Integrand size = 31, antiderivative size = 1016 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {b i \sqrt {1-c^2 x^2}}{c e^3}+\frac {5 b c d^3 i \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c d^2 (3 e h+4 d i) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c d \left (e^2 g+4 d e h-4 d^2 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c \left (e^3 f-2 d e^2 g+2 d^3 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b (e h-3 d i) \arcsin (c x)^2}{2 e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {5 b c^3 d^4 i \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c d^2 \left (3 c^2 d h+4 e i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c d \left (4 e^2 (e h-2 d i)+c^2 \left (d e^2 g+4 d^3 i\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c \left (2 e^4 g-6 d^2 e^2 i-c^2 \left (d e^3 f-4 d^4 i\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b (e h-3 d i) \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 (e h-3 d i) \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 (e h-3 d i) \arcsin (c x) \log (d+e x)}{e^4}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b (e h-3 d i) \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 (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \]
-1/2*I*b*(-3*d*i+e*h)*arcsin(c*x)^2/e^4+i*x*(a+b*arcsin(c*x))/e^3-1/2*(-d^ 3*i+d^2*e*h-d*e^2*g+e^3*f)*(a+b*arcsin(c*x))/e^4/(e*x+d)^2-(3*d^2*i-2*d*e* h+e^2*g)*(a+b*arcsin(c*x))/e^4/(e*x+d)+5/2*b*c^3*d^4*i*arctan((c^2*d*x+e)/ (c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^4/(c^2*d^2-e^2)^(3/2)-1/2*b*c*d^ 2*(3*c^2*d*h+4*e*i)*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1 /2))/e^3/(c^2*d^2-e^2)^(3/2)+1/2*b*c*d*(4*e^2*(-2*d*i+e*h)+c^2*(4*d^3*i+d* e^2*g))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^4/(c^ 2*d^2-e^2)^(3/2)-1/2*b*c*(2*e^4*g-6*d^2*e^2*i-c^2*(-4*d^4*i+d*e^3*f))*arct an((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^4/(c^2*d^2-e^2)^( 3/2)-b*(-3*d*i+e*h)*arcsin(c*x)*ln(e*x+d)/e^4+(-3*d*i+e*h)*(a+b*arcsin(c*x ))*ln(e*x+d)/e^4+b*(-3*d*i+e*h)*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+b*(-3*d*i+e*h)*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*(-3*d*i+e*h)* 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-I* b*(-3*d*i+e*h)*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+b*i*(-c^2*x^2+1)^(1/2)/c/e^3+5/2*b*c*d^3*i*(-c^2*x^2+1)^(1/2) /e^3/(c^2*d^2-e^2)/(e*x+d)-1/2*b*c*d^2*(4*d*i+3*e*h)*(-c^2*x^2+1)^(1/2)/e^ 3/(c^2*d^2-e^2)/(e*x+d)+1/2*b*c*d*(-4*d^2*i+4*d*e*h+e^2*g)*(-c^2*x^2+1)^(1 /2)/e^3/(c^2*d^2-e^2)/(e*x+d)+1/2*b*c*(2*d^3*i-2*d*e^2*g+e^3*f)*(-c^2*x^2+ 1)^(1/2)/e^3/(c^2*d^2-e^2)/(e*x+d)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.06 (sec) , antiderivative size = 1556, normalized size of antiderivative = 1.53 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {a i x}{e^3}+\frac {-a e^3 f+a d e^2 g-a d^2 e h+a d^3 i}{2 e^4 (d+e x)^2}+\frac {-a e^2 g+2 a d e h-3 a d^2 i}{e^4 (d+e x)}+b f \left (-\frac {c \sqrt {1+\frac {-d-\sqrt {\frac {1}{c^2}} e}{d+e x}} \sqrt {1+\frac {-d+\sqrt {\frac {1}{c^2}} e}{d+e x}} \operatorname {AppellF1}\left (2,\frac {1}{2},\frac {1}{2},3,-\frac {-d+\sqrt {\frac {1}{c^2}} e}{d+e x},-\frac {-d-\sqrt {\frac {1}{c^2}} e}{d+e x}\right )}{4 e^2 (d+e x) \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{2 e (d+e x)^2}\right )+\frac {(a e h-3 a d i) \log (d+e x)}{e^4}+b g \left (\frac {-\frac {\arcsin (c x)}{d+e x}+\frac {c \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}}{e^2}-\frac {d \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\arcsin (c x)}{e (d+e x)^2}-\frac {i c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right )}{2 e}\right )+b i \left (\frac {\sqrt {1-c^2 x^2}+c x \arcsin (c x)}{c e^3}+\frac {3 d^2 \left (-\frac {\arcsin (c x)}{d+e x}+\frac {c \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {d^3 \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\arcsin (c x)}{e (d+e x)^2}-\frac {i c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right )}{2 e^3}-\frac {3 d \left (-\frac {i \arcsin (c x)^2}{2 e}+\frac {\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}+\frac {\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}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}\right )}{e^3}\right )+b h \left (-\frac {2 d \left (-\frac {\arcsin (c x)}{d+e x}+\frac {c \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {d^2 \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\arcsin (c x)}{e (d+e x)^2}-\frac {i c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right )}{2 e^2}+\frac {-\frac {i \arcsin (c x)^2}{2 e}+\frac {\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}+\frac {\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}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}}{e^2}\right ) \]
(a*i*x)/e^3 + (-(a*e^3*f) + a*d*e^2*g - a*d^2*e*h + a*d^3*i)/(2*e^4*(d + e *x)^2) + (-(a*e^2*g) + 2*a*d*e*h - 3*a*d^2*i)/(e^4*(d + e*x)) + b*f*(-1/4* (c*Sqrt[1 + (-d - Sqrt[c^(-2)]*e)/(d + e*x)]*Sqrt[1 + (-d + Sqrt[c^(-2)]*e )/(d + e*x)]*AppellF1[2, 1/2, 1/2, 3, -((-d + Sqrt[c^(-2)]*e)/(d + e*x)), -((-d - Sqrt[c^(-2)]*e)/(d + e*x))])/(e^2*(d + e*x)*Sqrt[1 - c^2*x^2]) - A rcSin[c*x]/(2*e*(d + e*x)^2)) + ((a*e*h - 3*a*d*i)*Log[d + e*x])/e^4 + b*g *((-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2] *Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2])/e^2 - (d*((c*Sqrt[1 - c^2*x^2]) /((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*(Log [4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]* Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d ^2 - e^2])))/(2*e)) + b*i*((Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x])/(c*e^3) + (3*d^2*(-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2]))/e^4 - (d^3*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c ^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^ 2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*S qrt[c^2*d^2 - e^2])))/(2*e^3) - (3*d*(((-1/2*I)*ArcSin[c*x]^2)/e + (ArcSin [c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e + (A rcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]...
