Integrand size = 23, antiderivative size = 249 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=-\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \arcsin (c+d x))}+\frac {e^3 \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{2 b^3 d}-\frac {e^3 \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {4 a}{b}\right )}{b^3 d}-\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b^3 d}+\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^3 d} \]
-3/2*e^3*(d*x+c)^2/b^2/d/(a+b*arcsin(d*x+c))+2*e^3*(d*x+c)^4/b^2/d/(a+b*ar csin(d*x+c))-1/2*e^3*cos(2*a/b)*Si(2*(a+b*arcsin(d*x+c))/b)/b^3/d+e^3*cos( 4*a/b)*Si(4*(a+b*arcsin(d*x+c))/b)/b^3/d+1/2*e^3*Ci(2*(a+b*arcsin(d*x+c))/ b)*sin(2*a/b)/b^3/d-e^3*Ci(4*(a+b*arcsin(d*x+c))/b)*sin(4*a/b)/b^3/d-1/2*e ^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^2
Time = 0.89 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.73 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=\frac {e^3 \left (-\frac {b^2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{(a+b \arcsin (c+d x))^2}+\frac {b \left (-3 (c+d x)^2+4 (c+d x)^4\right )}{a+b \arcsin (c+d x)}+\operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {2 a}{b}\right )-2 \operatorname {CosIntegral}\left (4 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {4 a}{b}\right )-\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+2 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )}{2 b^3 d} \]
(e^3*(-((b^2*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^2) + (b*(-3*(c + d*x)^2 + 4*(c + d*x)^4))/(a + b*ArcSin[c + d*x]) + CosInteg ral[2*(a/b + ArcSin[c + d*x])]*Sin[(2*a)/b] - 2*CosIntegral[4*(a/b + ArcSi n[c + d*x])]*Sin[(4*a)/b] - Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d *x])] + 2*Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c + d*x])]))/(2*b^3*d)
Time = 1.31 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {5304, 27, 5144, 5222, 5146, 25, 4906, 27, 2009, 3042, 3784, 25, 3042, 3780, 3783}
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.
| \(\displaystyle \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {e^3 (c+d x)^3}{(a+b \arcsin (c+d x))^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int \frac {(c+d x)^3}{(a+b \arcsin (c+d x))^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5144 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \int \frac {(c+d x)^2}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}d(c+d x)}{2 b}-\frac {2 \int \frac {(c+d x)^4}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}d(c+d x)}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 5222 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (\frac {2 \int \frac {c+d x}{a+b \arcsin (c+d x)}d(c+d x)}{b}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {2 \left (\frac {4 \int \frac {(c+d x)^3}{a+b \arcsin (c+d x)}d(c+d x)}{b}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 5146 |
| \(\displaystyle \frac {e^3 \left (-\frac {2 \left (\frac {4 \int -\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}+\frac {3 \left (\frac {2 \int -\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^3 \left (-\frac {2 \left (-\frac {4 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}+\frac {3 \left (-\frac {2 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {2 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 (a+b \arcsin (c+d x))}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {2 \left (-\frac {4 \int \left (\frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{4 (a+b \arcsin (c+d x))}-\frac {\sin \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{8 (a+b \arcsin (c+d x))}\right )d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {2 \left (-\frac {4 \int \left (\frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{4 (a+b \arcsin (c+d x))}-\frac {\sin \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{8 (a+b \arcsin (c+d x))}\right )d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {2 \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {2 \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (\frac {-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))-\cos \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {2 \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (\frac {\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {2 \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (\frac {\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {2 \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {2 \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {2 \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
(e^3*(-1/2*((c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(b*(a + b*ArcSin[c + d*x])^ 2) + (3*(-((c + d*x)^2/(b*(a + b*ArcSin[c + d*x]))) + (-(CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b]*Sin[(2*a)/b]) + Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/b^2))/(2*b) - (2*(-((c + d*x)^4/(b*(a + b*ArcSin[ c + d*x]))) + (4*(-1/4*(CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b]*Sin[(2* a)/b]) + (CosIntegral[(4*(a + b*ArcSin[c + d*x]))/b]*Sin[(4*a)/b])/8 + (Co s[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/4 - (Cos[(4*a)/b]*S inIntegral[(4*(a + b*ArcSin[c + d*x]))/b])/8))/b^2))/b))/d
3.3.28.3.1 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[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Sim p[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n + 1)/Sqrt [1 - c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*ArcSi n[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[ m, 0] && LtQ[n, -2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[1 /(b*c^(m + 1)) Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^ 2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] - Simp[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b* ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2* d + e, 0] && LtQ[n, -1]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(506\) vs. \(2(239)=478\).
