Optimal. Leaf size=142 \[ -\frac {6 b^2 \log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^2}+\frac {3 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{c^2}+\frac {3 b \sqrt {x} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{c}+x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3-\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )}{c^2} \]
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Rubi [F] time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3 \, dx &=\int \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3 \, dx\\ \end {align*}
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Mathematica [A] time = 0.29, size = 201, normalized size = 1.42 \[ \frac {a \left (2 a^2 c^2 x+6 a b c \sqrt {x}+3 a b \log \left (1-c \sqrt {x}\right )-3 a b \log \left (c \sqrt {x}+1\right )+6 b^2 \log \left (1-c^2 x\right )\right )+6 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (a^2 c^2 x+2 a b c \sqrt {x}-2 b^2 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}+1\right )\right )+6 b^2 \left (c \sqrt {x}-1\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^2 \left (a c \sqrt {x}+a+b\right )+2 b^3 \left (c^2 x-1\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^3+6 b^3 \text {Li}_2\left (-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )}{2 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.35, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} \operatorname {artanh}\left (c \sqrt {x}\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c \sqrt {x}\right ) + a^{3}, x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 6235, normalized size = 43.91 \[ \text {output too large to display} \]
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 \[ \frac {3}{2} \, {\left (c {\left (\frac {2 \, \sqrt {x}}{c^{2}} - \frac {\log \left (c \sqrt {x} + 1\right )}{c^{3}} + \frac {\log \left (c \sqrt {x} - 1\right )}{c^{3}}\right )} + 2 \, x \operatorname {artanh}\left (c \sqrt {x}\right )\right )} a^{2} b + \frac {3}{4} \, {\left (4 \, c {\left (\frac {2 \, \sqrt {x}}{c^{2}} - \frac {\log \left (c \sqrt {x} + 1\right )}{c^{3}} + \frac {\log \left (c \sqrt {x} - 1\right )}{c^{3}}\right )} \operatorname {artanh}\left (c \sqrt {x}\right ) + 4 \, x \operatorname {artanh}\left (c \sqrt {x}\right )^{2} - \frac {2 \, {\left (\log \left (c \sqrt {x} - 1\right ) - 2\right )} \log \left (c \sqrt {x} + 1\right ) - \log \left (c \sqrt {x} + 1\right )^{2} - \log \left (c \sqrt {x} - 1\right )^{2} - 4 \, \log \left (c \sqrt {x} - 1\right )}{c^{2}}\right )} a b^{2} + a^{3} x - \frac {1}{32} \, b^{3} {\left (\frac {{\left (4 \, \log \left (-c \sqrt {x} + 1\right )^{3} - 6 \, \log \left (-c \sqrt {x} + 1\right )^{2} + 6 \, \log \left (-c \sqrt {x} + 1\right ) - 3\right )} {\left (c \sqrt {x} - 1\right )}^{2} + 8 \, {\left (\log \left (-c \sqrt {x} + 1\right )^{3} - 3 \, \log \left (-c \sqrt {x} + 1\right )^{2} + 6 \, \log \left (-c \sqrt {x} + 1\right ) - 6\right )} {\left (c \sqrt {x} - 1\right )}}{c^{2}} - 4 \, \int \log \left (c \sqrt {x} + 1\right )^{3} - 3 \, \log \left (c \sqrt {x} + 1\right )^{2} \log \left (-c \sqrt {x} + 1\right ) + 3 \, \log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x} + 1\right )^{2}\,{d x}\right )} \]
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 \[ \int {\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3 \,d x \]
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 \[ \int \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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