Integrand size = 19, antiderivative size = 132 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=-\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} e x^2 (a+b \arccos (c x))+\frac {b e \arcsin (c x)}{4 c^2}+\frac {1}{2} i b d \arcsin (c x)^2-b d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+d (a+b \arccos (c x)) \log (x)+b d \arcsin (c x) \log (x)+\frac {1}{2} i b d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \]
1/2*e*x^2*(a+b*arccos(c*x))+1/4*b*e*arcsin(c*x)/c^2+1/2*I*b*d*arcsin(c*x)^ 2-b*d*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+d*(a+b*arccos(c*x))*l n(x)+b*d*arcsin(c*x)*ln(x)+1/2*I*b*d*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^ 2)-1/4*b*e*x*(-c^2*x^2+1)^(1/2)/c
Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\frac {1}{2} a e x^2-\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} b e x^2 \arccos (c x)-\frac {1}{2} i b d \arccos (c x)^2+\frac {b e \arcsin (c x)}{4 c^2}+b d \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+a d \log (x)-\frac {1}{2} i b d \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) \]
(a*e*x^2)/2 - (b*e*x*Sqrt[1 - c^2*x^2])/(4*c) + (b*e*x^2*ArcCos[c*x])/2 - (I/2)*b*d*ArcCos[c*x]^2 + (b*e*ArcSin[c*x])/(4*c^2) + b*d*ArcCos[c*x]*Log[ 1 + E^((2*I)*ArcCos[c*x])] + a*d*Log[x] - (I/2)*b*d*PolyLog[2, -E^((2*I)*A rcCos[c*x])]
Time = 0.51 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5231, 27, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx\) |
| \(\Big \downarrow \) 5231 |
| \(\displaystyle b c \int \frac {e x^2+2 d \log (x)}{2 \sqrt {1-c^2 x^2}}dx+d \log (x) (a+b \arccos (c x))+\frac {1}{2} e x^2 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} b c \int \frac {e x^2+2 d \log (x)}{\sqrt {1-c^2 x^2}}dx+d \log (x) (a+b \arccos (c x))+\frac {1}{2} e x^2 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} b c \int \left (\frac {e x^2}{\sqrt {1-c^2 x^2}}+\frac {2 d \log (x)}{\sqrt {1-c^2 x^2}}\right )dx+d \log (x) (a+b \arccos (c x))+\frac {1}{2} e x^2 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d \log (x) (a+b \arccos (c x))+\frac {1}{2} e x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {e \arcsin (c x)}{2 c^3}+\frac {i d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c}+\frac {i d \arcsin (c x)^2}{c}-\frac {2 d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c}+\frac {2 d \log (x) \arcsin (c x)}{c}-\frac {e x \sqrt {1-c^2 x^2}}{2 c^2}\right )\) |
(e*x^2*(a + b*ArcCos[c*x]))/2 + d*(a + b*ArcCos[c*x])*Log[x] + (b*c*(-1/2* (e*x*Sqrt[1 - c^2*x^2])/c^2 + (e*ArcSin[c*x])/(2*c^3) + (I*d*ArcSin[c*x]^2 )/c - (2*d*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])])/c + (2*d*ArcSin[c*x ]*Log[x])/c + (I*d*PolyLog[2, E^((2*I)*ArcSin[c*x])])/c))/2
3.1.20.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp [(a + b*ArcCos[c*x]) u, x] + Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
Time = 4.93 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97
| method | result | size |
| parts | \(\frac {a e \,x^{2}}{2}+a d \ln \left (x \right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}-\frac {b e x \sqrt {-c^{2} x^{2}+1}}{4 c}+\frac {b \arccos \left (c x \right ) e \,x^{2}}{2}-\frac {b \arccos \left (c x \right ) e}{4 c^{2}}+b d \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b d \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(128\) |
| derivativedivides | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}-\frac {b e x \sqrt {-c^{2} x^{2}+1}}{4 c}+\frac {b \arccos \left (c x \right ) e \,x^{2}}{2}-\frac {b \arccos \left (c x \right ) e}{4 c^{2}}+b d \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b d \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(130\) |
| default | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}-\frac {b e x \sqrt {-c^{2} x^{2}+1}}{4 c}+\frac {b \arccos \left (c x \right ) e \,x^{2}}{2}-\frac {b \arccos \left (c x \right ) e}{4 c^{2}}+b d \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b d \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(130\) |
1/2*a*e*x^2+a*d*ln(x)-1/2*I*b*d*arccos(c*x)^2-1/4*b*e*x*(-c^2*x^2+1)^(1/2) /c+1/2*b*arccos(c*x)*e*x^2-1/4*b/c^2*arccos(c*x)*e+b*d*arccos(c*x)*ln(1+(c *x+I*(-c^2*x^2+1)^(1/2))^2)-1/2*I*b*d*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2) )^2)
\[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arccos \left (c x\right ) + a\right )}}{x} \,d x } \]
\[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \]
\[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arccos \left (c x\right ) + a\right )}}{x} \,d x } \]
1/2*a*e*x^2 + a*d*log(x) + integrate((b*e*x^2 + b*d)*arctan2(sqrt(c*x + 1) *sqrt(-c*x + 1), c*x)/x, x)
\[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arccos \left (c x\right ) + a\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x} \,d x \]