Optimal. Leaf size=97 \[ -\frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}-\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac{2 (a x+1)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]
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Rubi [A] time = 0.119516, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}-\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac{2 (a x+1)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]
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 )^{7/2}} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=-\left (c \int \frac{(1+a x)^2}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\right )\\ &=-\frac{2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac{5}{7} \int \frac{1}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=-\frac{2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac{4 \int \frac{1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{7 c}\\ &=-\frac{2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{8 \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{21 c^2}\\ &=-\frac{2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.053841, size = 96, normalized size = 0.99 \[ -\frac{\sqrt{1-a^2 x^2} \left (-8 a^5 x^5+16 a^4 x^4+4 a^3 x^3-24 a^2 x^2+9 a x+6\right )}{21 a c^3 (1-a x)^{7/2} (a x+1)^{3/2} \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 64, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax+1 \right ) ^{2} \left ( 8\,{x}^{5}{a}^{5}-16\,{x}^{4}{a}^{4}-4\,{x}^{3}{a}^{3}+24\,{a}^{2}{x}^{2}-9\,ax-6 \right ) }{21\,a} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{7}{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 = 3.02914, size = 250, normalized size = 2.58 \begin{align*} \frac{{\left (8 \, a^{5} x^{5} - 16 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 24 \, a^{2} x^{2} - 9 \, a x - 6\right )} \sqrt{-a^{2} c x^{2} + c}}{21 \,{\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x + 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{7}{2}} \left (a x - 1\right )}\, dx \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{7}{2}}{\left (a x - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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