Integrand size = 23, antiderivative size = 322 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^3} \, dx=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {2 e^4 (c+d x)^3}{b^2 d (a+b \arcsin (c+d x))}+\frac {5 e^4 (c+d x)^5}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {e^4 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{16 b^3 d}+\frac {27 e^4 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{32 b^3 d}-\frac {25 e^4 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{32 b^3 d}-\frac {e^4 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{16 b^3 d}+\frac {27 e^4 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{32 b^3 d}-\frac {25 e^4 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{32 b^3 d} \]
-2*e^4*(d*x+c)^3/b^2/d/(a+b*arcsin(d*x+c))+5/2*e^4*(d*x+c)^5/b^2/d/(a+b*ar csin(d*x+c))-1/16*e^4*Ci((a+b*arcsin(d*x+c))/b)*cos(a/b)/b^3/d+27/32*e^4*C i(3*(a+b*arcsin(d*x+c))/b)*cos(3*a/b)/b^3/d-25/32*e^4*Ci(5*(a+b*arcsin(d*x +c))/b)*cos(5*a/b)/b^3/d-1/16*e^4*Si((a+b*arcsin(d*x+c))/b)*sin(a/b)/b^3/d +27/32*e^4*Si(3*(a+b*arcsin(d*x+c))/b)*sin(3*a/b)/b^3/d-25/32*e^4*Si(5*(a+ b*arcsin(d*x+c))/b)*sin(5*a/b)/b^3/d-1/2*e^4*(d*x+c)^4*(1-(d*x+c)^2)^(1/2) /b/d/(a+b*arcsin(d*x+c))^2
Time = 1.53 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.98 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^3} \, dx=\frac {e^4 \left (-\frac {16 b^2 (c+d x)^4 \sqrt {1-(c+d x)^2}}{(a+b \arcsin (c+d x))^2}+\frac {16 b \left (-4 (c+d x)^3+5 (c+d x)^5\right )}{a+b \arcsin (c+d x)}+48 \left (\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right )-\cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )-\sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )-25 \left (2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right )-3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+\cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )-3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+\sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )\right )}{32 b^3 d} \]
(e^4*((-16*b^2*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^ 2 + (16*b*(-4*(c + d*x)^3 + 5*(c + d*x)^5))/(a + b*ArcSin[c + d*x]) + 48*( Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] - Cos[(3*a)/b]*CosIntegral[3*( a/b + ArcSin[c + d*x])] + Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] - Si n[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])]) - 25*(2*Cos[a/b]*CosInt egral[a/b + ArcSin[c + d*x]] - 3*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[ c + d*x])] + Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c + d*x])] + 2*Sin[a /b]*SinIntegral[a/b + ArcSin[c + d*x]] - 3*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])] + Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c + d*x])] )))/(32*b^3*d)
Time = 1.08 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5304, 27, 5144, 5222, 5146, 4906, 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.
| \(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \arcsin (c+d x))^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \arcsin (c+d x))^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5144 |
| \(\displaystyle \frac {e^4 \left (\frac {2 \int \frac {(c+d x)^3}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}d(c+d x)}{b}-\frac {5 \int \frac {(c+d x)^5}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}d(c+d x)}{2 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 5222 |
| \(\displaystyle \frac {e^4 \left (\frac {2 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \arcsin (c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {5 \left (\frac {5 \int \frac {(c+d x)^4}{a+b \arcsin (c+d x)}d(c+d x)}{b}-\frac {(c+d x)^5}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 5146 |
| \(\displaystyle \frac {e^4 \left (-\frac {5 \left (\frac {5 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^5}{b (a+b \arcsin (c+d x))}\right )}{2 b}+\frac {2 \left (\frac {3 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^3}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {e^4 \left (-\frac {5 \left (\frac {5 \int \left (\frac {\cos \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 (a+b \arcsin (c+d x))}-\frac {3 \cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 (a+b \arcsin (c+d x))}+\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{8 (a+b \arcsin (c+d x))}\right )d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^5}{b (a+b \arcsin (c+d x))}\right )}{2 b}+\frac {2 \left (\frac {3 \int \left (\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{4 (a+b \arcsin (c+d x))}-\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{4 (a+b \arcsin (c+d x))}\right )d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^3}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {2 \left (\frac {3 \left (\frac {1}{4} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )-\frac {1}{4} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{4} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )-\frac {1}{4} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^3}{b (a+b \arcsin (c+d x))}\right )}{b}-\frac {5 \left (\frac {5 \left (\frac {1}{8} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )-\frac {3}{16} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{16} \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{8} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )-\frac {3}{16} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{16} \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^5}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^4}{2 b (a+b \arcsin (c+d x))^2}\right )}{d}\) |
(e^4*(-1/2*((c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(b*(a + b*ArcSin[c + d*x])^ 2) + (2*(-((c + d*x)^3/(b*(a + b*ArcSin[c + d*x]))) + (3*((Cos[a/b]*CosInt egral[(a + b*ArcSin[c + d*x])/b])/4 - (Cos[(3*a)/b]*CosIntegral[(3*(a + b* ArcSin[c + d*x]))/b])/4 + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b] )/4 - (Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/4))/b^2))/ b - (5*(-((c + d*x)^5/(b*(a + b*ArcSin[c + d*x]))) + (5*((Cos[a/b]*CosInte gral[(a + b*ArcSin[c + d*x])/b])/8 - (3*Cos[(3*a)/b]*CosIntegral[(3*(a + b *ArcSin[c + d*x]))/b])/16 + (Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/16 + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/8 - (3* Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/16 + (Sin[(5*a)/b ]*SinIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/16))/b^2))/(2*b)))/d
3.3.27.3.1 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[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. \(719\) vs. \(2(304)=608\).
