Optimal. Leaf size=136 \[ b^2 c^2 \log \left (2-\frac {2}{c x^3+1}\right ) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac {1}{2} b c^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2+\frac {1}{6} c^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3-\frac {b c \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{2 x^3}-\frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3}{6 x^6}-\frac {1}{2} b^3 c^2 \text {Li}_2\left (\frac {2}{c x^3+1}-1\right ) \]
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Rubi [F] time = 1.57, 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 = {} \[ \int \frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3}{x^7} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3}{x^7} \, dx &=\int \left (\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^3}{8 x^7}+\frac {3 b \left (-2 a+b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{8 x^7}-\frac {3 b^2 \left (-2 a+b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{8 x^7}+\frac {b^3 \log ^3\left (1+c x^3\right )}{8 x^7}\right ) \, dx\\ &=\frac {1}{8} \int \frac {\left (2 a-b \log \left (1-c x^3\right )\right )^3}{x^7} \, dx+\frac {1}{8} (3 b) \int \frac {\left (-2 a+b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{x^7} \, dx-\frac {1}{8} \left (3 b^2\right ) \int \frac {\left (-2 a+b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{x^7} \, dx+\frac {1}{8} b^3 \int \frac {\log ^3\left (1+c x^3\right )}{x^7} \, dx\\ &=\frac {1}{24} \operatorname {Subst}\left (\int \frac {(2 a-b \log (1-c x))^3}{x^3} \, dx,x,x^3\right )+\frac {1}{8} b \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^3\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^3\right )+\frac {1}{24} b^3 \operatorname {Subst}\left (\int \frac {\log ^3(1+c x)}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 x^6}-\frac {b^3 \log ^3\left (1+c x^3\right )}{48 x^6}+\frac {1}{8} b \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^3\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^3\right )+\frac {1}{16} (b c) \operatorname {Subst}\left (\int \frac {(2 a-b \log (1-c x))^2}{x^2 (1-c x)} \, dx,x,x^3\right )+\frac {1}{16} \left (b^3 c\right ) \operatorname {Subst}\left (\int \frac {\log ^2(1+c x)}{x^2 (1+c x)} \, dx,x,x^3\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 x^6}-\frac {b^3 \log ^3\left (1+c x^3\right )}{48 x^6}-\frac {1}{16} b \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-c x^3\right )+\frac {1}{8} b \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^3\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^3\right )+\frac {1}{16} b^3 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+c x^3\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 x^6}-\frac {b^3 \log ^3\left (1+c x^3\right )}{48 x^6}-\frac {1}{16} b \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-c x^3\right )+\frac {1}{8} b \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^3\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^3\right )+\frac {1}{16} b^3 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+c x^3\right )-\frac {1}{16} (b c) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )} \, dx,x,1-c x^3\right )-\frac {1}{16} \left (b^3 c\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )} \, dx,x,1+c x^3\right )\\ &=-\frac {b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 x^3}-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 x^6}-\frac {b^3 c \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{16 x^3}-\frac {b^3 \log ^3\left (1+c x^3\right )}{48 x^6}+\frac {1}{8} b \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^3\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^3\right )-\frac {1}{16} (b c) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^3\right )-\frac {1}{8} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^3\right )-\frac {1}{16} \left (b^3 c\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+c x^3\right )+\frac {1}{8} \left (b^3 c\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+c x^3\right )-\frac {1}{16} \left (b c^2\right ) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x} \, dx,x,1-c x^3\right )+\frac {1}{16} \left (b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,1+c x^3\right )\\ &=\frac {3}{4} a b^2 c^2 \log (x)-\frac {b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 x^3}+\frac {1}{16} b c^2 \log \left (c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 x^6}-\frac {b^3 c \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{16 x^3}-\frac {1}{16} b^3 c^2 \log \left (-c x^3\right ) \log ^2\left (1+c x^3\right )-\frac {b^3 \log ^3\left (1+c x^3\right )}{48 x^6}-\frac {1}{8} b^3 c^2 \text {Li}_2\left (-c x^3\right )+\frac {1}{8} b \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^3\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^3\right )+\frac {1}{8} \left (b^3 c\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^3\right )+\frac {1}{16} c^2 \operatorname {Subst}\left (\int x^2 \, dx,x,2 a-b \log \left (1-c x^3\right )\right )+\frac {1}{8} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x) (2 a-b \log (x))}{x} \, dx,x,1-c