6} c^{9} x^{3} + 4 \, a^{4} c^{9} x + a^{3} c^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.54, size = 28, normalized size = 0.90 \[ \frac {4\,a\,x+1}{60\,a^3\,c^9\,{\left (a\,x-1\right )}^6\,{\left (a\,x+1\right )}^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.21, size = 182, normalized size = 5.87 \[ - \frac {- 4 a x - 1}{60 a^{19} c^{9} x^{16} + 240 a^{18} c^{9} x^{15} - 1200 a^{16} c^{9} x^{13} - 1200 a^{15} c^{9} x^{12} + 2160 a^{14} c^{9} x^{11} + 3840 a^{13} c^{9} x^{10} - 1200 a^{12} c^{9} x^{9} - 5400 a^{11} c^{9} x^{8} - 1200 a^{10} c^{9} x^{7} + 3840 a^{9} c^{9} x^{6} + 2160 a^{8} c^{9} x^{5} - 1200 a^{7} c^{9} x^{4} - 1200 a^{6} c^{9} x^{3} + 240 a^{4} c^{9} x + 60 a^{3} c^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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