Integrand size = 23, antiderivative size = 346 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^4} \, dx=-\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^3}-\frac {e^3 (c+d x)^2}{2 b^2 d (a+b \arcsin (c+d x))^2}+\frac {2 e^3 (c+d x)^4}{3 b^2 d (a+b \arcsin (c+d x))^2}-\frac {e^3 (c+d x) \sqrt {1-(c+d x)^2}}{b^3 d (a+b \arcsin (c+d x))}+\frac {8 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b^3 d (a+b \arcsin (c+d x))}-\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{3 b^4 d}+\frac {4 e^3 \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{3 b^4 d}-\frac {e^3 \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{3 b^4 d}+\frac {4 e^3 \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{3 b^4 d} \]
-1/2*e^3*(d*x+c)^2/b^2/d/(a+b*arcsin(d*x+c))^2+2/3*e^3*(d*x+c)^4/b^2/d/(a+ b*arcsin(d*x+c))^2-1/3*e^3*Ci(2*(a+b*arcsin(d*x+c))/b)*cos(2*a/b)/b^4/d+4/ 3*e^3*Ci(4*(a+b*arcsin(d*x+c))/b)*cos(4*a/b)/b^4/d-1/3*e^3*Si(2*(a+b*arcsi n(d*x+c))/b)*sin(2*a/b)/b^4/d+4/3*e^3*Si(4*(a+b*arcsin(d*x+c))/b)*sin(4*a/ b)/b^4/d-1/3*e^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^3-e ^3*(d*x+c)*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))+8/3*e^3*(d*x+c)^3 *(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))
Time = 1.26 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.92 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^4} \, dx=\frac {e^3 \left (-\frac {2 b^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{(a+b \arcsin (c+d x))^3}+\frac {b^2 \left (-3 (c+d x)^2+4 (c+d x)^4\right )}{(a+b \arcsin (c+d x))^2}+\frac {2 b \sqrt {1-(c+d x)^2} \left (-3 (c+d x)+8 (c+d x)^3\right )}{a+b \arcsin (c+d x)}+6 \log (a+b \arcsin (c+d x))+30 \left (\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )-\log (a+b \arcsin (c+d x))+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )+8 \left (-4 \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+\cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (4 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+3 \log (a+b \arcsin (c+d x))-4 \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+\sin \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )\right )}{6 b^4 d} \]
(e^3*((-2*b^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (b^2*(-3*(c + d*x)^2 + 4*(c + d*x)^4))/(a + b*ArcSin[c + d*x])^2 + (2*b *Sqrt[1 - (c + d*x)^2]*(-3*(c + d*x) + 8*(c + d*x)^3))/(a + b*ArcSin[c + d *x]) + 6*Log[a + b*ArcSin[c + d*x]] + 30*(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c + d*x])] - Log[a + b*ArcSin[c + d*x]] + Sin[(2*a)/b]*SinIntegra l[2*(a/b + ArcSin[c + d*x])]) + 8*(-4*Cos[(2*a)/b]*CosIntegral[2*(a/b + Ar cSin[c + d*x])] + Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c + d*x])] + 3* Log[a + b*ArcSin[c + d*x]] - 4*Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])] + Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c + d*x])])))/(6*b^4*d )
Time = 1.30 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {5304, 27, 5144, 5222, 5142, 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.
