Optimal. Leaf size=78 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{1-c \sqrt{x}}\right )}{c^2}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^2}+\frac{2 \log \left (\frac{2}{1-c \sqrt{x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^2} \]
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Rubi [A] time = 0.125208, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {5984, 5918, 2402, 2315} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{1-c \sqrt{x}}\right )}{c^2}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^2}+\frac{2 \log \left (\frac{2}{1-c \sqrt{x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^2} \]
Antiderivative was successfully verified.
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Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{1-c^2 x} \, dx &=2 \operatorname{Subst}\left (\int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^2}+\frac{2 \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx,x,\sqrt{x}\right )}{c}\\ &=-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^2}+\frac{2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1-c \sqrt{x}}\right )}{c^2}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{c}\\ &=-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^2}+\frac{2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1-c \sqrt{x}}\right )}{c^2}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c \sqrt{x}}\right )}{c^2}\\ &=-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^2}+\frac{2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1-c \sqrt{x}}\right )}{c^2}+\frac{b \text{Li}_2\left (1-\frac{2}{1-c \sqrt{x}}\right )}{c^2}\\ \end{align*}
Mathematica [A] time = 0.0968517, size = 75, normalized size = 0.96 \[ -\frac{b \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )-\tanh ^{-1}\left (c \sqrt{x}\right ) \left (\tanh ^{-1}\left (c \sqrt{x}\right )+2 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )\right )\right )}{c^2}-\frac{a \log \left (1-c^2 x\right )}{c^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.046, size = 186, normalized size = 2.4 \begin{align*} -{\frac{a}{{c}^{2}}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{a}{{c}^{2}}\ln \left ( 1+c\sqrt{x} \right ) }-{\frac{b}{{c}^{2}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) }-{\frac{b}{{c}^{2}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) }-{\frac{b}{4\,{c}^{2}} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}+{\frac{b}{{c}^{2}}{\it dilog} \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{b}{2\,{c}^{2}}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{b}{2\,{c}^{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}}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }+{\frac{b}{4\,{c}^{2}} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.07543, size = 136, normalized size = 1.74 \begin{align*} -\frac{{\left (\log \left (c \sqrt{x} + 1\right ) \log \left (-\frac{1}{2} \, c \sqrt{x} + \frac{1}{2}\right ) +{\rm Li}_2\left (\frac{1}{2} \, c \sqrt{x} + \frac{1}{2}\right )\right )} b}{c^{2}} - \frac{a \log \left (c^{2} x - 1\right )}{c^{2}} - \frac{b \log \left (c \sqrt{x} + 1\right )^{2} - 2 \, b \log \left (c \sqrt{x} + 1\right ) \log \left (-c \sqrt{x} + 1\right ) - b \log \left (-c \sqrt{x} + 1\right )^{2}}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b \operatorname{artanh}\left (c \sqrt{x}\right ) + a}{c^{2} x - 1}, x\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}{c^{2} x - 1}\, dx - \int \frac{b \operatorname{atanh}{\left (c \sqrt{x} \right )}}{c^{2} x - 1}\, dx \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{b \operatorname{artanh}\left (c \sqrt{x}\right ) + a}{c^{2} x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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