Optimal. Leaf size=189 \[ \frac{c^4 (1-a x)^{10} \sqrt{c-a^2 c x^2}}{10 a \sqrt{1-a^2 x^2}}-\frac{2 c^4 (1-a x)^9 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}}+\frac{3 c^4 (1-a x)^8 \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}-\frac{8 c^4 (1-a x)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.109391, antiderivative size = 189, 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 = {6143, 6140, 43} \[ \frac{c^4 (1-a x)^{10} \sqrt{c-a^2 c x^2}}{10 a \sqrt{1-a^2 x^2}}-\frac{2 c^4 (1-a x)^9 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}}+\frac{3 c^4 (1-a x)^8 \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}-\frac{8 c^4 (1-a x)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=\frac{\left (c^4 \sqrt{c-a^2 c x^2}\right ) \int e^{-3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{9/2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^4 \sqrt{c-a^2 c x^2}\right ) \int (1-a x)^6 (1+a x)^3 \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^4 \sqrt{c-a^2 c x^2}\right ) \int \left (8 (1-a x)^6-12 (1-a x)^7+6 (1-a x)^8-(1-a x)^9\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{8 c^4 (1-a x)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}}+\frac{3 c^4 (1-a x)^8 \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}-\frac{2 c^4 (1-a x)^9 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}}+\frac{c^4 (1-a x)^{10} \sqrt{c-a^2 c x^2}}{10 a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0538006, size = 68, normalized size = 0.36 \[ \frac{c^4 (a x-1)^7 \left (21 a^3 x^3+77 a^2 x^2+98 a x+44\right ) \sqrt{c-a^2 c x^2}}{210 a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 97, normalized size = 0.5 \begin{align*}{\frac{x \left ( 21\,{a}^{9}{x}^{9}-70\,{x}^{8}{a}^{8}+240\,{x}^{6}{a}^{6}-210\,{x}^{5}{a}^{5}-252\,{x}^{4}{a}^{4}+420\,{x}^{3}{a}^{3}-315\,ax+210 \right ) }{210\, \left ( ax+1 \right ) ^{6} \left ( ax-1 \right ) ^{6}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{9}{2}}} \left ( -{a}^{2}{x}^{2}+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 \frac{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{2}}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.88936, size = 266, normalized size = 1.41 \begin{align*} -\frac{{\left (21 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} + 240 \, a^{6} c^{4} x^{7} - 210 \, a^{5} c^{4} x^{6} - 252 \, a^{4} c^{4} x^{5} + 420 \, a^{3} c^{4} x^{4} - 315 \, a c^{4} x^{2} + 210 \, c^{4} x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{210 \,{\left (a^{2} x^{2} - 1\right )}} \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 \frac{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{2}}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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