Integrand size = 10, antiderivative size = 122 \[ \int \frac {\arccos (a x)^3}{x^2} \, dx=-\frac {\arccos (a x)^3}{x}-6 i a \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+6 i a \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-6 i a \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-6 a \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+6 a \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right ) \]
[Out]
Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4724, 4804, 4266, 2611, 2320, 6724} \[ \int \frac {\arccos (a x)^3}{x^2} \, dx=-6 i a \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+6 i a \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-6 i a \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-6 a \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+6 a \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )-\frac {\arccos (a x)^3}{x} \]
[In]
[Out]
Rule 2320
Rule 2611
Rule 4266
Rule 4724
Rule 4804
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)^3}{x}-(3 a) \int \frac {\arccos (a x)^2}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {\arccos (a x)^3}{x}+(3 a) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {\arccos (a x)^3}{x}-6 i a \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )-(6 a) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\arccos (a x)\right )+(6 a) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {\arccos (a x)^3}{x}-6 i a \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+6 i a \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-6 i a \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-(6 i a) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arccos (a x)\right )+(6 i a) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {\arccos (a x)^3}{x}-6 i a \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+6 i a \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-6 i a \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-(6 a) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arccos (a x)}\right )+(6 a) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arccos (a x)}\right ) \\ & = -\frac {\arccos (a x)^3}{x}-6 i a \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+6 i a \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-6 i a \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-6 a \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+6 a \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.14 \[ \int \frac {\arccos (a x)^3}{x^2} \, dx=-\frac {\arccos (a x)^3}{x}+3 a \left (\arccos (a x)^2 \left (\log \left (1-i e^{i \arccos (a x)}\right )-\log \left (1+i e^{i \arccos (a x)}\right )\right )+2 i \arccos (a x) \left (\operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )-2 \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right ) \]
[In]
[Out]
\[\int \frac {\arccos \left (a x \right )^{3}}{x^{2}}d x\]
[In]
[Out]
\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{x^2} \,d x \]
[In]
[Out]