Optimal. Leaf size=324 \[ \frac{c^3 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 x^6 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.156097, antiderivative size = 324, 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 = {6197, 6193, 88} \[ \frac{c^3 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 x^6 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 88
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \, dx &=\frac{\left (c^3 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{7/2} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c^3 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \frac{(-1+a x)^5 (1+a x)^2}{x^7} \, dx}{a^7 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c^3 \sqrt{c-\frac{c}{a^2 x^2}}\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}{a^7 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 \sqrt{1-\frac{1}{a^2 x^2}} x^6}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 \sqrt{1-\frac{1}{a^2 x^2}} x^5}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 \sqrt{1-\frac{1}{a^2 x^2}} x^4}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 \sqrt{1-\frac{1}{a^2 x^2}} x^3}-\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}} x}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}} \log (x)}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0801523, size = 94, normalized size = 0.29 \[ \frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} \left (\frac{60 a^7 x^7-60 a^5 x^5-150 a^4 x^4+100 a^3 x^3+15 a^2 x^2-36 a x+10}{60 x^6}-3 a^6 \log (x)\right )}{a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.237, size = 112, normalized size = 0.4 \begin{align*} -{\frac{ \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 ) x}{60\, \left ( ax-1 \right ) ^{3} \left ({a}^{2}{x}^{2}-1 \right ) ^{2}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{7}{2}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \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{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62321, size = 219, normalized size = 0.68 \begin{align*} \frac{{\left (60 \, a^{7} c^{3} x^{7} - 180 \, a^{6} c^{3} x^{6} \log \left (x\right ) - 60 \, a^{5} c^{3} x^{5} - 150 \, a^{4} c^{3} x^{4} + 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} c}}{60 \, a^{8} x^{6}} \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{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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