Optimal. Leaf size=181 \[ \frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}-\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}-\frac{3 a^5 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{4 \sqrt{1-a x} \sqrt{a x+1}} \]
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Rubi [A] time = 0.585254, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6167, 6159, 6129, 98, 151, 12, 92, 208} \[ \frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}-\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}-\frac{3 a^5 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{4 \sqrt{1-a x} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6159
Rule 6129
Rule 98
Rule 151
Rule 12
Rule 92
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^5} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^5} \, dx\\ &=-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{1-a x} \sqrt{1+a x}}{x^6} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1-a x)^{3/2}}{x^6 \sqrt{1+a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{10 a-9 a^2 x}{x^5 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{5 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{36 a^2-30 a^3 x}{x^4 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{20 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{90 a^3-72 a^4 x}{x^3 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{60 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}-\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{144 a^4-90 a^5 x}{x^2 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{120 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}-\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{90 a^5}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{120 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}-\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}+\frac{\left (3 a^5 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{4 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}-\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}-\frac{\left (3 a^6 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \operatorname{Subst}\left (\int \frac{1}{a-a x^2} \, dx,x,\sqrt{1-a x} \sqrt{1+a x}\right )}{4 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}-\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}-\frac{3 a^5 \sqrt{c-\frac{c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )}{4 \sqrt{1-a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 0.0890427, size = 102, normalized size = 0.56 \[ \frac{\sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (24 a^4 x^4-15 a^3 x^3+12 a^2 x^2-10 a x+4\right )+15 a^5 x^5 \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{20 x^4 \sqrt{a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.198, size = 447, normalized size = 2.5 \begin{align*} -{\frac{{a}^{2}}{20\,{x}^{4}c}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ( -40\,\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}\sqrt{-{\frac{c}{{a}^{2}}}}{x}^{6}{a}^{4}c+40\, \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}\sqrt{-{\frac{c}{{a}^{2}}}}{x}^{4}{a}^{4}-15\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{5}{a}^{3}c+40\,\sqrt{-{\frac{c}{{a}^{2}}}}{c}^{3/2}\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ){x}^{5}{a}^{2}-40\,\sqrt{-{\frac{c}{{a}^{2}}}}{c}^{3/2}\ln \left ({\frac{1}{\sqrt{c}} \left ( \sqrt{c}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}+cx \right ) } \right ){x}^{5}{a}^{2}+40\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}{x}^{5}{a}^{3}c-25\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{3}{a}^{3}-15\,\ln \left ( 2\,{\frac{1}{{a}^{2}x} \left ( \sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{a}^{2}-c \right ) } \right ){x}^{5}a{c}^{2}+16\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{2}{a}^{2}-10\,a \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}x\sqrt{-{\frac{c}{{a}^{2}}}}+4\, \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}\sqrt{-{\frac{c}{{a}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}}}{\frac{1}{\sqrt{-{\frac{c}{{a}^{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 (a x - 1\right )} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{{\left (a x + 1\right )} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65055, size = 521, normalized size = 2.88 \begin{align*} \left [\frac{15 \, a^{4} \sqrt{-c} x^{4} \log \left (-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \,{\left (24 \, a^{4} x^{4} - 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 10 \, a x + 4\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{40 \, x^{4}}, \frac{15 \, a^{4} \sqrt{c} x^{4} \arctan \left (\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) +{\left (24 \, a^{4} x^{4} - 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 10 \, a x + 4\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{20 \, x^{4}}\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{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )} \left (a x - 1\right )}{x^{5} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.14559, size = 489, normalized size = 2.7 \begin{align*} -\frac{1}{10} \,{\left (15 \, a^{3} \sqrt{c} \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}}{\sqrt{c}}\right ) \mathrm{sgn}\left (x\right ) - \frac{15 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{9} a^{3} c \mathrm{sgn}\left (x\right ) + 70 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{7} a^{3} c^{2} \mathrm{sgn}\left (x\right ) + 40 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{6} a^{2} c^{\frac{5}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 200 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{4} a^{2} c^{\frac{7}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 70 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{3} a^{3} c^{4} \mathrm{sgn}\left (x\right ) + 120 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} a^{2} c^{\frac{9}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 15 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )} a^{3} c^{5} \mathrm{sgn}\left (x\right ) + 24 \, a^{2} c^{\frac{11}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right )}{{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{5}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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