Optimal. Leaf size=121 \[ -2 a^2 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-\frac{1}{2} a^2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )-\frac{a c (4-a x) \sqrt{c-a^2 c x^2}}{2 x}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{2 x^2} \]
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Rubi [A] time = 0.28923, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6151, 1807, 813, 844, 217, 203, 266, 63, 208} \[ -2 a^2 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-\frac{1}{2} a^2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )-\frac{a c (4-a x) \sqrt{c-a^2 c x^2}}{2 x}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1807
Rule 813
Rule 844
Rule 217
Rule 203
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^3} \, dx &=c \int \frac{(1+a x)^2 \sqrt{c-a^2 c x^2}}{x^3} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{2 x^2}-\frac{1}{2} \int \frac{\left (-4 a c-a^2 c x\right ) \sqrt{c-a^2 c x^2}}{x^2} \, dx\\ &=-\frac{a c (4-a x) \sqrt{c-a^2 c x^2}}{2 x}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{2 x^2}+\frac{1}{4} \int \frac{2 a^2 c^2-8 a^3 c^2 x}{x \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{a c (4-a x) \sqrt{c-a^2 c x^2}}{2 x}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{2 x^2}+\frac{1}{2} \left (a^2 c^2\right ) \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx-\left (2 a^3 c^2\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{a c (4-a x) \sqrt{c-a^2 c x^2}}{2 x}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{2 x^2}+\frac{1}{4} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )-\left (2 a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=-\frac{a c (4-a x) \sqrt{c-a^2 c x^2}}{2 x}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{2 x^2}-2 a^2 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-\frac{1}{2} c \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 )\\ &=-\frac{a c (4-a x) \sqrt{c-a^2 c x^2}}{2 x}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{2 x^2}-2 a^2 c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-\frac{1}{2} a^2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.208506, size = 129, normalized size = 1.07 \[ \frac{1}{2} c \left (\frac{\left (2 a^2 x^2-4 a x-1\right ) \sqrt{c-a^2 c x^2}}{x^2}-a^2 \sqrt{c} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )+4 a^2 \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )+a^2 \sqrt{c} \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.046, size = 316, normalized size = 2.6 \begin{align*} -2\,{\frac{a \left ( -{a}^{2}c{x}^{2}+c \right ) ^{5/2}}{cx}}-2\,{a}^{3}x \left ( -{a}^{2}c{x}^{2}+c \right ) ^{3/2}-3\,{a}^{3}cx\sqrt{-{a}^{2}c{x}^{2}+c}-3\,{\frac{{a}^{3}{c}^{2}}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c{x}^{2}+c}}} \right ) }+{\frac{{a}^{2}}{6} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}}{2}{c}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ) }+{\frac{{a}^{2}c}{2}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{2\,{a}^{2}}{3} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{a}^{3}c\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }x+{{a}^{3}{c}^{2}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-{\frac{1}{2\,c{x}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{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^{2} c x^{2} + c\right )}^{\frac{3}{2}}{\left (a x + 1\right )}^{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 = 2.6843, size = 594, normalized size = 4.91 \begin{align*} \left [\frac{8 \, a^{2} c^{\frac{3}{2}} x^{2} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) + a^{2} c^{\frac{3}{2}} x^{2} \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 (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt{-a^{2} c x^{2} + c}}{4 \, x^{2}}, -\frac{a^{2} \sqrt{-c} c x^{2} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) - 2 \, a^{2} \sqrt{-c} c x^{2} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) -{\left (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt{-a^{2} c x^{2} + c}}{2 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.4378, size = 366, normalized size = 3.02 \begin{align*} a^{2} c \left (\begin{cases} i \sqrt{c} \sqrt{a^{2} x^{2} - 1} - \sqrt{c} \log{\left (a x \right )} + \frac{\sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2} + i \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt{c} \sqrt{- a^{2} x^{2} + 1} + \frac{\sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2} - \sqrt{c} \log{\left (\sqrt{- a^{2} x^{2} + 1} + 1 \right )} & \text{otherwise} \end{cases}\right ) + 2 a c \left (\begin{cases} - \frac{i a^{2} \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + i a \sqrt{c} \operatorname{acosh}{\left (a x \right )} + \frac{i \sqrt{c}}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{a^{2} \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - a \sqrt{c} \operatorname{asin}{\left (a x \right )} - \frac{\sqrt{c}}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right ) + c \left (\begin{cases} \frac{a^{2} \sqrt{c} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} + \frac{a \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac{i a^{2} \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18399, size = 370, normalized size = 3.06 \begin{align*} \frac{a^{2} c^{2} \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{2 \, a^{3} \sqrt{-c} c \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} + \sqrt{-a^{2} c x^{2} + c} a^{2} c - \frac{{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{3} a^{2} c^{2}{\left | a \right |} - 4 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt{-c} c^{2} +{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )} a^{2} c^{3}{\left | a \right |} + 4 \, a^{3} \sqrt{-c} c^{3}}{{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{2}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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