Optimal. Leaf size=213 \[ -\frac{5 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{2 (1-a x) (a x+1)}+\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{1-a x}+\frac{x (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{2 (1-a x)}+\frac{2 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}+\frac{a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{2 (1-a x)^{3/2} (a x+1)^{3/2}} \]
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Rubi [A] time = 0.459027, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {6167, 6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} \[ -\frac{5 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{2 (1-a x) (a x+1)}+\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{1-a x}+\frac{x (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{2 (1-a x)}+\frac{2 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}+\frac{a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{2 (1-a x)^{3/2} (a x+1)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6159
Rule 6129
Rule 97
Rule 149
Rule 154
Rule 157
Rule 41
Rule 216
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \, dx\\ &=-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{e^{2 \tanh ^{-1}(a x)} (1-a x)^{3/2} (1+a x)^{3/2}}{x^3} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{\sqrt{1-a x} (1+a x)^{5/2}}{x^3} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=\frac{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{(1+a x)^{3/2} \left (2 a-3 a^2 x\right )}{x^2 \sqrt{1-a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{\sqrt{1+a x} \left (a^2-5 a^3 x\right )}{x \sqrt{1-a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}-\frac{5 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}+\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{-a^3+4 a^4 x}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{2 a (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}-\frac{5 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}-\frac{\left (a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}+\frac{\left (2 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}-\frac{5 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}+\frac{\left (a^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \frac{1}{a-a x^2} \, dx,x,\sqrt{1-a x} \sqrt{1+a x}\right )}{2 (1-a x)^{3/2} (1+a x)^{3/2}}+\frac{\left (2 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}-\frac{5 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}+\frac{2 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3 \sin ^{-1}(a x)}{(1-a x)^{3/2} (1+a x)^{3/2}}+\frac{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3 \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.105637, size = 115, normalized size = 0.54 \[ \frac{c \sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (2 a^2 x^2-4 a x-1\right )+4 a^2 x^2 \log \left (\sqrt{a^2 x^2-1}+a x\right )+a^2 x^2 \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{2 a^2 x \sqrt{a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.183, size = 455, normalized size = 2.1 \begin{align*}{\frac{x}{6\,{a}^{2}c} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 12\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{3}{a}^{5}c-12\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{5/2}x{a}^{5}+4\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}} \right ) ^{3/2}{x}^{2}{a}^{4}c-\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{{\frac{3}{2}}}{x}^{2}{a}^{4}c+6\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}{x}^{3}{a}^{3}{c}^{2}-3\,{a}^{4} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{5/2}\sqrt{-{\frac{c}{{a}^{2}}}}-18\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{3}{a}^{3}{c}^{2}+18\,{c}^{5/2}\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) \sqrt{-{\frac{c}{{a}^{2}}}}{x}^{2}a-6\,{c}^{5/2}\sqrt{-{\frac{c}{{a}^{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}^{2}a+3\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{2}{a}^{2}{c}^{2}+3\,\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}^{2}{c}^{3} \right ){\frac{1}{\sqrt{-{\frac{c}{{a}^{2}}}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{-{\frac{3}{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 )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}}}{a x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7508, size = 694, normalized size = 3.26 \begin{align*} \left [-\frac{8 \, a \sqrt{-c} c x \arctan \left (\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - a \sqrt{-c} c x \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 (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, a^{2} x}, \frac{a c^{\frac{3}{2}} x \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 ) + 2 \, a c^{\frac{3}{2}} x \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) +{\left (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{2} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 12.2101, size = 376, normalized size = 1.77 \begin{align*} c \left (\begin{cases} \frac{\sqrt{c} \sqrt{a^{2} x^{2} - 1}}{a} - \frac{i \sqrt{c} \log{\left (a x \right )}}{a} + \frac{i \sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2 a} + \frac{\sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{i \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{a} + \frac{i \sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2 a} - \frac{i \sqrt{c} \log{\left (\sqrt{- a^{2} x^{2} + 1} + 1 \right )}}{a} & \text{otherwise} \end{cases}\right ) + \frac{2 c \left (\begin{cases} - \frac{a \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + \sqrt{c} \operatorname{acosh}{\left (a x \right )} + \frac{\sqrt{c}}{a x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{i a \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - i \sqrt{c} \operatorname{asin}{\left (a x \right )} - \frac{i \sqrt{c}}{a x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right )}{a} + \frac{c \left (\begin{cases} \frac{i a \sqrt{c} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} + \frac{i \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{i \sqrt{c}}{2 a^{2} x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac{a \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{\sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.58784, size = 359, normalized size = 1.69 \begin{align*} -{\left (\frac{c^{\frac{3}{2}} \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}}{\sqrt{c}}\right ) \mathrm{sgn}\left (x\right )}{a^{2}} + \frac{2 \, c^{\frac{3}{2}} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} - c} \right |}\right ) \mathrm{sgn}\left (x\right )}{a{\left | a \right |}} - \frac{\sqrt{a^{2} c x^{2} - c} c \mathrm{sgn}\left (x\right )}{a^{2}} - \frac{{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{3} c^{2}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 4 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} a c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) -{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )} c^{3}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 4 \, a c^{\frac{7}{2}} \mathrm{sgn}\left (x\right )}{{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a^{2}{\left | a \right |}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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