Time = 0.34 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.04
| method | result | size |
| derivativedivides | \(-\frac {e^{3} \left (16 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) b^{2}-16 \arcsin \left (d x +c \right )^{2} \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) b^{2}+8 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) b^{2}-8 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) b^{2}+32 \arcsin \left (d x +c \right ) \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a b -32 \arcsin \left (d x +c \right ) \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a b +16 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a b -16 \arcsin \left (d x +c \right ) \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a b -4 \arcsin \left (d x +c \right ) \cos \left (4 \arcsin \left (d x +c \right )\right ) b^{2}+4 \arcsin \left (d x +c \right ) \cos \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+16 \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a^{2}-16 \,\operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a^{2}+8 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a^{2}-8 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a^{2}-4 \cos \left (4 \arcsin \left (d x +c \right )\right ) a b +2 \sin \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+4 \cos \left (2 \arcsin \left (d x +c \right )\right ) a b -\sin \left (4 \arcsin \left (d x +c \right )\right ) b^{2}\right )}{16 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\) | \(507\) |
| default | \(-\frac {e^{3} \left (16 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) b^{2}-16 \arcsin \left (d x +c \right )^{2} \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) b^{2}+8 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) b^{2}-8 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) b^{2}+32 \arcsin \left (d x +c \right ) \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a b -32 \arcsin \left (d x +c \right ) \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a b +16 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a b -16 \arcsin \left (d x +c \right ) \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a b -4 \arcsin \left (d x +c \right ) \cos \left (4 \arcsin \left (d x +c \right )\right ) b^{2}+4 \arcsin \left (d x +c \right ) \cos \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+16 \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a^{2}-16 \,\operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a^{2}+8 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a^{2}-8 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a^{2}-4 \cos \left (4 \arcsin \left (d x +c \right )\right ) a b +2 \sin \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+4 \cos \left (2 \arcsin \left (d x +c \right )\right ) a b -\sin \left (4 \arcsin \left (d x +c \right )\right ) b^{2}\right )}{16 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\) | \(507\) |
-1/16/d*e^3*(16*arcsin(d*x+c)^2*sin(4*a/b)*Ci(4*arcsin(d*x+c)+4*a/b)*b^2-1 6*arcsin(d*x+c)^2*Si(4*arcsin(d*x+c)+4*a/b)*cos(4*a/b)*b^2+8*arcsin(d*x+c) ^2*cos(2*a/b)*Si(2*arcsin(d*x+c)+2*a/b)*b^2-8*arcsin(d*x+c)^2*sin(2*a/b)*C i(2*arcsin(d*x+c)+2*a/b)*b^2+32*arcsin(d*x+c)*sin(4*a/b)*Ci(4*arcsin(d*x+c )+4*a/b)*a*b-32*arcsin(d*x+c)*Si(4*arcsin(d*x+c)+4*a/b)*cos(4*a/b)*a*b+16* arcsin(d*x+c)*cos(2*a/b)*Si(2*arcsin(d*x+c)+2*a/b)*a*b-16*arcsin(d*x+c)*si n(2*a/b)*Ci(2*arcsin(d*x+c)+2*a/b)*a*b-4*arcsin(d*x+c)*cos(4*arcsin(d*x+c) )*b^2+4*arcsin(d*x+c)*cos(2*arcsin(d*x+c))*b^2+16*sin(4*a/b)*Ci(4*arcsin(d *x+c)+4*a/b)*a^2-16*Si(4*arcsin(d*x+c)+4*a/b)*cos(4*a/b)*a^2+8*cos(2*a/b)* Si(2*arcsin(d*x+c)+2*a/b)*a^2-8*sin(2*a/b)*Ci(2*arcsin(d*x+c)+2*a/b)*a^2-4 *cos(4*arcsin(d*x+c))*a*b+2*sin(2*arcsin(d*x+c))*b^2+4*cos(2*arcsin(d*x+c) )*a*b-sin(4*arcsin(d*x+c))*b^2)/(a+b*arcsin(d*x+c))^2/b^3
