Integrand size = 10, antiderivative size = 177 \[ \int \frac {\arccos (a+b x)}{x} \, dx=-\frac {1}{2} i \arccos (a+b x)^2+\arccos (a+b x) \log \left (1-\frac {e^{i \arccos (a+b x)}}{a-i \sqrt {1-a^2}}\right )+\arccos (a+b x) \log \left (1-\frac {e^{i \arccos (a+b x)}}{a+i \sqrt {1-a^2}}\right )-i \operatorname {PolyLog}\left (2,\frac {e^{i \arccos (a+b x)}}{a-i \sqrt {1-a^2}}\right )-i \operatorname {PolyLog}\left (2,\frac {e^{i \arccos (a+b x)}}{a+i \sqrt {1-a^2}}\right ) \]
[Out]
Time = 0.19 (sec) , antiderivative size = 177, 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 = {4890, 4826, 4618, 2221, 2317, 2438} \[ \int \frac {\arccos (a+b x)}{x} \, dx=-i \operatorname {PolyLog}\left (2,\frac {e^{i \arccos (a+b x)}}{a-i \sqrt {1-a^2}}\right )-i \operatorname {PolyLog}\left (2,\frac {e^{i \arccos (a+b x)}}{a+i \sqrt {1-a^2}}\right )+\arccos (a+b x) \log \left (1-\frac {e^{i \arccos (a+b x)}}{a-i \sqrt {1-a^2}}\right )+\arccos (a+b x) \log \left (1-\frac {e^{i \arccos (a+b x)}}{a+i \sqrt {1-a^2}}\right )-\frac {1}{2} i \arccos (a+b x)^2 \]
[In]
[Out]
Rule 2221
Rule 2317
Rule 2438
Rule 4618
Rule 4826
Rule 4890
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\arccos (x)}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \frac {x \sin (x)}{-\frac {a}{b}+\frac {\cos (x)}{b}} \, dx,x,\arccos (a+b x)\right )}{b} \\ & = -\frac {1}{2} i \arccos (a+b x)^2-\frac {\text {Subst}\left (\int \frac {e^{i x} x}{-\frac {i a}{b}-\frac {\sqrt {1-a^2}}{b}+\frac {i e^{i x}}{b}} \, dx,x,\arccos (a+b x)\right )}{b}-\frac {\text {Subst}\left (\int \frac {e^{i x} x}{-\frac {i a}{b}+\frac {\sqrt {1-a^2}}{b}+\frac {i e^{i x}}{b}} \, dx,x,\arccos (a+b x)\right )}{b} \\ & = -\frac {1}{2} i \arccos (a+b x)^2+\arccos (a+b x) \log \left (1-\frac {e^{i \arccos (a+b x)}}{a-i \sqrt {1-a^2}}\right )+\arccos (a+b x) \log \left (1-\frac {e^{i \arccos (a+b x)}}{a+i \sqrt {1-a^2}}\right )-\text {Subst}\left (\int \log \left (1+\frac {i e^{i x}}{\left (-\frac {i a}{b}-\frac {\sqrt {1-a^2}}{b}\right ) b}\right ) \, dx,x,\arccos (a+b x)\right )-\text {Subst}\left (\int \log \left (1+\frac {i e^{i x}}{\left (-\frac {i a}{b}+\frac {\sqrt {1-a^2}}{b}\right ) b}\right ) \, dx,x,\arccos (a+b x)\right ) \\ & = -\frac {1}{2} i \arccos (a+b x)^2+\arccos (a+b x) \log \left (1-\frac {e^{i \arccos (a+b x)}}{a-i \sqrt {1-a^2}}\right )+\arccos (a+b x) \log \left (1-\frac {e^{i \arccos (a+b x)}}{a+i \sqrt {1-a^2}}\right )+i \text {Subst}\left (\int \frac {\log \left (1+\frac {i x}{\left (-\frac {i a}{b}-\frac {\sqrt {1-a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{i \arccos (a+b x)}\right )+i \text {Subst}\left (\int \frac {\log \left (1+\frac {i x}{\left (-\frac {i a}{b}+\frac {\sqrt {1-a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{i \arccos (a+b x)}\right ) \\ & = -\frac {1}{2} i \arccos (a+b x)^2+\arccos (a+b x) \log \left (1-\frac {e^{i \arccos (a+b x)}}{a-i \sqrt {1-a^2}}\right )+\arccos (a+b x) \log \left (1-\frac {e^{i \arccos (a+b x)}}{a+i \sqrt {1-a^2}}\right )-i \operatorname {PolyLog}\left (2,\frac {e^{i \arccos (a+b x)}}{a-i \sqrt {1-a^2}}\right )-i \operatorname {PolyLog}\left (2,\frac {e^{i \arccos (a+b x)}}{a+i \sqrt {1-a^2}}\right ) \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.29 \[ \int \frac {\arccos (a+b x)}{x} \, dx=-\frac {1}{2} i \arccos (a+b x)^2-4 i \arcsin \left (\frac {\sqrt {1-a}}{\sqrt {2}}\right ) \arctan \left (\frac {(1+a) \tan \left (\frac {1}{2} \arccos (a+b x)\right )}{\sqrt {-1+a^2}}\right )+\left (\arccos (a+b x)-2 \arcsin \left (\frac {\sqrt {1-a}}{\sqrt {2}}\right )\right ) \log \left (1+\left (-a+\sqrt {-1+a^2}\right ) e^{i \arccos (a+b x)}\right )+\left (\arccos (a+b x)+2 \arcsin \left (\frac {\sqrt {1-a}}{\sqrt {2}}\right )\right ) \log \left (1-\left (a+\sqrt {-1+a^2}\right ) e^{i \arccos (a+b x)}\right )-i \left (\operatorname {PolyLog}\left (2,\left (a-\sqrt {-1+a^2}\right ) e^{i \arccos (a+b x)}\right )+\operatorname {PolyLog}\left (2,\left (a+\sqrt {-1+a^2}\right ) e^{i \arccos (a+b x)}\right )\right ) \]
[In]
[Out]
Time = 3.97 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.12
method | result | size |
derivativedivides | \(-\frac {i \arccos \left (b x +a \right )^{2}}{2}+\arccos \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}-b x -i \sqrt {1-\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}-1}}\right )+\arccos \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}+b x +i \sqrt {1-\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}-1}}\right )-i \operatorname {dilog}\left (\frac {\sqrt {a^{2}-1}-b x -i \sqrt {1-\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}-1}}\right )-i \operatorname {dilog}\left (\frac {\sqrt {a^{2}-1}+b x +i \sqrt {1-\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}-1}}\right )\) | \(199\) |
default | \(-\frac {i \arccos \left (b x +a \right )^{2}}{2}+\arccos \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}-b x -i \sqrt {1-\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}-1}}\right )+\arccos \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}+b x +i \sqrt {1-\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}-1}}\right )-i \operatorname {dilog}\left (\frac {\sqrt {a^{2}-1}-b x -i \sqrt {1-\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}-1}}\right )-i \operatorname {dilog}\left (\frac {\sqrt {a^{2}-1}+b x +i \sqrt {1-\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}-1}}\right )\) | \(199\) |
[In]
[Out]
\[ \int \frac {\arccos (a+b x)}{x} \, dx=\int { \frac {\arccos \left (b x + a\right )}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {\arccos (a+b x)}{x} \, dx=\int \frac {\operatorname {acos}{\left (a + b x \right )}}{x}\, dx \]
[In]
[Out]
\[ \int \frac {\arccos (a+b x)}{x} \, dx=\int { \frac {\arccos \left (b x + a\right )}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {\arccos (a+b x)}{x} \, dx=\int { \frac {\arccos \left (b x + a\right )}{x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\arccos (a+b x)}{x} \, dx=\int \frac {\mathrm {acos}\left (a+b\,x\right )}{x} \,d x \]
[In]
[Out]