Optimal. Leaf size=146 \[ \frac{c x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.117566, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6197, 6193, 75} \[ \frac{c x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 75
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \, dx &=\frac{\left (c \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \frac{(-1+a x) (1+a x)^2}{x^3} \, dx}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \left (a^3-\frac{1}{x^3}-\frac{a}{x^2}+\frac{a^2}{x}\right ) \, dx}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}} x}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}} \log (x)}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0449956, size = 64, normalized size = 0.44 \[ \frac{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} \left (a^3 x+a^2 \log (x)+\frac{3 a^2}{2}+\frac{a}{x}+\frac{1}{2 x^2}\right )}{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.237, size = 80, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,{x}^{3}{a}^{3}+2\,{a}^{2}\ln \left ( x \right ){x}^{2}+2\,ax+1 \right ) x}{ \left ( 2\,ax+2 \right ) \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \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 (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40766, size = 103, normalized size = 0.71 \begin{align*} \frac{{\left (2 \, a^{3} c x^{3} + 2 \, a^{2} c x^{2} \log \left (x\right ) + 2 \, a c x + c\right )} \sqrt{a^{2} c}}{2 \, a^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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