Optimal. Leaf size=142 \[ 6 b^2 c^2 \log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+3 b c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2+c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3-\frac {3 b c \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{\sqrt {x}}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{x}-3 b^3 c^2 \text {Li}_2\left (\frac {2}{\sqrt {x} c+1}-1\right ) \]
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Rubi [F] time = 0.02, 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 = {} \[ \int \frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{x^2} \, dx \]
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 )^3}{x^2} \, dx &=\int \frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{x^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.32, size = 230, normalized size = 1.62 \[ \frac {a \left (-2 a^2-3 a b c^2 x \log \left (1-c \sqrt {x}\right )+3 a b c^2 x \log \left (c \sqrt {x}+1\right )-6 a b c \sqrt {x}+12 b^2 c^2 x \log \left (\frac {c \sqrt {x}}{\sqrt {1-c^2 x}}\right )\right )-6 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (a^2+2 a b c \sqrt {x}-2 b^2 c^2 x \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )+6 b^2 \left (c \sqrt {x}-1\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^2 \left (a c \sqrt {x}+a+b c \sqrt {x}\right )-6 b^3 c^2 x \text {Li}_2\left (e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )+2 b^3 \left (c^2 x-1\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^3}{2 x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (c \sqrt {x}\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c \sqrt {x}\right ) + a^{3}}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.63, size = 5199, normalized size = 36.61 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.73, size = 528, normalized size = 3.72 \[ -3 \, {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b^{3} c^{2} - 3 \, {\left (\log \left (c \sqrt {x}\right ) \log \left (-c \sqrt {x} + 1\right ) + {\rm Li}_2\left (-c \sqrt {x} + 1\right )\right )} b^{3} c^{2} + 3 \, {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x}\right ) + {\rm Li}_2\left (c \sqrt {x} + 1\right )\right )} b^{3} c^{2} - 3 \, a b^{2} c^{2} \log \left (c \sqrt {x} - 1\right ) - \frac {3}{4} \, {\left ({\left (2 \, c \log \left (c \sqrt {x} - 1\right ) - c \log \relax (x) + \frac {2}{\sqrt {x}}\right )} c - \frac {2 \, \log \left (-c \sqrt {x} + 1\right )}{x}\right )} a^{2} b - \frac {a^{3}}{x} + \frac {3}{2} \, {\left (a^{2} b c^{2} - 2 \, a b^{2} c^{2}\right )} \log \left (c \sqrt {x} + 1\right ) - \frac {3}{4} \, {\left (a^{2} b c^{2} - 4 \, a b^{2} c^{2}\right )} \log \relax (x) - \frac {12 \, a^{2} b c \sqrt {x} - {\left (b^{3} c^{2} x - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{3} + {\left (b^{3} c^{2} x - b^{3}\right )} \log \left (-c \sqrt {x} + 1\right )^{3} + 6 \, {\left (b^{3} c \sqrt {x} + a b^{2} - {\left (a b^{2} c^{2} - b^{3} c^{2}\right )} x\right )} \log \left (c \sqrt {x} + 1\right )^{2} + 3 \, {\left (2 \, b^{3} c \sqrt {x} + 2 \, a b^{2} - 2 \, {\left (a b^{2} c^{2} + b^{3} c^{2}\right )} x - {\left (b^{3} c^{2} x - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )^{2} + 12 \, {\left (2 \, a b^{2} c \sqrt {x} + a^{2} b\right )} \log \left (c \sqrt {x} + 1\right ) - 3 \, {\left (8 \, a b^{2} c \sqrt {x} - {\left (b^{3} c^{2} x - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{2} + 4 \, {\left (b^{3} c \sqrt {x} + a b^{2} - {\left (a b^{2} c^{2} - b^{3} c^{2}\right )} x\right )} \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )}{8 \, x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{3}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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