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
integral((d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3)/(b^3*ar csin(d*x + c)^3 + 3*a*b^2*arcsin(d*x + c)^2 + 3*a^2*b*arcsin(d*x + c) + a^ 3), x)
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=e^{3} \left (\int \frac {c^{3}}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx\right ) \]
e**3*(Integral(c**3/(a**3 + 3*a**2*b*asin(c + d*x) + 3*a*b**2*asin(c + d*x )**2 + b**3*asin(c + d*x)**3), x) + Integral(d**3*x**3/(a**3 + 3*a**2*b*as in(c + d*x) + 3*a*b**2*asin(c + d*x)**2 + b**3*asin(c + d*x)**3), x) + Int egral(3*c*d**2*x**2/(a**3 + 3*a**2*b*asin(c + d*x) + 3*a*b**2*asin(c + d*x )**2 + b**3*asin(c + d*x)**3), x) + Integral(3*c**2*d*x/(a**3 + 3*a**2*b*a sin(c + d*x) + 3*a*b**2*asin(c + d*x)**2 + b**3*asin(c + d*x)**3), x))
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
1/2*(4*a*d^4*e^3*x^4 + 16*a*c*d^3*e^3*x^3 + 3*(8*a*c^2 - a)*d^2*e^3*x^2 + 2*(8*a*c^3 - 3*a*c)*d*e^3*x + (4*a*c^4 - 3*a*c^2)*e^3 - (b*d^3*e^3*x^3 + 3 *b*c*d^2*e^3*x^2 + 3*b*c^2*d*e^3*x + b*c^3*e^3)*sqrt(d*x + c + 1)*sqrt(-d* x - c + 1) + (4*b*d^4*e^3*x^4 + 16*b*c*d^3*e^3*x^3 + 3*(8*b*c^2 - b)*d^2*e ^3*x^2 + 2*(8*b*c^3 - 3*b*c)*d*e^3*x + (4*b*c^4 - 3*b*c^2)*e^3)*arctan2(d* x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) - 2*(b^4*d*arctan2(d*x + c, s qrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + 2*a*b^3*d*arctan2(d*x + c, sqrt(d *x + c + 1)*sqrt(-d*x - c + 1)) + a^2*b^2*d)*integrate((8*d^3*e^3*x^3 + 24 *c*d^2*e^3*x^2 + 3*(8*c^2 - 1)*d*e^3*x + (8*c^3 - 3*c)*e^3)/(b^3*arctan2(d *x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + a*b^2), x))/(b^4*d*arctan2 (d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + 2*a*b^3*d*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + a^2*b^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 2201 vs. \(2 (239) = 478\).
Time = 0.54 (sec) , antiderivative size = 2201, normalized size of antiderivative = 8.84 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=\text {Too large to display} \]
-8*b^2*e^3*arcsin(d*x + c)^2*cos(a/b)^3*cos_integral(4*a/b + 4*arcsin(d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2* b^3*d) + 8*b^2*e^3*arcsin(d*x + c)^2*cos(a/b)^4*sin_integral(4*a/b + 4*arc sin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b ^3*d) - 16*a*b*e^3*arcsin(d*x + c)*cos(a/b)^3*cos_integral(4*a/b + 4*arcsi n(d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 16*a*b*e^3*arcsin(d*x + c)*cos(a/b)^4*sin_integral(4*a/b + 4*arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 4*b^2*e^3*arcsin(d*x + c)^2*cos(a/b)*cos_integral(4*a/b + 4*a rcsin(d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 8*a^2*e^3*cos(a/b)^3*cos_integral(4*a/b + 4*arcsin(d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2* b^3*d) + b^2*e^3*arcsin(d*x + c)^2*cos(a/b)*cos_integral(2*a/b + 2*arcsin( d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 8*b^2*e^3*arcsin(d*x + c)^2*cos(a/b)^2*sin_integral(4*a/b + 4 *arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a ^2*b^3*d) + 8*a^2*e^3*cos(a/b)^4*sin_integral(4*a/b + 4*arcsin(d*x + c))/( b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - b^2*e^3 *arcsin(d*x + c)^2*cos(a/b)^2*sin_integral(2*a/b + 2*arcsin(d*x + c))/(b^5 *d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 8*a*b*e...
Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3} \,d x \]