Integrand size = 31, antiderivative size = 623 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{d+e x} \, dx=\frac {b i x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (2 e^2 i+9 c^2 \left (e^2 g-d e h+d^2 i\right )\right )+9 c^2 e (e h-d i) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}-\frac {b (e h-d i) \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {\left (e^2 g-d e h+d^2 i\right ) x (a+b \arcsin (c x))}{e^3}+\frac {(e h-d i) x^2 (a+b \arcsin (c x))}{2 e^2}+\frac {i x^3 (a+b \arcsin (c x))}{3 e}+\frac {b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \]
-1/4*b*(-d*i+e*h)*arcsin(c*x)/c^2/e^2-1/2*I*b*(-d^3*i+d^2*e*h-d*e^2*g+e^3* f)*arcsin(c*x)^2/e^4+(d^2*i-d*e*h+e^2*g)*x*(a+b*arcsin(c*x))/e^3+1/2*(-d*i +e*h)*x^2*(a+b*arcsin(c*x))/e^2+1/3*i*x^3*(a+b*arcsin(c*x))/e-b*(-d^3*i+d^ 2*e*h-d*e^2*g+e^3*f)*arcsin(c*x)*ln(e*x+d)/e^4+(-d^3*i+d^2*e*h-d*e^2*g+e^3 *f)*(a+b*arcsin(c*x))*ln(e*x+d)/e^4+b*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*arcsi n(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^4+ b*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^ (1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^4-I*b*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*p olylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^4-I*b *(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/( c*d+(c^2*d^2-e^2)^(1/2)))/e^4+1/9*b*i*x^2*(-c^2*x^2+1)^(1/2)/c/e+1/36*b*(8 *e^2*i+36*c^2*(d^2*i-d*e*h+e^2*g)+9*c^2*e*(-d*i+e*h)*x)*(-c^2*x^2+1)^(1/2) /c^3/e^3
Time = 0.87 (sec) , antiderivative size = 610, normalized size of antiderivative = 0.98 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{d+e x} \, dx=\frac {\frac {6 b e \left (e^2 g-d e h+d^2 i\right ) \sqrt {1-c^2 x^2}}{c}+\frac {3 b e^2 (e h-d i) x \sqrt {1-c^2 x^2}}{2 c}+\frac {2 b e^3 i \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )}{3 c^3}-\frac {3 b e^2 (e h-d i) \arcsin (c x)}{2 c^2}-3 i b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arcsin (c x)^2+6 e \left (e^2 g-d e h+d^2 i\right ) x (a+b \arcsin (c x))+3 e^2 (e h-d i) x^2 (a+b \arcsin (c x))+2 e^3 i x^3 (a+b \arcsin (c x))+6 b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arcsin (c x) \log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+6 b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-6 b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arcsin (c x) \log (d+e x)+6 \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x)) \log (d+e x)-6 i b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-6 i b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{6 e^4} \]
((6*b*e*(e^2*g - d*e*h + d^2*i)*Sqrt[1 - c^2*x^2])/c + (3*b*e^2*(e*h - d*i )*x*Sqrt[1 - c^2*x^2])/(2*c) + (2*b*e^3*i*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)) /(3*c^3) - (3*b*e^2*(e*h - d*i)*ArcSin[c*x])/(2*c^2) - (3*I)*b*(e^3*f - d* e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x]^2 + 6*e*(e^2*g - d*e*h + d^2*i)*x*(a + b*ArcSin[c*x]) + 3*e^2*(e*h - d*i)*x^2*(a + b*ArcSin[c*x]) + 2*e^3*i*x^3 *(a + b*ArcSin[c*x]) + 6*b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x] *Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + 6*b*(e^ 3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x] ))/(c*d + Sqrt[c^2*d^2 - e^2])] - 6*b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)* ArcSin[c*x]*Log[d + e*x] + 6*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*Ar cSin[c*x])*Log[d + e*x] - (6*I)*b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*Poly Log[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] - (6*I)*b*(e^3 *f - d*e^2*g + d^2*e*h - d^3*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(6*e^4)
Time = 1.57 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {5252, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x)) \left (f+g x+h x^2+i x^3\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 5252 |
