Integrand size = 14, antiderivative size = 521 \[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]
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Time = 0.53 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4758, 4826, 4618, 2221, 2317, 2438} \[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 4618
Rule 4758
Rule 4826
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-c} \arccos (a x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \arccos (a x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx \\ & = -\frac {\int \frac {\arccos (a x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\int \frac {\arccos (a x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}} \\ & = \frac {\text {Subst}\left (\int \frac {x \sin (x)}{a \sqrt {-c}-\sqrt {d} \cos (x)} \, dx,x,\arccos (a x)\right )}{2 \sqrt {-c}}+\frac {\text {Subst}\left (\int \frac {x \sin (x)}{a \sqrt {-c}+\sqrt {d} \cos (x)} \, dx,x,\arccos (a x)\right )}{2 \sqrt {-c}} \\ & = \frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}-i \sqrt {d} e^{i x}} \, dx,x,\arccos (a x)\right )}{2 \sqrt {-c}}+\frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}-i \sqrt {d} e^{i x}} \, dx,x,\arccos (a x)\right )}{2 \sqrt {-c}}+\frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}+i \sqrt {d} e^{i x}} \, dx,x,\arccos (a x)\right )}{2 \sqrt {-c}}+\frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}+i \sqrt {d} e^{i x}} \, dx,x,\arccos (a x)\right )}{2 \sqrt {-c}} \\ & = \frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {Subst}\left (\int \log \left (1-\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right ) \, dx,x,\arccos (a x)\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Subst}\left (\int \log \left (1+\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right ) \, dx,x,\arccos (a x)\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {Subst}\left (\int \log \left (1-\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right ) \, dx,x,\arccos (a x)\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Subst}\left (\int \log \left (1+\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right ) \, dx,x,\arccos (a x)\right )}{2 \sqrt {-c} \sqrt {d}} \\ & = \frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {d} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \arccos (a x)}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {d} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \arccos (a x)}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {d} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \arccos (a x)}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {d} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \arccos (a x)}\right )}{2 \sqrt {-c} \sqrt {d}} \\ & = \frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}} \\ \end{align*}
Time = 1.81 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.56 \[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\frac {4 \arcsin \left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (a \sqrt {c}-i \sqrt {d}\right ) \tan \left (\frac {1}{2} \arccos (a x)\right )}{\sqrt {a^2 c+d}}\right )-4 \arcsin \left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (a \sqrt {c}+i \sqrt {d}\right ) \tan \left (\frac {1}{2} \arccos (a x)\right )}{\sqrt {a^2 c+d}}\right )+i \arccos (a x) \log \left (1-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+2 i \arcsin \left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-i \arccos (a x) \log \left (1+\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-2 i \arcsin \left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-i \arccos (a x) \log \left (1-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+2 i \arcsin \left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+i \arccos (a x) \log \left (1+\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-2 i \arcsin \left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-\operatorname {PolyLog}\left (2,-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-\operatorname {PolyLog}\left (2,\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )}{2 \sqrt {c} \sqrt {d}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.62 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.41
\[-\frac {i a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 c \,a^{2}+d}\right )}{2}+\frac {i a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 c \,a^{2}+d \right )}\right )}{2}\]
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\[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\int { \frac {\arccos \left (a x\right )}{d x^{2} + c} \,d x } \]
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\[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\int \frac {\operatorname {acos}{\left (a x \right )}}{c + d x^{2}}\, dx \]
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\[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\int { \frac {\arccos \left (a x\right )}{d x^{2} + c} \,d x } \]
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\[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\int { \frac {\arccos \left (a x\right )}{d x^{2} + c} \,d x } \]
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Timed out. \[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{d\,x^2+c} \,d x \]
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