Integrand size = 14, antiderivative size = 89 \[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=-\frac {(a+b \arccos (c x))^2}{x}-4 i b c (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )+2 i b^2 c \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-2 i b^2 c \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4724, 4804, 4266, 2317, 2438} \[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=-4 i b c \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{x}+2 i b^2 c \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-2 i b^2 c \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) \]
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Rule 2317
Rule 2438
Rule 4266
Rule 4724
Rule 4804
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arccos (c x))^2}{x}-(2 b c) \int \frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {(a+b \arccos (c x))^2}{x}+(2 b c) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arccos (c x)) \\ & = -\frac {(a+b \arccos (c x))^2}{x}-4 i b c (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )-\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arccos (c x)\right )+\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arccos (c x)\right ) \\ & = -\frac {(a+b \arccos (c x))^2}{x}-4 i b c (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )+\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arccos (c x)}\right )-\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arccos (c x)}\right ) \\ & = -\frac {(a+b \arccos (c x))^2}{x}-4 i b c (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )+2 i b^2 c \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-2 i b^2 c \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.51 \[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=-\frac {a^2+2 a b \left (\arccos (c x)-c x \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )+b^2 \left (\arccos (c x)^2-2 c x \left (\arccos (c x) \left (\log \left (1-i e^{i \arccos (c x)}\right )-\log \left (1+i e^{i \arccos (c x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )\right )\right )}{x} \]
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Time = 0.70 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.01
method | result | size |
parts | \(-\frac {a^{2}}{x}+b^{2} c \left (-\frac {\arccos \left (c x \right )^{2}}{c x}-2 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )+2 a b c \left (-\frac {\arccos \left (c x \right )}{c x}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(179\) |
derivativedivides | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\arccos \left (c x \right )^{2}}{c x}-2 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )+2 a b \left (-\frac {\arccos \left (c x \right )}{c x}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(182\) |
default | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\arccos \left (c x \right )^{2}}{c x}-2 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )+2 a b \left (-\frac {\arccos \left (c x \right )}{c x}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(182\) |
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\[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
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\[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{x^2} \,d x \]
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