Optimal. Leaf size=123 \[ 3 b^2 c^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}-\frac {3}{2} b^3 c^2 \text {Li}_2\left (\frac {2}{c x+1}-1\right ) \]
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Rubi [A] time = 0.29, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5916, 5982, 5988, 5932, 2447, 5948} \[ -\frac {3}{2} b^3 c^2 \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+3 b^2 c^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x} \]
Antiderivative was successfully verified.
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Rule 2447
Rule 5916
Rule 5932
Rule 5948
Rule 5982
Rule 5988
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{x^3} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\frac {1}{2} (3 b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\frac {1}{2} (3 b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx+\frac {1}{2} \left (3 b c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\left (3 b^2 c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx\\ &=\frac {3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+\left (3 b^2 c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx\\ &=\frac {3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-\left (3 b^3 c^3\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac {3}{2} b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x}+\frac {1}{2} c^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 x^2}+3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-\frac {3}{2} b^3 c^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )\\ \end {align*}
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Mathematica [A] time = 0.29, size = 192, normalized size = 1.56 \[ \frac {a \left (-2 a^2-3 a b c^2 x^2 \log (1-c x)+3 a b c^2 x^2 \log (c x+1)-6 a b c x+12 b^2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )\right )-6 b \tanh ^{-1}(c x) \left (a^2+2 a b c x-2 b^2 c^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )+6 b^2 (c x-1) \tanh ^{-1}(c x)^2 (a c x+a+b c x)-6 b^3 c^2 x^2 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )+2 b^3 \left (c^2 x^2-1\right ) \tanh ^{-1}(c x)^3}{4 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c x\right ) + a^{3}}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.71, size = 5098, normalized size = 41.45 \[ \text {output too large to display} \]
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 \[ \frac {3}{4} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} a^{2} b + \frac {3}{8} \, {\left ({\left (2 \, {\left (\log \left (c x - 1\right ) - 2\right )} \log \left (c x + 1\right ) - \log \left (c x + 1\right )^{2} - \log \left (c x - 1\right )^{2} - 4 \, \log \left (c x - 1\right ) + 8 \, \log \relax (x)\right )} c^{2} + 4 \, {\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c \operatorname {artanh}\left (c x\right )\right )} a b^{2} - \frac {1}{16} \, b^{3} {\left (\frac {{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )^{3} + 3 \, {\left (2 \, c x - {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{x^{2}} + 2 \, \int -\frac {{\left (c x - 1\right )} \log \left (c x + 1\right )^{3} + 3 \, {\left (2 \, c^{2} x^{2} - {\left (c x - 1\right )} \log \left (c x + 1\right )^{2} - {\left (c^{3} x^{3} - c x\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{c x^{4} - x^{3}}\,{d x}\right )} - \frac {3 \, a b^{2} \operatorname {artanh}\left (c x\right )^{2}}{2 \, x^{2}} - \frac {a^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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