Optimal. Leaf size=183 \[ -\frac{c^3 (a x+1)^8 \sqrt{c-a^2 c x^2}}{8 a \sqrt{1-a^2 x^2}}+\frac{6 c^3 (a x+1)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}}-\frac{2 c^3 (a x+1)^6 \sqrt{c-a^2 c x^2}}{a \sqrt{1-a^2 x^2}}+\frac{8 c^3 (a x+1)^5 \sqrt{c-a^2 c x^2}}{5 a \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.105045, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6143, 6140, 43} \[ -\frac{c^3 (a x+1)^8 \sqrt{c-a^2 c x^2}}{8 a \sqrt{1-a^2 x^2}}+\frac{6 c^3 (a x+1)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}}-\frac{2 c^3 (a x+1)^6 \sqrt{c-a^2 c x^2}}{a \sqrt{1-a^2 x^2}}+\frac{8 c^3 (a x+1)^5 \sqrt{c-a^2 c x^2}}{5 a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx &=\frac{\left (c^3 \sqrt{c-a^2 c x^2}\right ) \int e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{7/2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^3 \sqrt{c-a^2 c x^2}\right ) \int (1-a x)^3 (1+a x)^4 \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^3 \sqrt{c-a^2 c x^2}\right ) \int \left (8 (1+a x)^4-12 (1+a x)^5+6 (1+a x)^6-(1+a x)^7\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{8 c^3 (1+a x)^5 \sqrt{c-a^2 c x^2}}{5 a \sqrt{1-a^2 x^2}}-\frac{2 c^3 (1+a x)^6 \sqrt{c-a^2 c x^2}}{a \sqrt{1-a^2 x^2}}+\frac{6 c^3 (1+a x)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}}-\frac{c^3 (1+a x)^8 \sqrt{c-a^2 c x^2}}{8 a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0463095, size = 68, normalized size = 0.37 \[ -\frac{c^3 (a x+1)^5 \left (35 a^3 x^3-135 a^2 x^2+185 a x-93\right ) \sqrt{c-a^2 c x^2}}{280 a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 97, normalized size = 0.5 \begin{align*}{\frac{x \left ( 35\,{a}^{7}{x}^{7}+40\,{x}^{6}{a}^{6}-140\,{x}^{5}{a}^{5}-168\,{x}^{4}{a}^{4}+210\,{x}^{3}{a}^{3}+280\,{a}^{2}{x}^{2}-140\,ax-280 \right ) }{280\, \left ( ax+1 \right ) ^{3} \left ( ax-1 \right ) ^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02817, size = 297, normalized size = 1.62 \begin{align*} -\frac{1}{7} \, a^{6} c^{\frac{7}{2}} x^{7} + \frac{3}{5} \, a^{4} c^{\frac{7}{2}} x^{5} - a^{2} c^{\frac{7}{2}} x^{3} + c^{\frac{7}{2}} x + \frac{1}{8} \,{\left (\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} a^{4} c^{3} x^{6} + \frac{6 \, a^{8} c^{6} \log \left (x^{2} - \frac{1}{a^{2}}\right )}{\left (a^{4} c\right )^{\frac{5}{2}}} + \frac{6 \, a^{6} c^{5} x^{2}}{\left (a^{4} c\right )^{\frac{3}{2}}} + \frac{3 \, a^{4} c^{4} x^{4}}{\sqrt{a^{4} c}} - 3 \, \sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} a^{2} c^{3} x^{4} - 6 \, c^{4} \sqrt{\frac{1}{a^{4} c}} \log \left (x^{2} - \frac{1}{a^{2}}\right ) - \frac{10 \, \sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} c^{3}}{a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18842, size = 263, normalized size = 1.44 \begin{align*} \frac{{\left (35 \, a^{7} c^{3} x^{8} + 40 \, a^{6} c^{3} x^{7} - 140 \, a^{5} c^{3} x^{6} - 168 \, a^{4} c^{3} x^{5} + 210 \, a^{3} c^{3} x^{4} + 280 \, a^{2} c^{3} x^{3} - 140 \, a c^{3} x^{2} - 280 \, c^{3} x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{280 \,{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{7}{2}} \left (a x + 1\right )}{\sqrt{- \left (a x - 1\right ) \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{{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{2}}{\left (a x + 1\right )}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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