Optimal. Leaf size=120 \[ -\frac{16 x}{45 c^4 \sqrt{c-a^2 c x^2}}-\frac{8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac{2 (a x+1)}{9 a \left (c-a^2 c x^2\right )^{9/2}} \]
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Rubi [A] time = 0.131384, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {6167, 6141, 653, 192, 191} \[ -\frac{16 x}{45 c^4 \sqrt{c-a^2 c x^2}}-\frac{8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac{2 (a x+1)}{9 a \left (c-a^2 c x^2\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6141
Rule 653
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\\ &=-\left (c \int \frac{(1+a x)^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx\right )\\ &=-\frac{2 (1+a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac{7}{9} \int \frac{1}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\\ &=-\frac{2 (1+a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac{x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac{2 \int \frac{1}{\left (c-a^2 c x^2\right )^{7/2}} \, dx}{3 c}\\ &=-\frac{2 (1+a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac{x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac{2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{8 \int \frac{1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{15 c^2}\\ &=-\frac{2 (1+a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac{x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac{2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{16 \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{45 c^3}\\ &=-\frac{2 (1+a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac{x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac{2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{16 x}{45 c^4 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0594047, size = 112, normalized size = 0.93 \[ \frac{\sqrt{1-a^2 x^2} \left (-16 a^7 x^7+32 a^6 x^6+24 a^5 x^5-80 a^4 x^4+10 a^3 x^3+60 a^2 x^2-25 a x-10\right )}{45 a c^4 (1-a x)^{9/2} (a x+1)^{5/2} \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 80, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax+1 \right ) ^{2} \left ( 16\,{a}^{7}{x}^{7}-32\,{x}^{6}{a}^{6}-24\,{x}^{5}{a}^{5}+80\,{x}^{4}{a}^{4}-10\,{x}^{3}{a}^{3}-60\,{a}^{2}{x}^{2}+25\,ax+10 \right ) }{45\,a} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 5.04126, size = 317, normalized size = 2.64 \begin{align*} \frac{{\left (16 \, a^{7} x^{7} - 32 \, a^{6} x^{6} - 24 \, a^{5} x^{5} + 80 \, a^{4} x^{4} - 10 \, a^{3} x^{3} - 60 \, a^{2} x^{2} + 25 \, a x + 10\right )} \sqrt{-a^{2} c x^{2} + c}}{45 \,{\left (a^{9} c^{5} x^{8} - 2 \, a^{8} c^{5} x^{7} - 2 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} - 6 \, a^{4} c^{5} x^{3} + 2 \, a^{3} c^{5} x^{2} + 2 \, a^{2} c^{5} x - a c^{5}\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{a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{2}}{\left (a x - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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