Optimal. Leaf size=301 \[ -\frac{a^7 x^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac{a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac{5 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{2 \left (1-a^2 x^2\right )^{7/2}}-\frac{5 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{3 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{4 \left (1-a^2 x^2\right )^{7/2}}+\frac{3 a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{5 \left (1-a^2 x^2\right )^{7/2}}-\frac{x \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{6 \left (1-a^2 x^2\right )^{7/2}}+\frac{3 a^6 x^7 \log (x) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}} \]
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Rubi [A] time = 0.189591, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6160, 6150, 88} \[ -\frac{a^7 x^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac{a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac{5 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{2 \left (1-a^2 x^2\right )^{7/2}}-\frac{5 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{3 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{4 \left (1-a^2 x^2\right )^{7/2}}+\frac{3 a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{5 \left (1-a^2 x^2\right )^{7/2}}-\frac{x \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{6 \left (1-a^2 x^2\right )^{7/2}}+\frac{3 a^6 x^7 \log (x) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \, dx &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{e^{-3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{7/2}}{x^7} \, dx}{\left (1-a^2 x^2\right )^{7/2}}\\ &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{(1-a x)^5 (1+a x)^2}{x^7} \, dx}{\left (1-a^2 x^2\right )^{7/2}}\\ &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \left (-a^7+\frac{1}{x^7}-\frac{3 a}{x^6}+\frac{a^2}{x^5}+\frac{5 a^3}{x^4}-\frac{5 a^4}{x^3}-\frac{a^5}{x^2}+\frac{3 a^6}{x}\right ) \, dx}{\left (1-a^2 x^2\right )^{7/2}}\\ &=-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}{6 \left (1-a^2 x^2\right )^{7/2}}+\frac{3 a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}{5 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}{4 \left (1-a^2 x^2\right )^{7/2}}-\frac{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{3 \left (1-a^2 x^2\right )^{7/2}}+\frac{5 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{2 \left (1-a^2 x^2\right )^{7/2}}+\frac{a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{\left (1-a^2 x^2\right )^{7/2}}-\frac{a^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^8}{\left (1-a^2 x^2\right )^{7/2}}+\frac{3 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7 \log (x)}{\left (1-a^2 x^2\right )^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0599441, size = 98, normalized size = 0.33 \[ \frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}} \left (60 a^7 x^7-60 a^5 x^5-150 a^4 x^4+100 a^3 x^3+15 a^2 x^2-180 a^6 x^6 \log (x)-36 a x+10\right )}{60 a^6 x^5 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.166, size = 102, normalized size = 0.3 \begin{align*}{\frac{x \left ( -60\,{a}^{7}{x}^{7}+180\,{a}^{6}\ln \left ( x \right ){x}^{6}+60\,{x}^{5}{a}^{5}+150\,{x}^{4}{a}^{4}-100\,{x}^{3}{a}^{3}-15\,{a}^{2}{x}^{2}+36\,ax-10 \right ) }{60\, \left ({a}^{2}{x}^{2}-1 \right ) ^{4}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{7}{2}}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30348, size = 1148, normalized size = 3.81 \begin{align*} \left [\frac{90 \,{\left (a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right )} \sqrt{-c} \log \left (\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} -{\left (a x^{5} - a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) -{\left (60 \, a^{7} c^{3} x^{7} - 60 \, a^{5} c^{3} x^{5} - 150 \, a^{4} c^{3} x^{4} -{\left (60 \, a^{7} - 60 \, a^{5} - 150 \, a^{4} + 100 \, a^{3} + 15 \, a^{2} - 36 \, a + 10\right )} c^{3} x^{6} + 100 \, a^{3} c^{3} x^{3} + 15 \, a^{2} c^{3} x^{2} - 36 \, a c^{3} x + 10 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \,{\left (a^{8} x^{7} - a^{6} x^{5}\right )}}, \frac{180 \,{\left (a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x^{3} + a x\right )} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} -{\left (a^{2} + 1\right )} c x^{2} + c}\right ) -{\left (60 \, a^{7} c^{3} x^{7} - 60 \, a^{5} c^{3} x^{5} - 150 \, a^{4} c^{3} x^{4} -{\left (60 \, a^{7} - 60 \, a^{5} - 150 \, a^{4} + 100 \, a^{3} + 15 \, a^{2} - 36 \, a + 10\right )} c^{3} x^{6} + 100 \, a^{3} c^{3} x^{3} + 15 \, a^{2} c^{3} x^{2} - 36 \, a c^{3} x + 10 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \,{\left (a^{8} x^{7} - a^{6} x^{5}\right )}}\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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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