Optimal. Leaf size=145 \[ -2 b \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+2 b \text {Li}_2\left (\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+4 \tanh ^{-1}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2+b^2 \text {Li}_3\left (1-\frac {2}{1-c \sqrt {x}}\right )-b^2 \text {Li}_3\left (\frac {2}{1-c \sqrt {x}}-1\right ) \]
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Rubi [A] time = 0.32, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6095, 5914, 6052, 5948, 6058, 6610} \[ -2 b \text {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+2 b \text {PolyLog}\left (2,\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+b^2 \text {PolyLog}\left (3,1-\frac {2}{1-c \sqrt {x}}\right )-b^2 \text {PolyLog}\left (3,\frac {2}{1-c \sqrt {x}}-1\right )+4 \tanh ^{-1}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2 \]
Antiderivative was successfully verified.
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Rule 5914
Rule 5948
Rule 6052
Rule 6058
Rule 6095
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{x} \, dx &=2 \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx,x,\sqrt {x}\right )\\ &=4 \tanh ^{-1}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2-(8 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )\\ &=4 \tanh ^{-1}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2+(4 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )-(4 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )\\ &=4 \tanh ^{-1}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2-2 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )+2 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \text {Li}_2\left (-1+\frac {2}{1-c \sqrt {x}}\right )+\left (2 b^2 c\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )-\left (2 b^2 c\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )\\ &=4 \tanh ^{-1}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2-2 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )+2 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \text {Li}_2\left (-1+\frac {2}{1-c \sqrt {x}}\right )+b^2 \text {Li}_3\left (1-\frac {2}{1-c \sqrt {x}}\right )-b^2 \text {Li}_3\left (-1+\frac {2}{1-c \sqrt {x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 164, normalized size = 1.13 \[ 4 \tanh ^{-1}\left (\frac {2}{c \sqrt {x}-1}+1\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2-b \left (-2 \text {Li}_2\left (\frac {\sqrt {x} c+1}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+2 \text {Li}_2\left (\frac {\sqrt {x} c+1}{c \sqrt {x}-1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+b \left (\text {Li}_3\left (\frac {\sqrt {x} c+1}{1-c \sqrt {x}}\right )-\text {Li}_3\left (\frac {\sqrt {x} c+1}{c \sqrt {x}-1}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2} + 2 \, a b \operatorname {artanh}\left (c \sqrt {x}\right ) + a^{2}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 742, normalized size = 5.12 \[ 2 a^{2} \ln \left (c \sqrt {x}\right )+2 b^{2} \ln \left (c \sqrt {x}\right ) \arctanh \left (c \sqrt {x}\right )^{2}-2 b^{2} \arctanh \left (c \sqrt {x}\right ) \polylog \left (2, -\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}\right )+b^{2} \polylog \left (3, -\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}\right )-2 b^{2} \arctanh \left (c \sqrt {x}\right )^{2} \ln \left (\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}-1\right )+2 b^{2} \arctanh \left (c \sqrt {x}\right )^{2} \ln \left (1-\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+4 b^{2} \arctanh \left (c \sqrt {x}\right ) \polylog \left (2, \frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )-4 b^{2} \polylog \left (3, \frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+2 b^{2} \arctanh \left (c \sqrt {x}\right )^{2} \ln \left (1+\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+4 b^{2} \arctanh \left (c \sqrt {x}\right ) \polylog \left (2, -\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )-4 b^{2} \polylog \left (3, -\frac {1+c \sqrt {x}}{\sqrt {-c^{2} x +1}}\right )+i b^{2} \pi \mathrm {csgn}\left (\frac {i \left (\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}-1\right )}{1+\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}}\right )^{3} \arctanh \left (c \sqrt {x}\right )^{2}-i b^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}-1\right )}{1+\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}}\right )^{2} \arctanh \left (c \sqrt {x}\right )^{2}+i b^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}-1\right )\right ) \mathrm {csgn}\left (\frac {i}{1+\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}-1\right )}{1+\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}}\right ) \arctanh \left (c \sqrt {x}\right )^{2}-i b^{2} \pi \,\mathrm {csgn}\left (\frac {i}{1+\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}-1\right )}{1+\frac {\left (1+c \sqrt {x}\right )^{2}}{-c^{2} x +1}}\right )^{2} \arctanh \left (c \sqrt {x}\right )^{2}+4 a b \ln \left (c \sqrt {x}\right ) \arctanh \left (c \sqrt {x}\right )-2 a b \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )-2 a b \dilog \left (c \sqrt {x}\right )-2 a b \dilog \left (1+c \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, b^{2} \int \frac {\log \left (c \sqrt {x} + 1\right )^{2}}{x}\,{d x} - \frac {1}{2} \, b^{2} \int \frac {\log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x} + 1\right )}{x}\,{d x} + \frac {1}{4} \, b^{2} \int \frac {\log \left (-c \sqrt {x} + 1\right )^{2}}{x}\,{d x} + a b \int \frac {\log \left (c \sqrt {x} + 1\right )}{x}\,{d x} - a b \int \frac {\log \left (-c \sqrt {x} + 1\right )}{x}\,{d x} + a^{2} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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