x^3\right )+\frac {1}{16} \left (b^3 c^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log \left (1+c x^3\right )\right )+\frac {1}{8} \left (b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x) \log (x)}{x} \, dx,x,1+c x^3\right )\\ &=\frac {3}{4} a b^2 c^2 \log (x)-\frac {b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 x^3}+\frac {1}{16} b c^2 \log \left (c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2+\frac {1}{48} c^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 x^6}-\frac {b^3 c \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{16 x^3}-\frac {1}{16} b^3 c^2 \log \left (-c x^3\right ) \log ^2\left (1+c x^3\right )+\frac {1}{48} b^3 c^2 \log ^3\left (1+c x^3\right )-\frac {b^3 \log ^3\left (1+c x^3\right )}{48 x^6}-\frac {1}{8} b^3 c^2 \text {Li}_2\left (-c x^3\right )+\frac {1}{8} b^3 c^2 \text {Li}_2\left (c x^3\right )-\frac {1}{8} b^2 c^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \text {Li}_2\left (1-c x^3\right )-\frac {1}{8} b^3 c^2 \log \left (1+c x^3\right ) \text {Li}_2\left (1+c x^3\right )+\frac {1}{8} b \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^3\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^3\right )-\frac {1}{8} \left (b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-c x^3\right )+\frac {1}{8} \left (b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1+c x^3\right )\\ &=\frac {3}{4} a b^2 c^2 \log (x)-\frac {b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 x^3}+\frac {1}{16} b c^2 \log \left (c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2+\frac {1}{48} c^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 x^6}-\frac {b^3 c \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{16 x^3}-\frac {1}{16} b^3 c^2 \log \left (-c x^3\right ) \log ^2\left (1+c x^3\right )+\frac {1}{48} b^3 c^2 \log ^3\left (1+c x^3\right )-\frac {b^3 \log ^3\left (1+c x^3\right )}{48 x^6}-\frac {1}{8} b^3 c^2 \text {Li}_2\left (-c x^3\right )+\frac {1}{8} b^3 c^2 \text {Li}_2\left (c x^3\right )-\frac {1}{8} b^2 c^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \text {Li}_2\left (1-c x^3\right )-\frac {1}{8} b^3 c^2 \log \left (1+c x^3\right ) \text {Li}_2\left (1+c x^3\right )-\frac {1}{8} b^3 c^2 \text {Li}_3\left (1-c x^3\right )+\frac {1}{8} b^3 c^2 \text {Li}_3\left (1+c x^3\right )+\frac {1}{8} b \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^3\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^3\right )\\ \end {align*}
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Mathematica [A] time = 0.30, size = 218, normalized size = 1.60 \[ \frac {a \left (-2 a^2-3 a b c^2 x^6 \log \left (1-c x^3\right )+3 a b c^2 x^6 \log \left (c x^3+1\right )-6 a b c x^3+12 b^2 c^2 x^6 \log \left (\frac {c x^3}{\sqrt {1-c^2 x^6}}\right )\right )-6 b \tanh ^{-1}\left (c x^3\right ) \left (a^2+2 a b c x^3-2 b^2 c^2 x^6 \log \left (1-e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )\right )+6 b^2 \left (c x^3-1\right ) \tanh ^{-1}\left (c x^3\right )^2 \left (a c x^3+a+b c x^3\right )-6 b^3 c^2 x^6 \text {Li}_2\left (e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )+2 b^3 \left (c^2 x^6-1\right ) \tanh ^{-1}\left (c x^3\right )^3}{12 x^6} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (c x^{3}\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c x^{3}\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c x^{3}\right ) + a^{3}}{x^{7}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3}}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctanh \left (c \,x^{3}\right )\right )^{3}}{x^{7}}\, dx \]
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 \[ \frac {1}{4} \, {\left ({\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac {2}{x^{3}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x^{3}\right )}{x^{6}}\right )} a^{2} b + \frac {1}{8} \, {\left ({\left (2 \, {\left (\log \left (c x^{3} - 1\right ) - 2\right )} \log \left (c x^{3} + 1\right ) - \log \left (c x^{3} + 1\right )^{2} - \log \left (c x^{3} - 1\right )^{2} - 4 \, \log \left (c x^{3} - 1\right ) + 24 \, \log \relax (x)\right )} c^{2} + 4 \, {\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac {2}{x^{3}}\right )} c \operatorname {artanh}\left (c x^{3}\right )\right )} a b^{2} - \frac {1}{48} \, b^{3} {\left (\frac {{\left (c^{2} x^{6} - 1\right )} \log \left (-c x^{3} + 1\right )^{3} + 3 \, {\left (2 \, c x^{3} - {\left (c^{2} x^{6} - 1\right )} \log \left (c x^{3} + 1\right )\right )} \log \left (-c x^{3} + 1\right )^{2}}{x^{6}} + 6 \, \int -\frac {{\left (c x^{3} - 1\right )} \log \left (c x^{3} + 1\right )^{3} + 3 \, {\left (2 \, c^{2} x^{6} - {\left (c x^{3} - 1\right )} \log \left (c x^{3} + 1\right )^{2} - {\left (c^{3} x^{9} - c x^{3}\right )} \log \left (c x^{3} + 1\right )\right )} \log \left (-c x^{3} + 1\right )}{c x^{10} - x^{7}}\,{d x}\right )} - \frac {a b^{2} \operatorname {artanh}\left (c x^{3}\right )^{2}}{2 \, x^{6}} - \frac {a^{3}}{6 \, x^{6}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^3\right )\right )}^3}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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