| \(\displaystyle -b c \int \frac {e x \left (6 i d^2-3 e (2 h+i x) d+e^2 \left (2 i x^2+3 h x+6 g\right )\right )+6 \left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) \log (d+e x)}{6 e^4 \sqrt {1-c^2 x^2}}dx+\frac {x (a+b \arcsin (c x)) \left (d^2 i-d e h+e^2 g\right )}{e^3}+\frac {\log (d+e x) (a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}+\frac {x^2 (e h-d i) (a+b \arcsin (c x))}{2 e^2}+\frac {i x^3 (a+b \arcsin (c x))}{3 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \int \frac {e x \left (6 i d^2-3 e (2 h+i x) d+e^2 \left (2 i x^2+3 h x+6 g\right )\right )+6 \left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}}dx}{6 e^4}+\frac {x (a+b \arcsin (c x)) \left (d^2 i-d e h+e^2 g\right )}{e^3}+\frac {\log (d+e x) (a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}+\frac {x^2 (e h-d i) (a+b \arcsin (c x))}{2 e^2}+\frac {i x^3 (a+b \arcsin (c x))}{3 e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {b c \int \left (\frac {e x \left (2 e^2 i x^2+3 e (e h-d i) x+6 \left (i d^2-e h d+e^2 g\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {6 \left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right )dx}{6 e^4}+\frac {x (a+b \arcsin (c x)) \left (d^2 i-d e h+e^2 g\right )}{e^3}+\frac {\log (d+e x) (a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}+\frac {x^2 (e h-d i) (a+b \arcsin (c x))}{2 e^2}+\frac {i x^3 (a+b \arcsin (c x))}{3 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x (a+b \arcsin (c x)) \left (d^2 i-d e h+e^2 g\right )}{e^3}+\frac {\log (d+e x) (a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}+\frac {x^2 (e h-d i) (a+b \arcsin (c x))}{2 e^2}+\frac {i x^3 (a+b \arcsin (c x))}{3 e}-\frac {b c \left (\frac {3 e^2 \arcsin (c x) (e h-d i)}{2 c^3}+\frac {6 i \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{c}+\frac {6 i \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{c}-\frac {6 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{c}-\frac {6 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{c}+\frac {3 i \arcsin (c x)^2 \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{c}+\frac {6 \arcsin (c x) \log (d+e x) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{c}+\frac {2 e \sqrt {1-c^2 x^2} \left (-e^2 \left (\frac {2 i}{c^2}+9 g\right )-9 d^2 i+9 d e h\right )}{3 c^2}-\frac {3 e^2 x \sqrt {1-c^2 x^2} (e h-d i)}{2 c^2}-\frac {2 e^3 i x^2 \sqrt {1-c^2 x^2}}{3 c^2}\right )}{6 e^4}\) |
((e^2*g - d*e*h + d^2*i)*x*(a + b*ArcSin[c*x]))/e^3 + ((e*h - d*i)*x^2*(a + b*ArcSin[c*x]))/(2*e^2) + (i*x^3*(a + b*ArcSin[c*x]))/(3*e) + ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 - (b*c*(( 2*e*(9*d*e*h - 9*d^2*i - e^2*(9*g + (2*i)/c^2))*Sqrt[1 - c^2*x^2])/(3*c^2) - (3*e^2*(e*h - d*i)*x*Sqrt[1 - c^2*x^2])/(2*c^2) - (2*e^3*i*x^2*Sqrt[1 - c^2*x^2])/(3*c^2) + (3*e^2*(e*h - d*i)*ArcSin[c*x])/(2*c^3) + ((3*I)*(e^3 *f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x]^2)/c - (6*(e^3*f - d*e^2*g + d ^2*e*h - d^3*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^ 2*d^2 - e^2])])/c - (6*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x]*Log [1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/c + (6*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x]*Log[d + e*x])/c + ((6*I)*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[ c^2*d^2 - e^2])])/c + ((6*I)*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*PolyLog[2 , (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/c))/(6*e^4)
3.2.9.3.1 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_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_ Symbol] :> With[{u = IntHide[Px*(d + e*x)^m, 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, m}, x] && PolynomialQ[Px, x]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3400 vs. \(2 (621 ) = 1242\).