| \(\displaystyle \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {e^3 (c+d x)^3}{(a+b \arcsin (c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int \frac {(c+d x)^3}{(a+b \arcsin (c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5144 |
| \(\displaystyle \frac {e^3 \left (\frac {\int \frac {(c+d x)^2}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}d(c+d x)}{b}-\frac {4 \int \frac {(c+d x)^4}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}d(c+d x)}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 5222 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\int \frac {c+d x}{(a+b \arcsin (c+d x))^2}d(c+d x)}{b}-\frac {(c+d x)^2}{2 b (a+b \arcsin (c+d x))^2}}{b}-\frac {4 \left (\frac {2 \int \frac {(c+d x)^3}{(a+b \arcsin (c+d x))^2}d(c+d x)}{b}-\frac {(c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 5142 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x) \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \arcsin (c+d x))^2}}{b}-\frac {4 \left (\frac {2 \left (\frac {\int \left (\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 (a+b \arcsin (c+d x))}-\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{2 (a+b \arcsin (c+d x))}\right )d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^3 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x) \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \arcsin (c+d x))^2}}{b}-\frac {4 \left (\frac {2 \left (\frac {\frac {1}{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{2} \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x) \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \arcsin (c+d x))^2}}{b}-\frac {4 \left (\frac {2 \left (\frac {\frac {1}{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{2} \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))-\sin \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x) \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \arcsin (c+d x))^2}}{b}-\frac {4 \left (\frac {2 \left (\frac {\frac {1}{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{2} \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))+\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x) \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \arcsin (c+d x))^2}}{b}-\frac {4 \left (\frac {2 \left (\frac {\frac {1}{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{2} \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))+\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x) \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \arcsin (c+d x))^2}}{b}-\frac {4 \left (\frac {2 \left (\frac {\frac {1}{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{2} \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x) \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \arcsin (c+d x))^2}}{b}-\frac {4 \left (\frac {2 \left (\frac {\frac {1}{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{2} \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x) \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \arcsin (c+d x))^2}}{b}-\frac {4 \left (\frac {2 \left (\frac {\frac {1}{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{2} \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{2} \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
(e^3*(-1/3*((c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(b*(a + b*ArcSin[c + d*x])^ 3) + (-1/2*(c + d*x)^2/(b*(a + b*ArcSin[c + d*x])^2) + (-(((c + d*x)*Sqrt[ 1 - (c + d*x)^2])/(b*(a + b*ArcSin[c + d*x]))) + (Cos[(2*a)/b]*CosIntegral [(2*(a + b*ArcSin[c + d*x]))/b] + Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSi n[c + d*x]))/b])/b^2)/b)/b - (4*(-1/2*(c + d*x)^4/(b*(a + b*ArcSin[c + d*x ])^2) + (2*(-(((c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(b*(a + b*ArcSin[c + d*x ]))) + ((Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/2 - (Cos [(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c + d*x]))/b])/2 + (Sin[(2*a)/b]*Si nIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/2 - (Sin[(4*a)/b]*SinIntegral[(4 *(a + b*ArcSin[c + d*x]))/b])/2)/b^2))/b))/(3*b)))/d
3.3.34.3.1 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[((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] - Simp [1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c* x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
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_)*((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. \(782\) vs. \(2(324)=648\).
