Integrand size = 23, antiderivative size = 416 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^4} \, dx=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^3}-\frac {2 e^4 (c+d x)^3}{3 b^2 d (a+b \arcsin (c+d x))^2}+\frac {5 e^4 (c+d x)^5}{6 b^2 d (a+b \arcsin (c+d x))^2}-\frac {2 e^4 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b^3 d (a+b \arcsin (c+d x))}+\frac {25 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{6 b^3 d (a+b \arcsin (c+d x))}-\frac {e^4 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{48 b^4 d}+\frac {27 e^4 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{32 b^4 d}-\frac {125 e^4 \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{96 b^4 d}+\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{48 b^4 d}-\frac {27 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{32 b^4 d}+\frac {125 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{96 b^4 d} \]
-2/3*e^4*(d*x+c)^3/b^2/d/(a+b*arcsin(d*x+c))^2+5/6*e^4*(d*x+c)^5/b^2/d/(a+ b*arcsin(d*x+c))^2+1/48*e^4*cos(a/b)*Si((a+b*arcsin(d*x+c))/b)/b^4/d-27/32 *e^4*cos(3*a/b)*Si(3*(a+b*arcsin(d*x+c))/b)/b^4/d+125/96*e^4*cos(5*a/b)*Si (5*(a+b*arcsin(d*x+c))/b)/b^4/d-1/48*e^4*Ci((a+b*arcsin(d*x+c))/b)*sin(a/b )/b^4/d+27/32*e^4*Ci(3*(a+b*arcsin(d*x+c))/b)*sin(3*a/b)/b^4/d-125/96*e^4* Ci(5*(a+b*arcsin(d*x+c))/b)*sin(5*a/b)/b^4/d-1/3*e^4*(d*x+c)^4*(1-(d*x+c)^ 2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^3-2*e^4*(d*x+c)^2*(1-(d*x+c)^2)^(1/2)/b^3 /d/(a+b*arcsin(d*x+c))+25/6*e^4*(d*x+c)^4*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*a rcsin(d*x+c))
Time = 1.69 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^4} \, dx=\frac {e^4 \left (-\frac {32 b^3 (c+d x)^4 \sqrt {1-(c+d x)^2}}{(a+b \arcsin (c+d x))^3}+\frac {16 b^2 \left (-4 (c+d x)^3+5 (c+d x)^5\right )}{(a+b \arcsin (c+d x))^2}+\frac {16 b \sqrt {1-(c+d x)^2} \left (-12 (c+d x)^2+25 (c+d x)^4\right )}{a+b \arcsin (c+d x)}+384 \left (-\operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right ) \sin \left (\frac {a}{b}\right )+\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+544 \left (3 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right ) \sin \left (\frac {a}{b}\right )-\operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )+\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )-125 \left (10 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right ) \sin \left (\frac {a}{b}\right )-5 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+\operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {5 a}{b}\right )-10 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )+5 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )-\cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )\right )}{96 b^4 d} \]
(e^4*((-32*b^3*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^ 3 + (16*b^2*(-4*(c + d*x)^3 + 5*(c + d*x)^5))/(a + b*ArcSin[c + d*x])^2 + (16*b*Sqrt[1 - (c + d*x)^2]*(-12*(c + d*x)^2 + 25*(c + d*x)^4))/(a + b*Arc Sin[c + d*x]) + 384*(-(CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b]) + Cos[ a/b]*SinIntegral[a/b + ArcSin[c + d*x]]) + 544*(3*CosIntegral[a/b + ArcSin [c + d*x]]*Sin[a/b] - CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] - 3*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] + Cos[(3*a)/b]*SinIntegral [3*(a/b + ArcSin[c + d*x])]) - 125*(10*CosIntegral[a/b + ArcSin[c + d*x]]* Sin[a/b] - 5*CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] + CosInte gral[5*(a/b + ArcSin[c + d*x])]*Sin[(5*a)/b] - 10*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] + 5*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x]) ] - Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c + d*x])])))/(96*b^4*d)
Time = 0.99 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5304, 27, 5144, 5222, 5142, 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.