Time = 0.87 (sec) , antiderivative size = 720, normalized size of antiderivative = 2.24
| method | result | size |
| derivativedivides | \(\frac {e^{4} \left (54 \arcsin \left (d x +c \right ) \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a b +54 \arcsin \left (d x +c \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a b -50 \arcsin \left (d x +c \right ) \sin \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a b -50 \arcsin \left (d x +c \right ) \cos \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a b -4 \arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b -4 \arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b +2 a b \left (d x +c \right )-25 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b^{2}-25 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b^{2}-2 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}-2 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}+27 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b^{2}+27 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b^{2}+3 \cos \left (3 \arcsin \left (d x +c \right )\right ) b^{2}-\cos \left (5 \arcsin \left (d x +c \right )\right ) b^{2}-2 \sqrt {1-\left (d x +c \right )^{2}}\, b^{2}+27 \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a^{2}+27 \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a^{2}-9 \sin \left (3 \arcsin \left (d x +c \right )\right ) a b +5 \arcsin \left (d x +c \right ) \sin \left (5 \arcsin \left (d x +c \right )\right ) b^{2}-25 \sin \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a^{2}-25 \cos \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a^{2}+5 \sin \left (5 \arcsin \left (d x +c \right )\right ) a b +2 \arcsin \left (d x +c \right ) b^{2} \left (d x +c \right )-2 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}-2 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}-9 \arcsin \left (d x +c \right ) \sin \left (3 \arcsin \left (d x +c \right )\right ) b^{2}\right )}{32 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\) | \(720\) |
| default | \(\frac {e^{4} \left (54 \arcsin \left (d x +c \right ) \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a b +54 \arcsin \left (d x +c \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a b -50 \arcsin \left (d x +c \right ) \sin \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a b -50 \arcsin \left (d x +c \right ) \cos \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a b -4 \arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b -4 \arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b +2 a b \left (d x +c \right )-25 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b^{2}-25 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b^{2}-2 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}-2 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}+27 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b^{2}+27 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b^{2}+3 \cos \left (3 \arcsin \left (d x +c \right )\right ) b^{2}-\cos \left (5 \arcsin \left (d x +c \right )\right ) b^{2}-2 \sqrt {1-\left (d x +c \right )^{2}}\, b^{2}+27 \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a^{2}+27 \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a^{2}-9 \sin \left (3 \arcsin \left (d x +c \right )\right ) a b +5 \arcsin \left (d x +c \right ) \sin \left (5 \arcsin \left (d x +c \right )\right ) b^{2}-25 \sin \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a^{2}-25 \cos \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a^{2}+5 \sin \left (5 \arcsin \left (d x +c \right )\right ) a b +2 \arcsin \left (d x +c \right ) b^{2} \left (d x +c \right )-2 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}-2 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}-9 \arcsin \left (d x +c \right ) \sin \left (3 \arcsin \left (d x +c \right )\right ) b^{2}\right )}{32 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\) | \(720\) |
