Optimal. Leaf size=31 \[ -\frac{1-4 a x}{60 a^3 c^9 (1-a x)^{10} (a x+1)^6} \]
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Rubi [A] time = 0.0890639, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 81} \[ -\frac{1-4 a x}{60 a^3 c^9 (1-a x)^{10} (a x+1)^6} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 81
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)^{11} (1+a x)^7} \, dx}{c^9}\\ &=-\frac{1-4 a x}{60 a^3 c^9 (1-a x)^{10} (1+a x)^6}\\ \end{align*}
Mathematica [A] time = 0.175405, size = 30, normalized size = 0.97 \[ \frac{4 a x-1}{60 a^3 c^9 (a x-1)^{10} (a x+1)^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 186, normalized size = 6. \begin{align*}{\frac{1}{{c}^{9}} \left ( -{\frac{1}{12288\,{a}^{3} \left ( ax+1 \right ) ^{6}}}-{\frac{7}{20480\,{a}^{3} \left ( ax+1 \right ) ^{5}}}-{\frac{11}{8192\,{a}^{3} \left ( ax+1 \right ) ^{3}}}-{\frac{121}{65536\,{a}^{3} \left ( ax+1 \right ) ^{2}}}-{\frac{143}{65536\,{a}^{3} \left ( ax+1 \right ) }}-{\frac{13}{16384\,{a}^{3} \left ( ax+1 \right ) ^{4}}}+{\frac{1}{1280\,{a}^{3} \left ( ax-1 \right ) ^{10}}}-{\frac{1}{768\,{a}^{3} \left ( ax-1 \right ) ^{9}}}-{\frac{7}{6144\,{a}^{3} \left ( ax-1 \right ) ^{6}}}+{\frac{21}{10240\,{a}^{3} \left ( ax-1 \right ) ^{5}}}-{\frac{21}{8192\,{a}^{3} \left ( ax-1 \right ) ^{4}}}+{\frac{11}{4096\,{a}^{3} \left ( ax-1 \right ) ^{3}}}-{\frac{165}{65536\,{a}^{3} \left ( ax-1 \right ) ^{2}}}+{\frac{143}{65536\,{a}^{3} \left ( ax-1 \right ) }}+{\frac{1}{1024\,{a}^{3} \left ( ax-1 \right ) ^{8}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11362, size = 228, normalized size = 7.35 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14771, size = 370, normalized size = 11.94 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 132.855, size = 178, normalized size = 5.74 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13442, size = 188, normalized size = 6.06 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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