Optimal. Leaf size=276 \[ \frac{\sqrt{1-a^2 x^2}}{8 a c^3 (1-a x) \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{16 a c^3 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{32 a c^3 (a x+1)^2 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{24 a c^3 (a x+1)^3 \sqrt{c-a^2 c x^2}}+\frac{5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a c^3 \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.138208, antiderivative size = 276, 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 = {6143, 6140, 44, 207} \[ \frac{\sqrt{1-a^2 x^2}}{8 a c^3 (1-a x) \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{16 a c^3 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{32 a c^3 (a x+1)^2 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{24 a c^3 (a x+1)^3 \sqrt{c-a^2 c x^2}}+\frac{5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a c^3 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 44
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{-\tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{(1-a x)^3 (1+a x)^4} \, dx}{c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (-\frac{1}{16 (-1+a x)^3}+\frac{1}{8 (-1+a x)^2}+\frac{1}{8 (1+a x)^4}+\frac{3}{16 (1+a x)^3}+\frac{3}{16 (1+a x)^2}-\frac{5}{16 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 a c^3 (1-a x) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{24 a c^3 (1+a x)^3 \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{32 a c^3 (1+a x)^2 \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{16 a c^3 (1+a x) \sqrt{c-a^2 c x^2}}-\frac{\left (5 \sqrt{1-a^2 x^2}\right ) \int \frac{1}{-1+a^2 x^2} \, dx}{16 c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 a c^3 (1-a x) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{24 a c^3 (1+a x)^3 \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{32 a c^3 (1+a x)^2 \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{16 a c^3 (1+a x) \sqrt{c-a^2 c x^2}}+\frac{5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a c^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0868012, size = 101, normalized size = 0.37 \[ \frac{\sqrt{1-a^2 x^2} \left (-15 a^4 x^4-15 a^3 x^3+25 a^2 x^2+25 a x+15 (a x-1)^2 (a x+1)^3 \tanh ^{-1}(a x)-8\right )}{48 a (a x-1)^2 (a c x+c)^3 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.095, size = 238, normalized size = 0.9 \begin{align*} -{\frac{15\,\ln \left ( ax+1 \right ){x}^{5}{a}^{5}-15\,\ln \left ( ax-1 \right ){x}^{5}{a}^{5}+15\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}-15\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}-30\,{x}^{4}{a}^{4}-30\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) +30\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-30\,{x}^{3}{a}^{3}-30\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+30\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+50\,{a}^{2}{x}^{2}+15\,ax\ln \left ( ax+1 \right ) -15\,\ln \left ( ax-1 \right ) xa+50\,ax+15\,\ln \left ( ax+1 \right ) -15\,\ln \left ( ax-1 \right ) -16}{ \left ( 96\,{a}^{2}{x}^{2}-96 \right ){c}^{4}a \left ( ax+1 \right ) ^{3} \left ( ax-1 \right ) ^{2}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0279, size = 182, normalized size = 0.66 \begin{align*} -\frac{15 \, a^{4} \sqrt{c} x^{4} + 15 \, a^{3} \sqrt{c} x^{3} - 25 \, a^{2} \sqrt{c} x^{2} - 25 \, a \sqrt{c} x + 8 \, \sqrt{c}}{48 \,{\left (a^{6} c^{4} x^{5} + a^{5} c^{4} x^{4} - 2 \, a^{4} c^{4} x^{3} - 2 \, a^{3} c^{4} x^{2} + a^{2} c^{4} x + a c^{4}\right )}} + \frac{5 \, \log \left (a x + 1\right )}{32 \, a c^{\frac{7}{2}}} - \frac{5 \, \log \left (a x - 1\right )}{32 \, a c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.64875, size = 1165, normalized size = 4.22 \begin{align*} \left [\frac{15 \,{\left (a^{7} x^{7} + a^{6} x^{6} - 3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \,{\left (a^{3} x^{3} + a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right ) - 4 \,{\left (8 \, a^{5} x^{5} - 7 \, a^{4} x^{4} - 31 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{192 \,{\left (a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x - a c^{4}\right )}}, \frac{15 \,{\left (a^{7} x^{7} + a^{6} x^{6} - 3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right ) - 2 \,{\left (8 \, a^{5} x^{5} - 7 \, a^{4} x^{4} - 31 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{96 \,{\left (a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x - a c^{4}\right )}}\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{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{\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{\sqrt{-a^{2} x^{2} + 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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