| \(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \arcsin (c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \arcsin (c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5144 |
| \(\displaystyle \frac {e^4 \left (\frac {4 \int \frac {(c+d x)^3}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}d(c+d x)}{3 b}-\frac {5 \int \frac {(c+d x)^5}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}d(c+d x)}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^4}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 5222 |
| \(\displaystyle \frac {e^4 \left (\frac {4 \left (\frac {3 \int \frac {(c+d x)^2}{(a+b \arcsin (c+d x))^2}d(c+d x)}{2 b}-\frac {(c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {5 \left (\frac {5 \int \frac {(c+d x)^4}{(a+b \arcsin (c+d x))^2}d(c+d x)}{2 b}-\frac {(c+d x)^5}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^4}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 5142 |
| \(\displaystyle \frac {e^4 \left (-\frac {5 \left (\frac {5 \left (\frac {\int \left (\frac {5 \sin \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 (a+b \arcsin (c+d x))}-\frac {9 \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 (a+b \arcsin (c+d x))}+\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{8 (a+b \arcsin (c+d x))}\right )d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^4 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {(c+d x)^5}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}+\frac {4 \left (\frac {3 \left (\frac {\int \left (\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{4 (a+b \arcsin (c+d x))}-\frac {3 \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{4 (a+b \arcsin (c+d x))}\right )d(a+b \arcsin (c+d x))}{b^2}-\frac {(c+d x)^2 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^4}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {4 \left (\frac {3 \left (\frac {\frac {1}{4} \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )-\frac {3}{4} \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{4} \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )+\frac {3}{4} \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {5 \left (\frac {5 \left (\frac {\frac {1}{8} \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )-\frac {9}{16} \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )+\frac {5}{16} \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{8} \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )+\frac {9}{16} \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )-\frac {5}{16} \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^4 \sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}\right )}{2 b}-\frac {(c+d x)^5}{2 b (a+b \arcsin (c+d x))^2}\right )}{3 b}-\frac {\sqrt {1-(c+d x)^2} (c+d x)^4}{3 b (a+b \arcsin (c+d x))^3}\right )}{d}\) |
(e^4*(-1/3*((c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(b*(a + b*ArcSin[c + d*x])^ 3) + (4*(-1/2*(c + d*x)^3/(b*(a + b*ArcSin[c + d*x])^2) + (3*(-(((c + d*x) ^2*Sqrt[1 - (c + d*x)^2])/(b*(a + b*ArcSin[c + d*x]))) + ((CosIntegral[(a + b*ArcSin[c + d*x])/b]*Sin[a/b])/4 - (3*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b]*Sin[(3*a)/b])/4 - (Cos[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/ b])/4 + (3*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/4)/b^2 ))/(2*b)))/(3*b) - (5*(-1/2*(c + d*x)^5/(b*(a + b*ArcSin[c + d*x])^2) + (5 *(-(((c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(b*(a + b*ArcSin[c + d*x]))) + ((C osIntegral[(a + b*ArcSin[c + d*x])/b]*Sin[a/b])/8 - (9*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b]*Sin[(3*a)/b])/16 + (5*CosIntegral[(5*(a + b*ArcSin [c + d*x]))/b]*Sin[(5*a)/b])/16 - (Cos[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/8 + (9*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/ 16 - (5*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/16)/b^2)) /(2*b)))/(3*b)))/d
3.3.33.3.1 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_.))^(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. \(1137\) vs. \(2(390)=780\).
