Optimal. Leaf size=156 \[ \frac{4}{3} a^3 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{7 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{8 x}+\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 x^3}+\frac{7 a^4 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{8 \sqrt{1-a x} \sqrt{a x+1}} \]
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Rubi [A] time = 0.556423, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{4}{3} a^3 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{7 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{8 x}+\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 x^3}+\frac{7 a^4 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{8 \sqrt{1-a x} \sqrt{a x+1}} \]
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^4} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^4} \, 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^5} \, 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^5 \sqrt{1-a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 x^3}+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{-8 a-7 a^2 x}{x^4 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{4 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 x^3}+\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{21 a^2+16 a^3 x}{x^3 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{12 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 x^3}+\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{7 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{8 x}+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{-32 a^3-21 a^4 x}{x^2 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{24 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{4}{3} a^3 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 x^3}+\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{7 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{8 x}-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{21 a^4}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{24 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{4}{3} a^3 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 x^3}+\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{7 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{8 x}-\frac{\left (7 a^4 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{8 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{4}{3} a^3 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 x^3}+\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{7 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{8 x}+\frac{\left (7 a^5 \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 )}{8 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{4}{3} a^3 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 x^3}+\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{7 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{8 x}+\frac{7 a^4 \sqrt{c-\frac{c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )}{8 \sqrt{1-a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 0.0934193, size = 94, normalized size = 0.6 \[ \frac{\sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (32 a^3 x^3+21 a^2 x^2+16 a x+6\right )-21 a^4 x^4 \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{24 x^3 \sqrt{a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.198, size = 410, normalized size = 2.6 \begin{align*} -{\frac{{a}^{2}}{24\,c{x}^{3}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ( -48\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{5}{a}^{3}c+48\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{3}{a}^{3}+48\,\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}^{4}a-48\,\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}^{4}a-48\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}{x}^{4}{a}^{2}c+21\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{4}{a}^{2}c+27\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{2}{a}^{2}+21\,\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}^{4}{c}^{2}+16\,a \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}x\sqrt{-{\frac{c}{{a}^{2}}}}+6\, \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}{{a}^{2}}}}}}{\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71248, size = 487, normalized size = 3.12 \begin{align*} \left [\frac{21 \, a^{3} \sqrt{-c} x^{3} \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 (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, x^{3}}, -\frac{21 \, a^{3} \sqrt{c} x^{3} \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 (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, x^{3}}\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^{4} \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 = 2.97763, size = 427, normalized size = 2.74 \begin{align*} \frac{1}{12} \,{\left (21 \, a^{2} \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{21 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{7} a^{2} c \mathrm{sgn}\left (x\right ) + 45 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{5} a^{2} c^{2} \mathrm{sgn}\left (x\right ) - 96 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{4} a c^{\frac{5}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 45 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{3} a^{2} c^{3} \mathrm{sgn}\left (x\right ) - 128 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} a c^{\frac{7}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 21 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )} a^{2} c^{4} \mathrm{sgn}\left (x\right ) - 32 \, a c^{\frac{9}{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 )}^{4}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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