Time = 2.56 (sec) , antiderivative size = 3401, normalized size of antiderivative = 5.46
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(3401\) |
| default | \(\text {Expression too large to display}\) | \(3401\) |
| parts | \(\text {Expression too large to display}\) | \(3403\) |
1/c*(a/c^2*(1/e^3*(c^3*d^2*i*x-c^3*d*e*h*x+c^3*e^2*g*x-1/2*c^3*d*e*i*x^2+1 /2*c^3*e^2*h*x^2+1/3*i*c^3*x^3*e^2)-c^3*(d^3*i-d^2*e*h+d*e^2*g-e^3*f)/e^4* ln(c*e*x+c*d))-1/4*b/c/e*arcsin(c*x)*cos(2*arcsin(c*x))*h-1/12*b/c^2*i*arc sin(c*x)/e*sin(3*arcsin(c*x))-1/2*I*b*c*arcsin(c*x)^2/e*f+b*arcsin(c*x)/e* g*c*x-I*b*c*d*g/(c^2*d^2-e^2)*dilog((-I*d*c-(I*c*x+(-c^2*x^2+1)^(1/2))*e+( -c^2*d^2+e^2)^(1/2))/(-I*d*c+(-c^2*d^2+e^2)^(1/2)))-1/2*I*b*c*arcsin(c*x)^ 2/e^3*d^2*h+1/2*I*b*c*arcsin(c*x)^2/e^4*d^3*i+1/2*I*b*c*arcsin(c*x)^2/e^2* d*g-I*b*c*d*g/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^ 2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-1/8*b/c/e^2*sin(2*arcsin(c *x))*d*i+1/4*b/c^2/e*(-c^2*x^2+1)^(1/2)*i+b/e^3*(-c^2*x^2+1)^(1/2)*d^2*i-b /e^2*(-c^2*x^2+1)^(1/2)*d*h-1/36*b/c^2*i/e*cos(3*arcsin(c*x))+1/8*b/c/e*si n(2*arcsin(c*x))*h+1/4*b/c/e*arcsin(c*x)*i*x+b*arcsin(c*x)/e^3*d^2*i*c*x-b *arcsin(c*x)/e^2*d*h*c*x-b*c*e*f*arcsin(c*x)/(c^2*d^2-e^2)*ln((-I*d*c-(I*c *x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(-I*d*c+(-c^2*d^2+e^2)^(1/2 )))+b*c*d*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2)) *e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-b*c*e*f*arcsin(c*x) /(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2) )/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+I*b*c*e*f/(c^2*d^2-e^2)*dilog((-I*d*c-(I*c *x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(-I*d*c+(-c^2*d^2+e^2)^(1/2 )))+I*b*c*e*f/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(...
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{d+e x} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{e x + d} \,d x } \]
integral((a*i*x^3 + a*h*x^2 + a*g*x + a*f + (b*i*x^3 + b*h*x^2 + b*g*x + b *f)*arcsin(c*x))/(e*x + d), x)
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{d+e x} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2} + i x^{3}\right )}{d + e x}\, dx \]
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{d+e x} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{e x + d} \,d x } \]
a*g*(x/e - d*log(e*x + d)/e^2) - 1/6*a*i*(6*d^3*log(e*x + d)/e^4 - (2*e^2* x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3) + 1/2*a*h*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) + a*f*log(e*x + d)/e + integrate((b*i*x^3 + b*h*x^2 + b*g*x + b*f)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(e*x + d), x)
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{d+e x} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{e x + d} \,d x } \]
Timed out. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{d+e x} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (i\,x^3+h\,x^2+g\,x+f\right )}{d+e\,x} \,d x \]