Integrand size = 31, antiderivative size = 1278 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {11 b c d^3 i \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (2 e h+27 d i) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d \left (e^2 g-6 d e h-18 d^2 i\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {11 b c^3 d^4 i \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (18 e^2 i+c^2 d (2 e h+9 d i)\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (e h+6 d i)-c^2 d \left (e^2 g-2 d e h+6 d^2 i\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {i b i \arcsin (c x)^2}{2 e^4}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{3 e^4 (d+e x)^3}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {(e h-3 d i) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}-\frac {11 b c^3 d^3 \left (2 c^2 d^2+e^2\right ) i \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e^4 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c^3 d^2 \left (4 c^2 d^2 h+e (2 e h+81 d i)\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c d \left (2 c^4 d^2 g-36 e^2 i+c^2 \left (e^2 g-18 d e h-18 d^2 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 )}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b 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 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 i \arcsin (c x) \log (d+e x)}{e^4}+\frac {i (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b 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 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*i*arcsin(c*x)^2/e^4-1/3*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*(a+b*arcsi n(c*x))/e^4/(e*x+d)^3-1/2*(3*d^2*i-2*d*e*h+e^2*g)*(a+b*arcsin(c*x))/e^4/(e *x+d)^2-(-3*d*i+e*h)*(a+b*arcsin(c*x))/e^4/(e*x+d)+1/12*b*c*(4*c^4*d^2*f+1 2*e^2*h+c^2*(6*d^2*h-9*d*e*g+2*e^2*f))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1 /2)/(-c^2*x^2+1)^(1/2))/e/(c^2*d^2-e^2)^(5/2)-11/12*b*c^3*d^3*(2*c^2*d^2+e ^2)*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)^(5/2)+1/12*b*c^3*d^2*(4*c^2*d^2*h+e*(81*d*i+2*e*h))*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)^(5/2)+1/12 *b*c*d*(2*c^4*d^2*g-36*e^2*i+c^2*(-18*d^2*i-18*d*e*h+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^2/(c^2*d^2-e^2)^(5/2)-b*i* arcsin(c*x)*ln(e*x+d)/e^4+i*(a+b*arcsin(c*x))*ln(e*x+d)/e^4+b*i*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*i*a rcsin(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*i*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*i*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+1/12*b*c*(6*d^2*h-3*d*e*g+2*e^2*f)*(-c^2*x^2+1)^(1/2)/e^2/(c^2 *d^2-e^2)/(e*x+d)^2-11/12*b*c*d^3*i*(-c^2*x^2+1)^(1/2)/e^3/(c^2*d^2-e^2)/( e*x+d)^2+1/12*b*c*d^2*(27*d*i+2*e*h)*(-c^2*x^2+1)^(1/2)/e^3/(c^2*d^2-e^2)/ (e*x+d)^2+1/12*b*c*d*(-18*d^2*i-6*d*e*h+e^2*g)*(-c^2*x^2+1)^(1/2)/e^3/(c^2 *d^2-e^2)/(e*x+d)^2-1/4*b*c*(2*e^2*(-4*d*h+e*g)-c^2*d*(-2*d^2*h-d*e*g+2...
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 8.96 (sec) , antiderivative size = 1921, normalized size of antiderivative = 1.50 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^4} \, dx =\text {Too large to display} \]
(-(a*e^3*f) + a*d*e^2*g - a*d^2*e*h + a*d^3*i)/(3*e^4*(d + e*x)^3) + (-(a* e^2*g) + 2*a*d*e*h - 3*a*d^2*i)/(2*e^4*(d + e*x)^2) + (-(a*e*h) + 3*a*d*i) /(e^4*(d + e*x)) + b*f*(-1/9*(c*Sqrt[1 + (-d - Sqrt[c^(-2)]*e)/(d + e*x)]* Sqrt[1 + (-d + Sqrt[c^(-2)]*e)/(d + e*x)]*AppellF1[3, 1/2, 1/2, 4, -((-d + Sqrt[c^(-2)]*e)/(d + e*x)), -((-d - Sqrt[c^(-2)]*e)/(d + e*x))])/(e^2*(d + e*x)^2*Sqrt[1 - c^2*x^2]) - ArcSin[c*x]/(3*e*(d + e*x)^3)) + (a*i*Log[d + e*x])/e^4 + b*h*((-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sq rt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2])/e^3 - (d*((c*S qrt[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 + Sqr t[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])))/e^2 + (d^2*((Sqrt[1 - c^2*x^2]*(-(c*e^2) + c ^3*d*(4*d + 3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcSin[c*x])/ (e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)*Log[d + e*x])/(e*(-(c*d) + e)^2*( c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (c^3*(2*c^2*d^2 + e^2)*Log[e + c^2*d* x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e) ^2*Sqrt[-(c^2*d^2) + e^2])))/(6*e^2)) + b*g*(((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] + L og[(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 -...