Time = 2.78 (sec) , antiderivative size = 965, normalized size of antiderivative = 0.95, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x)) \left (f+g x+h x^2+i x^3\right )}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 5252 |
| \(\displaystyle -b c \int -\frac {5 i d^3-e (3 h-4 i x) d^2+e^2 (g-4 x (h+i x)) d+e^3 \left (-2 i x^3+2 g x+f\right )-2 (e h-3 d i) (d+e x)^2 \log (d+e x)}{2 e^4 (d+e x)^2 \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4 (d+e x)}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{2 e^4 (d+e x)^2}+\frac {(e h-3 d i) \log (d+e x) (a+b \arcsin (c x))}{e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {5 i d^3-e (3 h-4 i x) d^2+e^2 (g-4 x (h+i x)) d+e^3 \left (-2 i x^3+2 g x+f\right )-2 (e h-3 d i) (d+e x)^2 \log (d+e x)}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 e^4}-\frac {(a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4 (d+e x)}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{2 e^4 (d+e x)^2}+\frac {(e h-3 d i) \log (d+e x) (a+b \arcsin (c x))}{e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {b c \int \left (\frac {5 i d^3}{(d+e x)^2 \sqrt {1-c^2 x^2}}+\frac {e (4 i x-3 h) d^2}{(d+e x)^2 \sqrt {1-c^2 x^2}}-\frac {e^2 \left (4 i x^2+4 h x-g\right ) d}{(d+e x)^2 \sqrt {1-c^2 x^2}}-\frac {e^3 \left (2 i x^3-2 g x-f\right )}{(d+e x)^2 \sqrt {1-c^2 x^2}}-\frac {2 (e h-3 d i) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right )dx}{2 e^4}-\frac {(a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4 (d+e x)}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{2 e^4 (d+e x)^2}+\frac {(e h-3 d i) \log (d+e x) (a+b \arcsin (c x))}{e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i x (a+b \arcsin (c x))}{e^3}+\frac {(e h-3 d i) \log (d+e x) (a+b \arcsin (c x))}{e^4}-\frac {\left (3 i d^2-2 e h d+e^2 g\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}-\frac {\left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}+\frac {b c \left (\frac {5 c^2 i \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^4}{\left (c^2 d^2-e^2\right )^{3/2}}+\frac {5 e i \sqrt {1-c^2 x^2} d^3}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {e \left (3 d h c^2+4 e i\right ) \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^2}{\left (c^2 d^2-e^2\right )^{3/2}}-\frac {e (3 e h+4 d i) \sqrt {1-c^2 x^2} d^2}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {\left (\left (4 i d^3+e^2 g d\right ) c^2+4 e^2 (e h-2 d i)\right ) \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d}{\left (c^2 d^2-e^2\right )^{3/2}}+\frac {e \left (-4 i d^2+4 e h d+e^2 g\right ) \sqrt {1-c^2 x^2} d}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i (e h-3 d i) \arcsin (c x)^2}{c}-\frac {\left (2 g e^4-6 d^2 i e^2-c^2 \left (d e^3 f-4 d^4 i\right )\right ) \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\left (c^2 d^2-e^2\right )^{3/2}}+\frac {2 (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}+\frac {2 (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {2 (e h-3 d i) \arcsin (c x) \log (d+e x)}{c}-\frac {2 i (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {2 i (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{c}+\frac {2 e i \sqrt {1-c^2 x^2}}{c^2}+\frac {e \left (2 i d^3-2 e^2 g d+e^3 f\right ) \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 e^4}\) |
(i*x*(a + b*ArcSin[c*x]))/e^3 - ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(2*e^4*(d + e*x)^2) - ((e^2*g - 2*d*e*h + 3*d^2*i)*(a + b* ArcSin[c*x]))/(e^4*(d + e*x)) + ((e*h - 3*d*i)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 + (b*c*((2*e*i*Sqrt[1 - c^2*x^2])/c^2 + (5*d^3*e*i*Sqrt[1 - c^2 *x^2])/((c^2*d^2 - e^2)*(d + e*x)) - (d^2*e*(3*e*h + 4*d*i)*Sqrt[1 - c^2*x ^2])/((c^2*d^2 - e^2)*(d + e*x)) + (d*e*(e^2*g + 4*d*e*h - 4*d^2*i)*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) + (e*(e^3*f - 2*d*e^2*g + 2*d^3*i )*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - (I*(e*h - 3*d*i)*ArcSin [c*x]^2)/c + (5*c^2*d^4*i*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3/2) - (d^2*e*(3*c^2*d*h + 4*e*i)*ArcTan[( e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3/ 2) + (d*(4*e^2*(e*h - 2*d*i) + c^2*(d*e^2*g + 4*d^3*i))*ArcTan[(e + c^2*d* x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3/2) - ((2*e ^4*g - 6*d^2*e^2*i - c^2*(d*e^3*f - 4*d^4*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c ^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3/2) + (2*(e*h - 3*d*i )*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] )/c + (2*(e*h - 3*d*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/c - (2*(e*h - 3*d*i)*ArcSin[c*x]*Log[d + e*x])/c - ((2*I)*(e*h - 3*d*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^ 2 - e^2])])/c - ((2*I)*(e*h - 3*d*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))...
