Optimal. Leaf size=234 \[ 4 b^2 c^4 \log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b^2 c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 x}+\frac {1}{2} c^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3+2 b c^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2-\frac {3 b c^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 \sqrt {x}}-\frac {b c \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 x^{3/2}}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{2 x^2}-2 b^3 c^4 \text {Li}_2\left (\frac {2}{\sqrt {x} c+1}-1\right )+\frac {1}{2} b^3 c^4 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {b^3 c^3}{2 \sqrt {x}} \]
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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 = {} \[ \int \frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx \]
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^3} \, dx &=\int \frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx\\ \end {align*}
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Mathematica [A] time = 0.72, size = 333, normalized size = 1.42 \[ -\frac {2 a^3+2 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (3 a^2+2 a b c \sqrt {x} \left (3 c^2 x+1\right )-8 b^2 c^4 x^2 \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )+b^2 c^2 x \left (1-c^2 x\right )\right )+3 a^2 b c^4 x^2 \log \left (1-c \sqrt {x}\right )-3 a^2 b c^4 x^2 \log \left (c \sqrt {x}+1\right )+6 a^2 b c^3 x^{3/2}+2 a^2 b c \sqrt {x}-2 a b^2 c^4 x^2+2 a b^2 c^2 x-16 a b^2 c^4 x^2 \log \left (\frac {c \sqrt {x}}{\sqrt {1-c^2 x}}\right )-2 b^2 \tanh ^{-1}\left (c \sqrt {x}\right )^2 \left (3 a \left (c^4 x^2-1\right )+b c \sqrt {x} \left (4 c^3 x^{3/2}-3 c^2 x-1\right )\right )+8 b^3 c^4 x^2 \text {Li}_2\left (e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )-2 b^3 \left (c^4 x^2-1\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^3+2 b^3 c^3 x^{3/2}}{4 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (c \sqrt {x}\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c \sqrt {x}\right ) + a^{3}}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.59, size = 1365, normalized size = 5.83 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.96, size = 703, normalized size = 3.00 \[ -2 \, {\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^{4} - 2 \, {\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^{4} + 2 \, {\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^{4} - \frac {1}{8} \, {\left ({\left (6 \, c^{3} \log \left (c \sqrt {x} - 1\right ) - 3 \, c^{3} \log \relax (x) + \frac {6 \, c^{2} x + 3 \, c \sqrt {x} + 2}{x^{\frac {3}{2}}}\right )} c - \frac {6 \, \log \left (-c \sqrt {x} + 1\right )}{x^{2}}\right )} a^{2} b + \frac {1}{4} \, {\left (3 \, a^{2} b c^{4} - 8 \, a b^{2} c^{4} + b^{3} c^{4}\right )} \log \left (c \sqrt {x} + 1\right ) - \frac {1}{4} \, {\left (8 \, a b^{2} c^{4} + b^{3} c^{4}\right )} \log \left (c \sqrt {x} - 1\right ) - \frac {1}{8} \, {\left (3 \, a^{2} b c^{4} - 16 \, a b^{2} c^{4}\right )} \log \relax (x) - \frac {a^{3}}{2 \, x^{2}} - \frac {4 \, a^{2} b c \sqrt {x} - {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{3} + {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (-c \sqrt {x} + 1\right )^{3} + 2 \, {\left (3 \, b^{3} c^{3} x^{\frac {3}{2}} + b^{3} c \sqrt {x} + 3 \, a b^{2} - {\left (3 \, a b^{2} c^{4} - 4 \, b^{3} c^{4}\right )} x^{2}\right )} \log \left (c \sqrt {x} + 1\right )^{2} + {\left (6 \, b^{3} c^{3} x^{\frac {3}{2}} + 2 \, b^{3} c \sqrt {x} + 6 \, a b^{2} - 2 \, {\left (3 \, a b^{2} c^{4} + 4 \, b^{3} c^{4}\right )} x^{2} - 3 \, {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )^{2} + 4 \, {\left (3 \, a^{2} b c^{3} + 2 \, b^{3} c^{3}\right )} x^{\frac {3}{2}} - 2 \, {\left (3 \, a^{2} b c^{2} - 4 \, a b^{2} c^{2}\right )} x + 4 \, {\left (6 \, a b^{2} c^{3} x^{\frac {3}{2}} + b^{3} c^{2} x + 2 \, a b^{2} c \sqrt {x} + 3 \, a^{2} b\right )} \log \left (c \sqrt {x} + 1\right ) - {\left (24 \, a b^{2} c^{3} x^{\frac {3}{2}} + 4 \, b^{3} c^{2} x + 8 \, a b^{2} c \sqrt {x} - 3 \, {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{2} + 4 \, {\left (3 \, b^{3} c^{3} x^{\frac {3}{2}} + b^{3} c \sqrt {x} + 3 \, a b^{2} - {\left (3 \, a b^{2} c^{4} - 4 \, b^{3} c^{4}\right )} x^{2}\right )} \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )}{16 \, x^{2}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{3}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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