Optimal. Leaf size=98 \[ \frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac{2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]
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Rubi [A] time = 0.08145, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6142, 653, 192, 191} \[ \frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac{2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6142
Rule 653
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=c \int \frac{(1-a x)^2}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\\ &=-\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 = 0.0544708, size = 96, normalized size = 0.98 \[ -\frac{\sqrt{1-a^2 x^2} \left (8 a^5 x^5+16 a^4 x^4-4 a^3 x^3-24 a^2 x^2-9 a x+6\right )}{21 a c^3 (1-a x)^{3/2} (a x+1)^{7/2} \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, 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 = 4.60181, size = 251, normalized size = 2.56 \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}{- a^{7} c^{3} x^{7} \sqrt{- a^{2} c x^{2} + c} - a^{6} c^{3} x^{6} \sqrt{- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt{- a^{2} c x^{2} + c} + 3 a^{4} c^{3} x^{4} \sqrt{- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt{- a^{2} c x^{2} + c} - 3 a^{2} c^{3} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{3} x \sqrt{- a^{2} c x^{2} + c} + c^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int - \frac{1}{- a^{7} c^{3} x^{7} \sqrt{- a^{2} c x^{2} + c} - a^{6} c^{3} x^{6} \sqrt{- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt{- a^{2} c x^{2} + c} + 3 a^{4} c^{3} x^{4} \sqrt{- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt{- a^{2} c x^{2} + c} - 3 a^{2} c^{3} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{3} x \sqrt{- a^{2} c x^{2} + c} + c^{3} \sqrt{- a^{2} c x^{2} + c}}\, 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^{2} x^{2} - 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{2}}{\left (a x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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