Optimal. Leaf size=322 \[ \frac{c^4 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{8 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{7 a^8 x^7 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^9 x^8 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^4 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.155947, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6197, 6193, 88} \[ \frac{c^4 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{8 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{7 a^8 x^7 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^9 x^8 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^4 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 88
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{9/2} \, dx &=\frac{\left (c^4 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c^4 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \frac{(-1+a x)^6 (1+a x)^3}{x^9} \, dx}{a^9 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c^4 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \left (a^9+\frac{1}{x^9}-\frac{3 a}{x^8}+\frac{8 a^3}{x^6}-\frac{6 a^4}{x^5}-\frac{6 a^5}{x^4}+\frac{8 a^6}{x^3}-\frac{3 a^8}{x}\right ) \, dx}{a^9 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=-\frac{c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^9 \sqrt{1-\frac{1}{a^2 x^2}} x^8}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{7 a^8 \sqrt{1-\frac{1}{a^2 x^2}} x^7}-\frac{8 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 \sqrt{1-\frac{1}{a^2 x^2}} x^5}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^5 \sqrt{1-\frac{1}{a^2 x^2}} x^4}+\frac{2 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 \sqrt{1-\frac{1}{a^2 x^2}} x^3}-\frac{4 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}+\frac{c^4 \sqrt{c-\frac{c}{a^2 x^2}} x}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}} \log (x)}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0809334, size = 97, normalized size = 0.3 \[ \frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} \left (-\frac{4 a^6}{x^2}+\frac{2 a^5}{x^3}+\frac{3 a^4}{2 x^4}-\frac{8 a^3}{5 x^5}+a^9 x-3 a^8 \log (x)+\frac{3 a}{7 x^7}-\frac{1}{8 x^8}\right )}{a^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.243, size = 112, normalized size = 0.4 \begin{align*} -{\frac{ \left ( -280\,{a}^{9}{x}^{9}+840\,{a}^{8}\ln \left ( x \right ){x}^{8}+1120\,{x}^{6}{a}^{6}-560\,{x}^{5}{a}^{5}-420\,{x}^{4}{a}^{4}+448\,{x}^{3}{a}^{3}-120\,ax+35 \right ) x}{280\, \left ( ax-1 \right ) ^{3} \left ({a}^{2}{x}^{2}-1 \right ) ^{3}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{9}{2}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{9}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56452, size = 228, normalized size = 0.71 \begin{align*} \frac{{\left (280 \, a^{9} c^{4} x^{9} - 840 \, a^{8} c^{4} x^{8} \log \left (x\right ) - 1120 \, a^{6} c^{4} x^{6} + 560 \, a^{5} c^{4} x^{5} + 420 \, a^{4} c^{4} x^{4} - 448 \, a^{3} c^{4} x^{3} + 120 \, a c^{4} x - 35 \, c^{4}\right )} \sqrt{a^{2} c}}{280 \, a^{10} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{9}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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