Optimal. Leaf size=224 \[ 3 b^2 \text {Li}_3\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-3 b^2 \text {Li}_3\left (\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-3 b \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2+3 b \text {Li}_2\left (\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2+4 \tanh ^{-1}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3-\frac {3}{2} b^3 \text {Li}_4\left (1-\frac {2}{1-c \sqrt {x}}\right )+\frac {3}{2} b^3 \text {Li}_4\left (\frac {2}{1-c \sqrt {x}}-1\right ) \]
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Rubi [A] time = 0.51, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6095, 5914, 6052, 5948, 6058, 6062, 6610} \[ 3 b^2 \text {PolyLog}\left (3,1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-3 b^2 \text {PolyLog}\left (3,\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-3 b \text {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2+3 b \text {PolyLog}\left (2,\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2-\frac {3}{2} b^3 \text {PolyLog}\left (4,1-\frac {2}{1-c \sqrt {x}}\right )+\frac {3}{2} b^3 \text {PolyLog}\left (4,\frac {2}{1-c \sqrt {x}}-1\right )+4 \tanh ^{-1}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3 \]
Antiderivative was successfully verified.
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Rule 5914
Rule 5948
Rule 6052
Rule 6058
Rule 6062
Rule 6095
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{x} \, dx &=2 \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{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 )^3-(12 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \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 )^3+(6 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )-(6 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \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 )^3-3 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2 \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )+3 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2 \text {Li}_2\left (-1+\frac {2}{1-c \sqrt {x}}\right )+\left (6 b^2 c\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )-\left (6 b^2 c\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \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 )^3-3 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2 \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )+3 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2 \text {Li}_2\left (-1+\frac {2}{1-c \sqrt {x}}\right )+3 b^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \text {Li}_3\left (1-\frac {2}{1-c \sqrt {x}}\right )-3 b^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \text {Li}_3\left (-1+\frac {2}{1-c \sqrt {x}}\right )-\left (3 b^3 c\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )+\left (3 b^3 c\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\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 )^3-3 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2 \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )+3 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2 \text {Li}_2\left (-1+\frac {2}{1-c \sqrt {x}}\right )+3 b^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \text {Li}_3\left (1-\frac {2}{1-c \sqrt {x}}\right )-3 b^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \text {Li}_3\left (-1+\frac {2}{1-c \sqrt {x}}\right )-\frac {3}{2} b^3 \text {Li}_4\left (1-\frac {2}{1-c \sqrt {x}}\right )+\frac {3}{2} b^3 \text {Li}_4\left (-1+\frac {2}{1-c \sqrt {x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.21, size = 248, normalized size = 1.11 \[ \frac {3}{2} b \left (2 \text {Li}_2\left (\frac {\sqrt {x} c+1}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2-2 \text {Li}_2\left (\frac {\sqrt {x} c+1}{c \sqrt {x}-1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2+b \left (-2 \text {Li}_3\left (\frac {\sqrt {x} c+1}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+2 \text {Li}_3\left (\frac {\sqrt {x} c+1}{c \sqrt {x}-1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+b \left (\text {Li}_4\left (\frac {\sqrt {x} c+1}{1-c \sqrt {x}}\right )-\text {Li}_4\left (\frac {\sqrt {x} c+1}{c \sqrt {x}-1}\right )\right )\right )\right )+4 \tanh ^{-1}\left (\frac {2}{c \sqrt {x}-1}+1\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3 \]
Antiderivative was successfully verified.
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fricas [F] time = 1.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (c \sqrt {x}\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c \sqrt {x}\right ) + a^{3}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 1542, normalized size = 6.88 \[ \text {result too large to display} \]
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 \[ \frac {1}{8} \, b^{3} \int \frac {\log \left (c \sqrt {x} + 1\right )^{3}}{x}\,{d x} - \frac {3}{8} \, b^{3} \int \frac {\log \left (c \sqrt {x} + 1\right )^{2} \log \left (-c \sqrt {x} + 1\right )}{x}\,{d x} + \frac {3}{8} \, b^{3} \int \frac {\log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x} + 1\right )^{2}}{x}\,{d x} - \frac {1}{8} \, b^{3} \int \frac {\log \left (-c \sqrt {x} + 1\right )^{3}}{x}\,{d x} + \frac {3}{4} \, a b^{2} \int \frac {\log \left (c \sqrt {x} + 1\right )^{2}}{x}\,{d x} - \frac {3}{2} \, a b^{2} \int \frac {\log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x} + 1\right )}{x}\,{d x} + \frac {3}{4} \, a b^{2} \int \frac {\log \left (-c \sqrt {x} + 1\right )^{2}}{x}\,{d x} + \frac {3}{2} \, a^{2} b \int \frac {\log \left (c \sqrt {x} + 1\right )}{x}\,{d x} - \frac {3}{2} \, a^{2} b \int \frac {\log \left (-c \sqrt {x} + 1\right )}{x}\,{d x} + a^{3} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{3}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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