Optimal. Leaf size=234 \[ \frac{(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{32 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{8 (1-a x)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}} \]
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Rubi [A] time = 0.211301, antiderivative size = 234, 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 = {6192, 6193, 43} \[ \frac{(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{32 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{8 (1-a x)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=\frac{\left (c-a^2 c x^2\right )^{9/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9 \, dx}{\left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{\left (c-a^2 c x^2\right )^{9/2} \int (-1+a x)^5 (1+a x)^4 \, dx}{a^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{\left (c-a^2 c x^2\right )^{9/2} \int \left (16 (-1+a x)^5+32 (-1+a x)^6+24 (-1+a x)^7+8 (-1+a x)^8+(-1+a x)^9\right ) \, dx}{a^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{8 (1-a x)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}-\frac{32 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}+\frac{3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}-\frac{8 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}+\frac{(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}\\ \end{align*}
Mathematica [A] time = 0.0609177, size = 79, normalized size = 0.34 \[ \frac{c^4 (a x-1)^6 \left (63 a^4 x^4+308 a^3 x^3+588 a^2 x^2+528 a x+193\right ) \sqrt{c-a^2 c x^2}}{630 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 116, normalized size = 0.5 \begin{align*}{\frac{x \left ( 63\,{a}^{9}{x}^{9}-70\,{x}^{8}{a}^{8}-315\,{a}^{7}{x}^{7}+360\,{x}^{6}{a}^{6}+630\,{x}^{5}{a}^{5}-756\,{x}^{4}{a}^{4}-630\,{x}^{3}{a}^{3}+840\,{a}^{2}{x}^{2}+315\,ax-630 \right ) }{630\, \left ( ax+1 \right ) ^{4} \left ( ax-1 \right ) ^{5}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{9}{2}}}\sqrt{{\frac{ax-1}{ax+1}}}} \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 (-a^{2} c x^{2} + c\right )}^{\frac{9}{2}} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8097, size = 261, normalized size = 1.12 \begin{align*} \frac{{\left (63 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} - 315 \, a^{7} c^{4} x^{8} + 360 \, a^{6} c^{4} x^{7} + 630 \, a^{5} c^{4} x^{6} - 756 \, a^{4} c^{4} x^{5} - 630 \, a^{3} c^{4} x^{4} + 840 \, a^{2} c^{4} x^{3} + 315 \, a c^{4} x^{2} - 630 \, c^{4} x\right )} \sqrt{-a^{2} c}}{630 \, a} \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 (-a^{2} c x^{2} + c\right )}^{\frac{9}{2}} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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