Integrand size = 14, antiderivative size = 151 \[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=-\frac {(a+b \arccos (c x))^3}{x}-6 i b c (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )+6 i b^2 c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-6 i b^2 c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )-6 b^3 c \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )+6 b^3 c \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4724, 4804, 4266, 2611, 2320, 6724} \[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=-6 i b c \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2+6 i b^2 c \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-6 i b^2 c \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^3}{x}-6 b^3 c \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )+6 b^3 c \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right ) \]
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Rule 2320
Rule 2611
Rule 4266
Rule 4724
Rule 4804
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arccos (c x))^3}{x}-(3 b c) \int \frac {(a+b \arccos (c x))^2}{x \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {(a+b \arccos (c x))^3}{x}+(3 b c) \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\arccos (c x)\right ) \\ & = -\frac {(a+b \arccos (c x))^3}{x}-6 i b c (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )-\left (6 b^2 c\right ) \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\arccos (c x)\right )+\left (6 b^2 c\right ) \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\arccos (c x)\right ) \\ & = -\frac {(a+b \arccos (c x))^3}{x}-6 i b c (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )+6 i b^2 c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-6 i b^2 c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )-\left (6 i b^3 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arccos (c x)\right )+\left (6 i b^3 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arccos (c x)\right ) \\ & = -\frac {(a+b \arccos (c x))^3}{x}-6 i b c (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )+6 i b^2 c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-6 i b^2 c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )-\left (6 b^3 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arccos (c x)}\right )+\left (6 b^3 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arccos (c x)}\right ) \\ & = -\frac {(a+b \arccos (c x))^3}{x}-6 i b c (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )+6 i b^2 c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-6 i b^2 c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )-6 b^3 c \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )+6 b^3 c \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right ) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(308\) vs. \(2(151)=302\).
Time = 0.22 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=-\frac {a^3}{x}-\frac {3 a^2 b \arccos (c x)}{x}-3 a^2 b c \log (x)+3 a^2 b c \log \left (1+\sqrt {1-c^2 x^2}\right )+3 a b^2 c \left (-\frac {\arccos (c x)^2}{c x}+2 \left (\arccos (c x) \left (\log \left (1-i e^{i \arccos (c x)}\right )-\log \left (1+i e^{i \arccos (c x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )\right )\right )+b^3 c \left (-\frac {\arccos (c x)^3}{c x}+3 \left (\arccos (c x)^2 \left (\log \left (1-i e^{i \arccos (c x)}\right )-\log \left (1+i e^{i \arccos (c x)}\right )\right )+2 i \arccos (c x) \left (\operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )-2 \left (\operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )\right )\right )\right ) \]
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\[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{3}}{x^{2}}d x\]
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\[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{x^{2}} \,d x } \]
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\[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \]
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\[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{x^{2}} \,d x } \]
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\[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arccos (c x))^3}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3}{x^2} \,d x \]
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