Time = 2.86 (sec) , antiderivative size = 1244, normalized size of antiderivative = 0.97, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x)) \left (f+g x+h x^2+i x^3\right )}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 5252 |
| \(\displaystyle -b c \int \frac {11 i d^3-e (2 h-27 i x) d^2-e^2 (g+6 x (h-3 i x)) d-e^3 (2 f+3 x (g+2 h x))+6 i (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3 \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{2 e^4 (d+e x)^2}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{3 e^4 (d+e x)^3}-\frac {(e h-3 d i) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {i \log (d+e x) (a+b \arcsin (c x))}{e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \int \frac {11 i d^3-e (2 h-27 i x) d^2-e^2 (g+6 x (h-3 i x)) d-e^3 (2 f+3 x (g+2 h x))+6 i (d+e x)^3 \log (d+e x)}{(d+e x)^3 \sqrt {1-c^2 x^2}}dx}{6 e^4}-\frac {(a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{2 e^4 (d+e x)^2}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{3 e^4 (d+e x)^3}-\frac {(e h-3 d i) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {i \log (d+e x) (a+b \arcsin (c x))}{e^4}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {b c \int \left (\frac {11 i d^3}{(d+e x)^3 \sqrt {1-c^2 x^2}}+\frac {e (27 i x-2 h) d^2}{(d+e x)^3 \sqrt {1-c^2 x^2}}+\frac {e^2 \left (18 i x^2-6 h x-g\right ) d}{(d+e x)^3 \sqrt {1-c^2 x^2}}-\frac {e^3 \left (6 h x^2+3 g x+2 f\right )}{(d+e x)^3 \sqrt {1-c^2 x^2}}+\frac {6 i \log (d+e x)}{\sqrt {1-c^2 x^2}}\right )dx}{6 e^4}-\frac {(a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{2 e^4 (d+e x)^2}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{3 e^4 (d+e x)^3}-\frac {(e h-3 d i) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {i \log (d+e x) (a+b \arcsin (c x))}{e^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i \log (d+e x) (a+b \arcsin (c x))}{e^4}-\frac {(e h-3 d i) (a+b \arcsin (c x))}{e^4 (d+e x)}-\frac {\left (3 i d^2-2 e h d+e^2 g\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) (a+b \arcsin (c x))}{3 e^4 (d+e x)^3}-\frac {b c \left (\frac {33 c^2 e i \sqrt {1-c^2 x^2} d^4}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {11 c^2 \left (2 c^2 d^2+e^2\right ) i \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^3}{2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {11 e i \sqrt {1-c^2 x^2} d^3}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^2 e \left (4 c^2 h d^2+e (2 e h+81 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^2}{2 \left (c^2 d^2-e^2\right )^{5/2}}-\frac {3 e \left (d (2 e h+9 d i) c^2+18 e^2 i\right ) \sqrt {1-c^2 x^2} d^2}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {e (2 e h+27 d i) \sqrt {1-c^2 x^2} d^2}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {e^2 \left (2 d^2 g c^4+\left (-18 i d^2-18 e h d+e^2 g\right ) c^2-36 e^2 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 )^{5/2}}+\frac {3 e \left (4 e^2 (e h+6 d i)-c^2 d \left (6 i d^2-2 e h d+e^2 g\right )\right ) \sqrt {1-c^2 x^2} d}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {e \left (-18 i d^2-6 e h d+e^2 g\right ) \sqrt {1-c^2 x^2} d}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {3 i i \arcsin (c x)^2}{c}-\frac {e^3 \left (4 d^2 f c^4+\left (6 h d^2-9 e g d+2 e^2 f\right ) c^2+12 e^2 h\right ) \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 \left (c^2 d^2-e^2\right )^{5/2}}-\frac {6 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 {6 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 {6 i \arcsin (c x) \log (d+e x)}{c}+\frac {6 i 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 {6 i 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 {3 e^2 \left (2 e^2 (e g-4 d h)-c^2 d \left (-2 h d^2-e g d+2 e^2 f\right )\right ) \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {e^2 \left (6 h d^2-3 e g d+2 e^2 f\right ) \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}\right )}{6 e^4}\) |
