Integrand size = 19, antiderivative size = 132 \[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x} \, dx=\frac {b e x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e \arcsin (c x)}{4 c^2}-\frac {1}{2} i b d \arcsin (c x)^2+\frac {1}{2} e x^2 (a+b \arcsin (c x))+b d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-b d \arcsin (c x) \log (x)+d (a+b \arcsin (c x)) \log (x)-\frac {1}{2} i b d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \]
-1/4*b*e*arcsin(c*x)/c^2-1/2*I*b*d*arcsin(c*x)^2+1/2*e*x^2*(a+b*arcsin(c*x ))+b*d*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-b*d*arcsin(c*x)*ln(x )+d*(a+b*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 = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86 \[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x} \, dx=\frac {1}{2} a e x^2+\frac {b e x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e \arcsin (c x)}{4 c^2}+\frac {1}{2} b e x^2 \arcsin (c x)+b d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+a d \log (x)-\frac {1}{2} i b d \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right ) \]
(a*e*x^2)/2 + (b*e*x*Sqrt[1 - c^2*x^2])/(4*c) - (b*e*ArcSin[c*x])/(4*c^2) + (b*e*x^2*ArcSin[c*x])/2 + b*d*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + a*d*Log[x] - (I/2)*b*d*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x] )])
Time = 0.50 (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 = {5230, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x} \, dx\) |
| \(\Big \downarrow \) 5230 |
| \(\displaystyle -b c \int \frac {e x^2+2 d \log (x)}{2 \sqrt {1-c^2 x^2}}dx+d \log (x) (a+b \arcsin (c x))+\frac {1}{2} e x^2 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{2} b c \int \frac {e x^2+2 d \log (x)}{\sqrt {1-c^2 x^2}}dx+d \log (x) (a+b \arcsin (c x))+\frac {1}{2} e x^2 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} b c \int \left (\frac {e x^2}{\sqrt {1-c^2 x^2}}+\frac {2 d \log (x)}{\sqrt {1-c^2 x^2}}\right )dx+d \log (x) (a+b \arcsin (c x))+\frac {1}{2} e x^2 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d \log (x) (a+b \arcsin (c x))+\frac {1}{2} e x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {e \arcsin (c x)}{2 c^3}+\frac {i d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c}+\frac {i d \arcsin (c x)^2}{c}-\frac {2 d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c}+\frac {2 d \log (x) \arcsin (c x)}{c}-\frac {e x \sqrt {1-c^2 x^2}}{2 c^2}\right )\) |
(e*x^2*(a + b*ArcSin[c*x]))/2 + d*(a + b*ArcSin[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
3.7.1.3.1 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_.) + ArcSin[(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*ArcSin[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 = 0.67 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.22
| method | result | size |
| parts | \(\frac {a e \,x^{2}}{2}+a d \ln \left (x \right )+b \left (-\frac {i d \arcsin \left (c x \right )^{2}}{2}+d \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+d \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right ) e \cos \left (2 \arcsin \left (c x \right )\right )}{4 c^{2}}+\frac {e \sin \left (2 \arcsin \left (c x \right )\right )}{8 c^{2}}\right )\) | \(161\) |
| derivativedivides | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {i b d \arcsin \left (c x \right )^{2}}{2}+b d \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i b d \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+b d \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i b d \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {b e \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4 c^{2}}+\frac {b e \sin \left (2 \arcsin \left (c x \right )\right )}{8 c^{2}}\) | \(167\) |
| default | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {i b d \arcsin \left (c x \right )^{2}}{2}+b d \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i b d \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+b d \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i b d \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {b e \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4 c^{2}}+\frac {b e \sin \left (2 \arcsin \left (c x \right )\right )}{8 c^{2}}\) | \(167\) |
1/2*a*e*x^2+a*d*ln(x)+b*(-1/2*I*d*arcsin(c*x)^2+d*arcsin(c*x)*ln(1+I*c*x+( -c^2*x^2+1)^(1/2))+d*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-I*d*polylo g(2,-I*c*x-(-c^2*x^2+1)^(1/2))-I*d*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-1/4 /c^2*arcsin(c*x)*e*cos(2*arcsin(c*x))+1/8*e/c^2*sin(2*arcsin(c*x)))
\[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x} \,d x } \]
\[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \]
\[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x} \,d x } \]
1/2*a*e*x^2 + a*d*log(x) + integrate((b*e*x^2 + b*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x, x)
\[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x} \,d x \]