1/32/d*e^4*(54*arcsin(d*x+c)*sin(3*a/b)*Si(3*arcsin(d*x+c)+3*a/b)*a*b+54*a rcsin(d*x+c)*cos(3*a/b)*Ci(3*arcsin(d*x+c)+3*a/b)*a*b-50*arcsin(d*x+c)*sin (5*a/b)*Si(5*arcsin(d*x+c)+5*a/b)*a*b-50*arcsin(d*x+c)*cos(5*a/b)*Ci(5*arc sin(d*x+c)+5*a/b)*a*b-4*arcsin(d*x+c)*sin(a/b)*Si(arcsin(d*x+c)+a/b)*a*b-4 *arcsin(d*x+c)*cos(a/b)*Ci(arcsin(d*x+c)+a/b)*a*b+2*a*b*(d*x+c)-25*arcsin( d*x+c)^2*sin(5*a/b)*Si(5*arcsin(d*x+c)+5*a/b)*b^2-25*arcsin(d*x+c)^2*cos(5 *a/b)*Ci(5*arcsin(d*x+c)+5*a/b)*b^2-2*arcsin(d*x+c)^2*sin(a/b)*Si(arcsin(d *x+c)+a/b)*b^2-2*arcsin(d*x+c)^2*cos(a/b)*Ci(arcsin(d*x+c)+a/b)*b^2+27*arc sin(d*x+c)^2*sin(3*a/b)*Si(3*arcsin(d*x+c)+3*a/b)*b^2+27*arcsin(d*x+c)^2*c os(3*a/b)*Ci(3*arcsin(d*x+c)+3*a/b)*b^2+3*cos(3*arcsin(d*x+c))*b^2-cos(5*a rcsin(d*x+c))*b^2-2*(1-(d*x+c)^2)^(1/2)*b^2+27*sin(3*a/b)*Si(3*arcsin(d*x+ c)+3*a/b)*a^2+27*cos(3*a/b)*Ci(3*arcsin(d*x+c)+3*a/b)*a^2-9*sin(3*arcsin(d *x+c))*a*b+5*arcsin(d*x+c)*sin(5*arcsin(d*x+c))*b^2-25*sin(5*a/b)*Si(5*arc sin(d*x+c)+5*a/b)*a^2-25*cos(5*a/b)*Ci(5*arcsin(d*x+c)+5*a/b)*a^2+5*sin(5* arcsin(d*x+c))*a*b+2*arcsin(d*x+c)*b^2*(d*x+c)-2*sin(a/b)*Si(arcsin(d*x+c) +a/b)*a^2-2*cos(a/b)*Ci(arcsin(d*x+c)+a/b)*a^2-9*arcsin(d*x+c)*sin(3*arcsi n(d*x+c))*b^2)/(a+b*arcsin(d*x+c))^2/b^3
\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
integral((d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4* x + c^4*e^4)/(b^3*arcsin(d*x + c)^3 + 3*a*b^2*arcsin(d*x + c)^2 + 3*a^2*b* arcsin(d*x + c) + a^3), x)
\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^3} \, dx=e^{4} \left (\int \frac {c^{4}}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx\right ) \]
e**4*(Integral(c**4/(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**4*x**4/(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(4*c*d**3*x**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(6*c**2*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(4*c**3*d*x/(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))
\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
1/2*(5*a*d^5*e^4*x^5 + 25*a*c*d^4*e^4*x^4 + 2*(25*a*c^2 - 2*a)*d^3*e^4*x^3 + 2*(25*a*c^3 - 6*a*c)*d^2*e^4*x^2 + (25*a*c^4 - 12*a*c^2)*d*e^4*x + (5*a *c^5 - 4*a*c^3)*e^4 - (b*d^4*e^4*x^4 + 4*b*c*d^3*e^4*x^3 + 6*b*c^2*d^2*e^4 *x^2 + 4*b*c^3*d*e^4*x + b*c^4*e^4)*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1) + (5*b*d^5*e^4*x^5 + 25*b*c*d^4*e^4*x^4 + 2*(25*b*c^2 - 2*b)*d^3*e^4*x^3 + 2*(25*b*c^3 - 6*b*c)*d^2*e^4*x^2 + (25*b*c^4 - 12*b*c^2)*d*e^4*x + (5*b*c^ 5 - 4*b*c^3)*e^4)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) - 2*(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)*i ntegrate(1/2*(25*d^4*e^4*x^4 + 100*c*d^3*e^4*x^3 + 6*(25*c^2 - 2)*d^2*e^4* x^2 + 4*(25*c^3 - 6*c)*d*e^4*x + (25*c^4 - 12*c^2)*e^4)/(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. 3180 vs. \(2 (304) = 608\).
Time = 0.61 (sec) , antiderivative size = 3180, normalized size of antiderivative = 9.88 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^3} \, dx=\text {Too large to display} \]
-25/2*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^5*cos_integral(5*a/b + 5*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) - 25/2*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^4*sin(a/b)*sin_integral(5*a/b + 5*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) - 25*a*b*e^4*arcsin(d*x + c)*cos(a/b)^5*cos_integral(5*a/b + 5 *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) - 25*a*b*e^4*arcsin(d*x + c)*cos(a/b)^4*sin(a/b)*sin_integral(5* a/b + 5*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) + 125/8*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^3*cos_integral (5*a/b + 5*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) - 25/2*a^2*e^4*cos(a/b)^5*cos_integral(5*a/b + 5*arcsi n(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) + 27/8*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^3*cos_integral(3*a/b + 3*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) + 75/8*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^2*sin(a/b)*sin_integral(5* a/b + 5*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) - 25/2*a^2*e^4*cos(a/b)^4*sin(a/b)*sin_integral(5*a/b + 5 *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) + 27/8*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^2*sin(a/b)*sin_integra l(3*a/b + 3*arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsi...
Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^3} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3} \,d x \]