Optimal. Leaf size=357 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{75 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{59 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (a x+1)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1)^4 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{9 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{201 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}} \]
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Rubi [A] time = 0.198547, antiderivative size = 357, 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} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{75 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{59 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (a x+1)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1)^4 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{9 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{201 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \, dx &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{7/2}} \, dx}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^7 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \frac{x^7}{(-1+a x)^2 (1+a x)^5} \, dx}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^7 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \left (\frac{1}{a^7}+\frac{1}{32 a^7 (-1+a x)^2}+\frac{9}{64 a^7 (-1+a x)}-\frac{1}{4 a^7 (1+a x)^5}+\frac{3}{2 a^7 (1+a x)^4}-\frac{59}{16 a^7 (1+a x)^3}+\frac{75}{16 a^7 (1+a x)^2}-\frac{201}{64 a^7 (1+a x)}\right ) \, dx}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)^4}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)^3}+\frac{59 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)^2}-\frac{75 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)}+\frac{9 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{201 \sqrt{1-\frac{1}{a^2 x^2}} \log (1+a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.179346, size = 105, normalized size = 0.29 \[ \frac{\left (1-\frac{1}{a^2 x^2}\right )^{7/2} \left (2 \left (32 a x+\frac{1}{1-a x}-\frac{150}{a x+1}+\frac{59}{(a x+1)^2}-\frac{16}{(a x+1)^3}+\frac{2}{(a x+1)^4}\right )+9 \log (1-a x)-201 \log (a x+1)\right )}{64 a \left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.263, size = 247, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) \left ( -64\,{x}^{6}{a}^{6}+201\,\ln \left ( ax+1 \right ){x}^{5}{a}^{5}-9\,\ln \left ( ax-1 \right ){x}^{5}{a}^{5}-192\,{x}^{5}{a}^{5}+603\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}-27\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}+174\,{x}^{4}{a}^{4}+402\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -18\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}+618\,{x}^{3}{a}^{3}-402\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+18\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+118\,{a}^{2}{x}^{2}-603\,ax\ln \left ( ax+1 \right ) +27\,\ln \left ( ax-1 \right ) xa-414\,ax-201\,\ln \left ( ax+1 \right ) +9\,\ln \left ( ax-1 \right ) -208 \right ) }{64\,{a}^{8}{x}^{7}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{-{\frac{7}{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 \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65509, size = 460, normalized size = 1.29 \begin{align*} \frac{{\left (64 \, a^{6} x^{6} + 192 \, a^{5} x^{5} - 174 \, a^{4} x^{4} - 618 \, a^{3} x^{3} - 118 \, a^{2} x^{2} + 414 \, a x - 201 \,{\left (a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right )} \log \left (a x + 1\right ) + 9 \,{\left (a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right )} \log \left (a x - 1\right ) + 208\right )} \sqrt{a^{2} c}}{64 \,{\left (a^{7} c^{4} x^{5} + 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} - 2 \, a^{4} c^{4} x^{2} - 3 \, a^{3} c^{4} x - a^{2} c^{4}\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 (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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