Optimal. Leaf size=200 \[ -b^2 c^3 \text {Li}_2\left (\frac {2}{c x+1}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2+b c^3 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{2} b^3 c^3 \text {Li}_3\left (\frac {2}{c x+1}-1\right )+b^3 c^3 \log (x)-\frac {1}{2} b^3 c^3 \log \left (1-c^2 x^2\right ) \]
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Rubi [A] time = 0.50, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5916, 5982, 266, 36, 29, 31, 5948, 5988, 5932, 6056, 6610} \[ -b^2 c^3 \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{2} b^3 c^3 \text {PolyLog}\left (3,\frac {2}{c x+1}-1\right )-\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2+b c^3 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}-\frac {1}{2} b^3 c^3 \log \left (1-c^2 x^2\right )+b^3 c^3 \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 5916
Rule 5932
Rule 5948
Rule 5982
Rule 5988
Rule 6056
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{x^4} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+(b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+(b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx+\left (b c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+\left (b^2 c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (b c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x (1+c x)} \, dx\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )+\left (b^2 c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (b^2 c^4\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx-\left (2 b^2 c^4\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )-b^2 c^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\left (b^3 c^3\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx+\left (b^3 c^4\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )-b^2 c^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )-\frac {1}{2} b^3 c^3 \text {Li}_3\left (-1+\frac {2}{1+c x}\right )+\frac {1}{2} \left (b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )-b^2 c^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )-\frac {1}{2} b^3 c^3 \text {Li}_3\left (-1+\frac {2}{1+c x}\right )+\frac {1}{2} \left (b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (b^3 c^5\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+b^3 c^3 \log (x)-\frac {1}{2} b^3 c^3 \log \left (1-c^2 x^2\right )+b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )-b^2 c^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )-\frac {1}{2} b^3 c^3 \text {Li}_3\left (-1+\frac {2}{1+c x}\right )\\ \end {align*}
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Mathematica [C] time = 0.94, size = 323, normalized size = 1.62 \[ -\frac {8 a^3-24 a^2 b c^3 x^3 \log (x)+12 a^2 b c^3 x^3 \log \left (1-c^2 x^2\right )+12 a^2 b c x+24 a^2 b \tanh ^{-1}(c x)+24 a b^2 \left (c^3 x^3 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )+\left (1-c^3 x^3\right ) \tanh ^{-1}(c x)^2+c^2 x^2-c x \tanh ^{-1}(c x) \left (c^2 x^2+2 c^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-1\right )\right )+b^3 \left (-24 c^3 x^3 \tanh ^{-1}(c x) \text {Li}_2\left (e^{2 \tanh ^{-1}(c x)}\right )+12 c^3 x^3 \text {Li}_3\left (e^{2 \tanh ^{-1}(c x)}\right )-i \pi ^3 c^3 x^3+8 c^3 x^3 \tanh ^{-1}(c x)^3-12 c^3 x^3 \tanh ^{-1}(c x)^2-24 c^3 x^3 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+24 c^2 x^2 \tanh ^{-1}(c x)-24 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+12 c x \tanh ^{-1}(c x)^2+8 \tanh ^{-1}(c x)^3\right )}{24 x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c x\right ) + a^{3}}{x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.94, size = 1838, normalized size = 9.19 \[ \text {result too large to display} \]
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 \[ -\frac {1}{2} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} a^{2} b - \frac {a^{3}}{3 \, x^{3}} - \frac {{\left (b^{3} c^{3} x^{3} - b^{3}\right )} \log \left (-c x + 1\right )^{3} + 3 \, {\left (b^{3} c x + 2 \, a b^{2} + {\left (b^{3} c^{3} x^{3} + b^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{24 \, x^{3}} - \int -\frac {{\left (b^{3} c x - b^{3}\right )} \log \left (c x + 1\right )^{3} + 6 \, {\left (a b^{2} c x - a b^{2}\right )} \log \left (c x + 1\right )^{2} + {\left (2 \, b^{3} c^{2} x^{2} + 4 \, a b^{2} c x - 3 \, {\left (b^{3} c x - b^{3}\right )} \log \left (c x + 1\right )^{2} + 2 \, {\left (b^{3} c^{4} x^{4} + 6 \, a b^{2} - {\left (6 \, a b^{2} c - b^{3} c\right )} x\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \, {\left (c x^{5} - x^{4}\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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