Time = 0.33 (sec) , antiderivative size = 783, normalized size of antiderivative = 2.26
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(783\) |
| default | \(\text {Expression too large to display}\) | \(783\) |
-1/24/d*e^3*(8*arcsin(d*x+c)^2*sin(4*arcsin(d*x+c))*b^3+2*arcsin(d*x+c)*co s(2*arcsin(d*x+c))*b^3-2*arcsin(d*x+c)*cos(4*arcsin(d*x+c))*b^3+8*cos(2*a/ b)*Ci(2*arcsin(d*x+c)+2*a/b)*a^3+8*Si(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b)*a^ 3-32*sin(4*a/b)*Si(4*arcsin(d*x+c)+4*a/b)*a^3-32*cos(4*a/b)*Ci(4*arcsin(d* x+c)+4*a/b)*a^3-4*sin(2*arcsin(d*x+c))*a^2*b+2*cos(2*arcsin(d*x+c))*a*b^2+ 8*sin(4*arcsin(d*x+c))*a^2*b-2*cos(4*arcsin(d*x+c))*a*b^2-4*arcsin(d*x+c)^ 2*sin(2*arcsin(d*x+c))*b^3+24*arcsin(d*x+c)^2*cos(2*a/b)*Ci(2*arcsin(d*x+c )+2*a/b)*a*b^2+24*arcsin(d*x+c)^2*Si(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b)*a*b ^2-96*arcsin(d*x+c)^2*sin(4*a/b)*Si(4*arcsin(d*x+c)+4*a/b)*a*b^2-96*arcsin (d*x+c)^2*cos(4*a/b)*Ci(4*arcsin(d*x+c)+4*a/b)*a*b^2+24*arcsin(d*x+c)*cos( 2*a/b)*Ci(2*arcsin(d*x+c)+2*a/b)*a^2*b+24*arcsin(d*x+c)*Si(2*arcsin(d*x+c) +2*a/b)*sin(2*a/b)*a^2*b-96*arcsin(d*x+c)*sin(4*a/b)*Si(4*arcsin(d*x+c)+4* a/b)*a^2*b-96*arcsin(d*x+c)*cos(4*a/b)*Ci(4*arcsin(d*x+c)+4*a/b)*a^2*b+8*a rcsin(d*x+c)^3*cos(2*a/b)*Ci(2*arcsin(d*x+c)+2*a/b)*b^3+8*arcsin(d*x+c)^3* Si(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b)*b^3-32*arcsin(d*x+c)^3*sin(4*a/b)*Si( 4*arcsin(d*x+c)+4*a/b)*b^3-32*arcsin(d*x+c)^3*cos(4*a/b)*Ci(4*arcsin(d*x+c )+4*a/b)*b^3-8*arcsin(d*x+c)*sin(2*arcsin(d*x+c))*a*b^2+16*arcsin(d*x+c)*s in(4*arcsin(d*x+c))*a*b^2-sin(4*arcsin(d*x+c))*b^3+2*sin(2*arcsin(d*x+c))* b^3)/(a+b*arcsin(d*x+c))^3/b^4
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
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^4*ar csin(d*x + c)^4 + 4*a*b^3*arcsin(d*x + c)^3 + 6*a^2*b^2*arcsin(d*x + c)^2 + 4*a^3*b*arcsin(d*x + c) + a^4), x)
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^4} \, dx=e^{3} \left (\int \frac {c^{3}}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx\right ) \]
e**3*(Integral(c**3/(a**4 + 4*a**3*b*asin(c + d*x) + 6*a**2*b**2*asin(c + d*x)**2 + 4*a*b**3*asin(c + d*x)**3 + b**4*asin(c + d*x)**4), x) + Integra l(d**3*x**3/(a**4 + 4*a**3*b*asin(c + d*x) + 6*a**2*b**2*asin(c + d*x)**2 + 4*a*b**3*asin(c + d*x)**3 + b**4*asin(c + d*x)**4), x) + Integral(3*c*d* *2*x**2/(a**4 + 4*a**3*b*asin(c + d*x) + 6*a**2*b**2*asin(c + d*x)**2 + 4* a*b**3*asin(c + d*x)**3 + b**4*asin(c + d*x)**4), x) + Integral(3*c**2*d*x /(a**4 + 4*a**3*b*asin(c + d*x) + 6*a**2*b**2*asin(c + d*x)**2 + 4*a*b**3* asin(c + d*x)**3 + b**4*asin(c + d*x)**4), x))
Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 4040 vs. \(2 (324) = 648\).
Time = 0.76 (sec) , antiderivative size = 4040, normalized size of antiderivative = 11.68 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^4} \, dx=\text {Too large to display} \]
32/3*b^3*e^3*arcsin(d*x + c)^3*cos(a/b)^4*cos_integral(4*a/b + 4*arcsin(d* x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5 *d*arcsin(d*x + c) + a^3*b^4*d) + 32/3*b^3*e^3*arcsin(d*x + c)^3*cos(a/b)^ 3*sin(a/b)*sin_integral(4*a/b + 4*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^ 3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 32*a*b^2*e^3*arcsin(d*x + c)^2*cos(a/b)^4*cos_integral(4*a/b + 4*arcsin (d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2* b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 32*a*b^2*e^3*arcsin(d*x + c)^2*cos(a/ b)^3*sin(a/b)*sin_integral(4*a/b + 4*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4 *d) - 32/3*b^3*e^3*arcsin(d*x + c)^3*cos(a/b)^2*cos_integral(4*a/b + 4*arc sin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a ^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 32*a^2*b*e^3*arcsin(d*x + c)*cos(a /b)^4*cos_integral(4*a/b + 4*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3 *a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 2/ 3*b^3*e^3*arcsin(d*x + c)^3*cos(a/b)^2*cos_integral(2*a/b + 2*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d* arcsin(d*x + c) + a^3*b^4*d) - 16/3*b^3*e^3*arcsin(d*x + c)^3*cos(a/b)*sin (a/b)*sin_integral(4*a/b + 4*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3 *a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) +...
Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^4} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4} \,d x \]