Optimal. Leaf size=142 \[ -\frac{2 a \sqrt{c-a^2 c x^2}}{c^2 x}-\frac{\sqrt{c-a^2 c x^2}}{2 c^2 x^2}-\frac{7 a^2 \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )}{2 c^{3/2}}+\frac{a^2 (10 a x+9)}{3 c \sqrt{c-a^2 c x^2}}+\frac{2 a^2 (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.419906, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6151, 1805, 1807, 807, 266, 63, 208} \[ -\frac{2 a \sqrt{c-a^2 c x^2}}{c^2 x}-\frac{\sqrt{c-a^2 c x^2}}{2 c^2 x^2}-\frac{7 a^2 \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )}{2 c^{3/2}}+\frac{a^2 (10 a x+9)}{3 c \sqrt{c-a^2 c x^2}}+\frac{2 a^2 (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1805
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac{(1+a x)^2}{x^3 \left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac{2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{1}{3} \int \frac{-3-6 a x-6 a^2 x^2-4 a^3 x^3}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^2 (9+10 a x)}{3 c \sqrt{c-a^2 c x^2}}+\frac{\int \frac{3+6 a x+9 a^2 x^2}{x^3 \sqrt{c-a^2 c x^2}} \, dx}{3 c}\\ &=\frac{2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^2 (9+10 a x)}{3 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{c-a^2 c x^2}}{2 c^2 x^2}-\frac{\int \frac{-12 a c-21 a^2 c x}{x^2 \sqrt{c-a^2 c x^2}} \, dx}{6 c^2}\\ &=\frac{2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^2 (9+10 a x)}{3 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{c-a^2 c x^2}}{2 c^2 x^2}-\frac{2 a \sqrt{c-a^2 c x^2}}{c^2 x}+\frac{\left (7 a^2\right ) \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx}{2 c}\\ &=\frac{2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^2 (9+10 a x)}{3 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{c-a^2 c x^2}}{2 c^2 x^2}-\frac{2 a \sqrt{c-a^2 c x^2}}{c^2 x}+\frac{\left (7 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )}{4 c}\\ &=\frac{2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^2 (9+10 a x)}{3 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{c-a^2 c x^2}}{2 c^2 x^2}-\frac{2 a \sqrt{c-a^2 c x^2}}{c^2 x}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c-a^2 c x^2}\right )}{2 c^2}\\ &=\frac{2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^2 (9+10 a x)}{3 c \sqrt{c-a^2 c x^2}}-\frac{\sqrt{c-a^2 c x^2}}{2 c^2 x^2}-\frac{2 a \sqrt{c-a^2 c x^2}}{c^2 x}-\frac{7 a^2 \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )}{2 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.144843, size = 105, normalized size = 0.74 \[ -\frac{\left (32 a^3 x^3-43 a^2 x^2+6 a x+3\right ) \sqrt{c-a^2 c x^2}}{6 c^2 x^2 (a x-1)^2}-\frac{7 a^2 \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )}{2 c^{3/2}}+\frac{7 a^2 \log (x)}{2 c^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.043, size = 205, normalized size = 1.4 \begin{align*} -2\,{\frac{a}{cx\sqrt{-{a}^{2}c{x}^{2}+c}}}+4\,{\frac{x{a}^{3}}{\sqrt{-{a}^{2}c{x}^{2}+c}c}}+{\frac{7\,{a}^{2}}{2\,c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}}-{\frac{7\,{a}^{2}}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}-{\frac{2\,a}{3\,c} \left ( x-{a}^{-1} \right ) ^{-1}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}}+{\frac{4\,x{a}^{3}}{3\,c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}}-{\frac{1}{2\,c{x}^{2}}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \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 )}^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}{\left (a^{2} x^{2} - 1\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.84559, size = 570, normalized size = 4.01 \begin{align*} \left [\frac{21 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) - 2 \,{\left (32 \, a^{3} x^{3} - 43 \, a^{2} x^{2} + 6 \, a x + 3\right )} \sqrt{-a^{2} c x^{2} + c}}{12 \,{\left (a^{2} c^{2} x^{4} - 2 \, a c^{2} x^{3} + c^{2} x^{2}\right )}}, -\frac{21 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) +{\left (32 \, a^{3} x^{3} - 43 \, a^{2} x^{2} + 6 \, a x + 3\right )} \sqrt{-a^{2} c x^{2} + c}}{6 \,{\left (a^{2} c^{2} x^{4} - 2 \, a c^{2} x^{3} + c^{2} x^{2}\right )}}\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{a x}{- a^{3} c x^{6} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{5} \sqrt{- a^{2} c x^{2} + c} + a c x^{4} \sqrt{- a^{2} c x^{2} + c} - c x^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{- a^{3} c x^{6} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{5} \sqrt{- a^{2} c x^{2} + c} + a c x^{4} \sqrt{- a^{2} c x^{2} + c} - c x^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24404, size = 275, normalized size = 1.94 \begin{align*} a^{4} c^{2}{\left (\frac{7 \, \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{3}} - \frac{{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{3} a - 4 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} \sqrt{-c}{\left | a \right |} +{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )} a c + 4 \, \sqrt{-c} c{\left | a \right |}}{{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{2} a^{3} c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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