-1/3*((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(e^4*(d + e *x)^3) - ((e^2*g - 2*d*e*h + 3*d^2*i)*(a + b*ArcSin[c*x]))/(2*e^4*(d + e*x )^2) - ((e*h - 3*d*i)*(a + b*ArcSin[c*x]))/(e^4*(d + e*x)) + (i*(a + b*Arc Sin[c*x])*Log[d + e*x])/e^4 - (b*c*(-1/2*(e^2*(2*e^2*f - 3*d*e*g + 6*d^2*h )*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)^2) + (11*d^3*e*i*Sqrt[1 - c^2*x^2])/(2*(c^2*d^2 - e^2)*(d + e*x)^2) - (d^2*e*(2*e*h + 27*d*i)*Sqrt[1 - c^2*x^2])/(2*(c^2*d^2 - e^2)*(d + e*x)^2) - (d*e*(e^2*g - 6*d*e*h - 18* d^2*i)*Sqrt[1 - c^2*x^2])/(2*(c^2*d^2 - e^2)*(d + e*x)^2) + (3*e^2*(2*e^2* (e*g - 4*d*h) - c^2*d*(2*e^2*f - d*e*g - 2*d^2*h))*Sqrt[1 - c^2*x^2])/(2*( c^2*d^2 - e^2)^2*(d + e*x)) + (33*c^2*d^4*e*i*Sqrt[1 - c^2*x^2])/(2*(c^2*d ^2 - e^2)^2*(d + e*x)) - (3*d^2*e*(18*e^2*i + c^2*d*(2*e*h + 9*d*i))*Sqrt[ 1 - c^2*x^2])/(2*(c^2*d^2 - e^2)^2*(d + e*x)) + (3*d*e*(4*e^2*(e*h + 6*d*i ) - c^2*d*(e^2*g - 2*d*e*h + 6*d^2*i))*Sqrt[1 - c^2*x^2])/(2*(c^2*d^2 - e^ 2)^2*(d + e*x)) + ((3*I)*i*ArcSin[c*x]^2)/c - (e^3*(4*c^4*d^2*f + 12*e^2*h + c^2*(2*e^2*f - 9*d*e*g + 6*d^2*h))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*(c^2*d^2 - e^2)^(5/2)) + (11*c^2*d^3*(2*c^2* d^2 + e^2)*i*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])] )/(2*(c^2*d^2 - e^2)^(5/2)) - (c^2*d^2*e*(4*c^2*d^2*h + e*(2*e*h + 81*d*i) )*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*(c^2*d ^2 - e^2)^(5/2)) - (d*e^2*(2*c^4*d^2*g - 36*e^2*i + c^2*(e^2*g - 18*d*e...
3.2.12.3.1 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. 3756 vs. \(2 (1225 ) = 2450\).
Time = 11.16 (sec) , antiderivative size = 3757, normalized size of antiderivative = 2.94
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(3757\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3777\) |
| default | \(\text {Expression too large to display}\) | \(3777\) |
a*(i/e^4*ln(e*x+d)-(-3*d*i+e*h)/e^4/(e*x+d)-1/3*(-d^3*i+d^2*e*h-d*e^2*g+e^ 3*f)/e^4/(e*x+d)^3-1/2*(3*d^2*i-2*d*e*h+e^2*g)/e^4/(e*x+d)^2)+b/c*(-I/e^4/ (c^2*d^2-e^2)^3*c^7*i*d^6*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^2/(c^2*d^2-e^2)^2*c^3* i*d^2*arcsin(c*x)^2-I/e^4/(c^2*d^2-e^2)^2*c^5*i*d^4*arcsin(c*x)^2+3*I/e^2/ (c^2*d^2-e^2)^3*c^5*i*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)))+1/e^4/(c^2*d^2-e^2)^3*c^7*i* d^6*arcsin(c*x)*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)))+1/e^4/(c^2*d^2-e^2)^3*c^7*i*d^6*arcsin(c* x)*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)))-3/e^2/(c^2*d^2-e^2)^3*c^5*i*d^4*arcsin(c*x)*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)))-3/e^2/(c^2*d^2-e^2)^3*c^5*i*d^4*arcsin(c*x)*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)))+3*I/e^ 2/(c^2*d^2-e^2)^3*c^5*i*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)))+3/(c^2*d^2-e^2)^3*c^3*i*ar csin(c*x)*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)))*d^2+I*e^2/(c^2*d^2-e^2)^3*c*i*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^2/(c^2*d^2-e^2)^3*c*i*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-...
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{4}} \,d x } \]
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^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4) , x)
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2} + i x^{3}\right )}{\left (d + e x\right )^{4}}\, dx \]
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{4}} \,d x } \]
1/6*a*i*((18*d*e^2*x^2 + 27*d^2*e*x + 11*d^3)/(e^7*x^3 + 3*d*e^6*x^2 + 3*d ^2*e^5*x + d^3*e^4) + 6*log(e*x + d)/e^4) - 1/6*(3*e*x + d)*a*g/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2) - 1/3*(3*e^2*x^2 + 3*d*e*x + d^2)*a* h/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3) - 1/3*a*f/(e^4*x^3 + 3*d *e^3*x^2 + 3*d^2*e^2*x + d^3*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^4*x^4 + 4*d*e^3*x^3 + 6* d^2*e^2*x^2 + 4*d^3*e*x + d^4), x)
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^4} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (i\,x^3+h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^4} \,d x \]