3.2.11.3.1 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. 3609 vs. \(2 (979 ) = 1958\).
Time = 7.05 (sec) , antiderivative size = 3610, normalized size of antiderivative = 3.55
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(3610\) |
| default | \(\text {Expression too large to display}\) | \(3610\) |
| parts | \(\text {Expression too large to display}\) | \(3617\) |
1/c*(a*(i/e^3*c*x+1/2*c^3*(d^3*i-d^2*e*h+d*e^2*g-e^3*f)/e^4/(c*e*x+c*d)^2- c^2/e^4*(3*d^2*i-2*d*e*h+e^2*g)/(c*e*x+c*d)-c/e^4*(3*d*i-e*h)*ln(c*e*x+c*d ))+b*(2*I/(c^2*d^2-e^2)^(3/2)*c^2*g*arctanh(1/2*(2*I*e*(I*c*x+(-c^2*x^2+1) ^(1/2))-2*d*c)/(c^2*d^2-e^2)^(1/2))-1/2*c^2*(-(-c^2*x^2+1)^(1/2)*c^2*d*e^4 *f-e^4*c*g*arcsin(c*x)*d+3*e^3*c*d^2*h*arcsin(c*x)-5*e^2*c*d^3*i*arcsin(c* x)-3*e*c^3*d^4*h*arcsin(c*x)+e^2*c^3*d^3*g*arcsin(c*x)+e^3*c^3*d^2*f*arcsi n(c*x)-2*arcsin(c*x)*e^5*g*c*x-I*c^3*d^3*e^2*g+I*c^3*d^4*e*h+I*c^3*d^2*e^3 *f+(-c^2*x^2+1)^(1/2)*c^2*d^4*e*i-(-c^2*x^2+1)^(1/2)*c^2*d^3*e^2*h+(-c^2*x ^2+1)^(1/2)*c^2*d^2*e^3*g+I*c^3*d^2*e^3*h*x^2+(-c^2*x^2+1)^(1/2)*c^2*d^3*e ^2*i*x-(-c^2*x^2+1)^(1/2)*c^2*d^2*e^3*h*x+(-c^2*x^2+1)^(1/2)*c^2*d*e^4*g*x +I*c^3*e^5*f*x^2-(-c^2*x^2+1)^(1/2)*c^2*e^5*f*x-6*arcsin(c*x)*d^2*e^3*i*c* x+4*arcsin(c*x)*d*e^4*h*c*x-I*c^3*d^5*i-e^5*c*f*arcsin(c*x)+5*c^3*d^5*i*ar csin(c*x)+2*I*c^3*d^3*e^2*h*x-2*I*c^3*d^2*e^3*g*x+2*I*c^3*d*e^4*f*x-I*c^3* d^3*e^2*i*x^2-I*c^3*d*e^4*g*x^2-2*I*c^3*d^4*e*i*x+6*arcsin(c*x)*c^3*d^4*e* i*x-4*arcsin(c*x)*c^3*d^3*e^2*h*x+2*arcsin(c*x)*c^3*d^2*e^3*g*x)/(c^2*d^2- e^2)/(c*e*x+c*d)^2/e^4-I/e^3/(c^2*d^2-e^2)^2*c^5*h*d^4*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/e^3/(c^2*d^2-e^2)*c^3*h*d^2*arcsin(c*x)^2-I/e^3/(c^2*d^2-e^2)^2*c^5*h *d^4*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)))+2*I/e/(c^2*d^2-e^2)^2*c^3*h*dilog((I*d*c+(I*c*...
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{3}} \,d x } \]
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^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2} + i x^{3}\right )}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((e-c*d)*(e+c*d)>0)', see `assume ?` for mor
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (i\,x^3+h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^3} \,d x \]