Integrand size = 19, antiderivative size = 119 \[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} i b e \arcsin (c x)^2-\frac {d (a+b \arcsin (c x))}{2 x^2}+b e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-b e \arcsin (c x) \log (x)+e (a+b \arcsin (c x)) \log (x)-\frac {1}{2} i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \]
-1/2*I*b*e*arcsin(c*x)^2-1/2*d*(a+b*arcsin(c*x))/x^2+b*e*arcsin(c*x)*ln(1- (I*c*x+(-c^2*x^2+1)^(1/2))^2)-b*e*arcsin(c*x)*ln(x)+e*(a+b*arcsin(c*x))*ln (x)-1/2*I*b*e*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*b*c*d*(-c^2*x^2+ 1)^(1/2)/x
Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.85 \[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=-\frac {a d}{2 x^2}-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {b d \arcsin (c x)}{2 x^2}+b e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+a e \log (x)-\frac {1}{2} i b e \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right ) \]
-1/2*(a*d)/x^2 - (b*c*d*Sqrt[1 - c^2*x^2])/(2*x) - (b*d*ArcSin[c*x])/(2*x^ 2) + b*e*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + a*e*Log[x] - (I/2)*b *e*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x])])
Time = 0.49 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.07, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx\) |
| \(\Big \downarrow \) 5230 |
| \(\displaystyle -b c \int -\frac {\frac {d}{x^2}-2 e \log (x)}{2 \sqrt {1-c^2 x^2}}dx-\frac {d (a+b \arcsin (c x))}{2 x^2}+e \log (x) (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} b c \int \frac {\frac {d}{x^2}-2 e \log (x)}{\sqrt {1-c^2 x^2}}dx-\frac {d (a+b \arcsin (c x))}{2 x^2}+e \log (x) (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} b c \int \left (\frac {d}{x^2 \sqrt {1-c^2 x^2}}-\frac {2 e \log (x)}{\sqrt {1-c^2 x^2}}\right )dx-\frac {d (a+b \arcsin (c x))}{2 x^2}+e \log (x) (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d (a+b \arcsin (c x))}{2 x^2}+e \log (x) (a+b \arcsin (c x))+\frac {1}{2} b c \left (-\frac {i e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c}-\frac {i e \arcsin (c x)^2}{c}+\frac {2 e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c}-\frac {2 e \log (x) \arcsin (c x)}{c}-\frac {d \sqrt {1-c^2 x^2}}{x}\right )\) |
-1/2*(d*(a + b*ArcSin[c*x]))/x^2 + e*(a + b*ArcSin[c*x])*Log[x] + (b*c*(-( (d*Sqrt[1 - c^2*x^2])/x) - (I*e*ArcSin[c*x]^2)/c + (2*e*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])])/c - (2*e*ArcSin[c*x]*Log[x])/c - (I*e*PolyLog[2, E^((2*I)*ArcSin[c*x])])/c))/2
3.7.3.3.1 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 = 0.52 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.52
| method | result | size |
| derivativedivides | \(c^{2} \left (-\frac {a d}{2 c^{2} x^{2}}+\frac {a e \ln \left (c x \right )}{c^{2}}+\frac {b \left (-\frac {i \arcsin \left (c x \right )^{2} e}{2}-\frac {d \left (-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 x^{2}}+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) e \arcsin \left (c x \right )+\ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right ) e \arcsin \left (c x \right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right ) e -i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right ) e \right )}{c^{2}}\right )\) | \(181\) |
| default | \(c^{2} \left (-\frac {a d}{2 c^{2} x^{2}}+\frac {a e \ln \left (c x \right )}{c^{2}}+\frac {b \left (-\frac {i \arcsin \left (c x \right )^{2} e}{2}-\frac {d \left (-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 x^{2}}+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) e \arcsin \left (c x \right )+\ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right ) e \arcsin \left (c x \right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right ) e -i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right ) e \right )}{c^{2}}\right )\) | \(181\) |
| parts | \(-\frac {a d}{2 x^{2}}+a e \ln \left (x \right )+b \,c^{2} \left (-\frac {i \arcsin \left (c x \right )^{2} e}{2 c^{2}}-\frac {d \left (-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}+\frac {e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}-\frac {i e \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}+\frac {e \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}-\frac {i e \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}\right )\) | \(187\) |
c^2*(-1/2*a*d/c^2/x^2+a/c^2*e*ln(c*x)+b/c^2*(-1/2*I*arcsin(c*x)^2*e-1/2*d* (-I*c^2*x^2+c*x*(-c^2*x^2+1)^(1/2)+arcsin(c*x))/x^2+ln(1+I*c*x+(-c^2*x^2+1 )^(1/2))*e*arcsin(c*x)+ln(1-I*c*x-(-c^2*x^2+1)^(1/2))*e*arcsin(c*x)-I*poly log(2,-I*c*x-(-c^2*x^2+1)^(1/2))*e-I*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))*e ))
\[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
\[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{3}}\, dx \]
\[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
-1/2*b*d*(sqrt(-c^2*x^2 + 1)*c/x + arcsin(c*x)/x^2) + b*e*integrate(arctan 2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x, x) + a*e*log(x) - 1/2*a*d/x^2
\[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x^3} \,d x \]