Optimal. Leaf size=85 \[ c^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2-\frac{2 b c \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{\sqrt{x}}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{x}+b^2 c^2 \log (x)-b^2 c^2 \log \left (1-c^2 x\right ) \]
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Rubi [F] time = 0.0238263, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{x^2} \, dx &=\int \frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{x^2} \, dx\\ \end{align*}
Mathematica [A] time = 0.110467, size = 129, normalized size = 1.52 \[ -\frac{a^2-a b c^2 x \log \left (c \sqrt{x}+1\right )+b c^2 x (a+b) \log \left (1-c \sqrt{x}\right )+2 a b c \sqrt{x}+2 b \tanh ^{-1}\left (c \sqrt{x}\right ) \left (a+b c \sqrt{x}\right )+b^2 c^2 x \log \left (c \sqrt{x}+1\right )-b^2 c^2 x \log (x)-b^2 \left (c^2 x-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^2}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 292, normalized size = 3.4 \begin{align*} -{\frac{{a}^{2}}{x}}-{\frac{{b}^{2}}{x} \left ({\it Artanh} \left ( c\sqrt{x} \right ) \right ) ^{2}}-2\,{\frac{{b}^{2}c{\it Artanh} \left ( c\sqrt{x} \right ) }{\sqrt{x}}}-{c}^{2}{b}^{2}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) +{c}^{2}{b}^{2}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) +{\frac{{b}^{2}{c}^{2}}{2}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }-{\frac{{b}^{2}{c}^{2}}{4} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}-{c}^{2}{b}^{2}\ln \left ( c\sqrt{x}-1 \right ) +2\,{c}^{2}{b}^{2}\ln \left ( c\sqrt{x} \right ) -{c}^{2}{b}^{2}\ln \left ( 1+c\sqrt{x} \right ) +{\frac{{b}^{2}{c}^{2}}{2}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }-{\frac{{b}^{2}{c}^{2}}{2}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }-{\frac{{b}^{2}{c}^{2}}{4} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}}-2\,{\frac{ab{\it Artanh} \left ( c\sqrt{x} \right ) }{x}}-2\,{\frac{cab}{\sqrt{x}}}-{c}^{2}ab\ln \left ( c\sqrt{x}-1 \right ) +{c}^{2}ab\ln \left ( 1+c\sqrt{x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01967, size = 235, normalized size = 2.76 \begin{align*}{\left ({\left (c \log \left (c \sqrt{x} + 1\right ) - c \log \left (c \sqrt{x} - 1\right ) - \frac{2}{\sqrt{x}}\right )} c - \frac{2 \, \operatorname{artanh}\left (c \sqrt{x}\right )}{x}\right )} a b + \frac{1}{4} \,{\left ({\left (2 \,{\left (\log \left (c \sqrt{x} - 1\right ) - 2\right )} \log \left (c \sqrt{x} + 1\right ) - \log \left (c \sqrt{x} + 1\right )^{2} - \log \left (c \sqrt{x} - 1\right )^{2} - 4 \, \log \left (c \sqrt{x} - 1\right ) + 4 \, \log \left (x\right )\right )} c^{2} + 4 \,{\left (c \log \left (c \sqrt{x} + 1\right ) - c \log \left (c \sqrt{x} - 1\right ) - \frac{2}{\sqrt{x}}\right )} c \operatorname{artanh}\left (c \sqrt{x}\right )\right )} b^{2} - \frac{b^{2} \operatorname{artanh}\left (c \sqrt{x}\right )^{2}}{x} - \frac{a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.83377, size = 375, normalized size = 4.41 \begin{align*} \frac{8 \, b^{2} c^{2} x \log \left (\sqrt{x}\right ) + 4 \,{\left (a b - b^{2}\right )} c^{2} x \log \left (c \sqrt{x} + 1\right ) - 4 \,{\left (a b + b^{2}\right )} c^{2} x \log \left (c \sqrt{x} - 1\right ) - 8 \, a b c \sqrt{x} +{\left (b^{2} c^{2} x - b^{2}\right )} \log \left (-\frac{c^{2} x + 2 \, c \sqrt{x} + 1}{c^{2} x - 1}\right )^{2} - 4 \, a^{2} - 4 \,{\left (b^{2} c \sqrt{x} + a b\right )} \log \left (-\frac{c^{2} x + 2 \, c \sqrt{x} + 1}{c^{2} x - 1}\right )}{4 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 42.9305, size = 680, normalized size = 8. \begin{align*} \text{result too large to display} \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 (b \operatorname{artanh}\left (c \sqrt{x}\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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