Time = 0.91 (sec) , antiderivative size = 1138, normalized size of antiderivative = 2.74
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1138\) |
| default | \(\text {Expression too large to display}\) | \(1138\) |
1/96/d*e^4*(2*arcsin(d*x+c)*b^3*(d*x+c)-2*cos(5*arcsin(d*x+c))*b^3-4*(1-(d *x+c)^2)^(1/2)*b^3+6*arcsin(d*x+c)*cos(a/b)*Si(arcsin(d*x+c)+a/b)*a^2*b-6* arcsin(d*x+c)*sin(a/b)*Ci(arcsin(d*x+c)+a/b)*a^2*b+243*arcsin(d*x+c)^2*sin (3*a/b)*Ci(3*arcsin(d*x+c)+3*a/b)*a*b^2-243*arcsin(d*x+c)^2*Si(3*arcsin(d* x+c)+3*a/b)*cos(3*a/b)*a*b^2+243*arcsin(d*x+c)*sin(3*a/b)*Ci(3*arcsin(d*x+ c)+3*a/b)*a^2*b-243*arcsin(d*x+c)*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)*a^2 *b+375*arcsin(d*x+c)^2*Si(5*arcsin(d*x+c)+5*a/b)*cos(5*a/b)*a*b^2-375*arcs in(d*x+c)^2*Ci(5*arcsin(d*x+c)+5*a/b)*sin(5*a/b)*a*b^2+375*arcsin(d*x+c)*S i(5*arcsin(d*x+c)+5*a/b)*cos(5*a/b)*a^2*b-375*arcsin(d*x+c)*Ci(5*arcsin(d* x+c)+5*a/b)*sin(5*a/b)*a^2*b+6*arcsin(d*x+c)^2*cos(a/b)*Si(arcsin(d*x+c)+a /b)*a*b^2-6*arcsin(d*x+c)^2*sin(a/b)*Ci(arcsin(d*x+c)+a/b)*a*b^2+6*cos(3*a rcsin(d*x+c))*b^3+81*arcsin(d*x+c)^3*sin(3*a/b)*Ci(3*arcsin(d*x+c)+3*a/b)* b^3-81*arcsin(d*x+c)^3*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)*b^3-54*arcsin( d*x+c)*cos(3*arcsin(d*x+c))*a*b^2+125*arcsin(d*x+c)^3*Si(5*arcsin(d*x+c)+5 *a/b)*cos(5*a/b)*b^3-125*arcsin(d*x+c)^3*Ci(5*arcsin(d*x+c)+5*a/b)*sin(5*a /b)*b^3+50*arcsin(d*x+c)*cos(5*arcsin(d*x+c))*a*b^2+2*arcsin(d*x+c)^3*cos( a/b)*Si(arcsin(d*x+c)+a/b)*b^3-2*arcsin(d*x+c)^3*sin(a/b)*Ci(arcsin(d*x+c) +a/b)*b^3+4*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2)*a*b^2-9*arcsin(d*x+c)*sin(3* arcsin(d*x+c))*b^3+81*sin(3*a/b)*Ci(3*arcsin(d*x+c)+3*a/b)*a^3-81*Si(3*arc sin(d*x+c)+3*a/b)*cos(3*a/b)*a^3-9*sin(3*arcsin(d*x+c))*a*b^2-27*cos(3*...
\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
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^4*arcsin(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)
\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^4} \, dx=e^{4} \left (\int \frac {c^{4}}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{4} + 4 a^{3} b \operatorname {asin}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}\, dx\right ) \]
e**4*(Integral(c**4/(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**4*x**4/(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(4*c*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(6*c**2*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(4*c**3*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*a sin(c + d*x)**3 + b**4*asin(c + d*x)**4), x))
Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 5870 vs. \(2 (390) = 780\).
Time = 0.79 (sec) , antiderivative size = 5870, normalized size of antiderivative = 14.11 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^4} \, dx=\text {Too large to display} \]
-125/6*b^3*e^4*arcsin(d*x + c)^3*cos(a/b)^4*cos_integral(5*a/b + 5*arcsin( d*x + c))*sin(a/b)/(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) + 125/6*b^3*e^4*arcsin(d*x + c) ^3*cos(a/b)^5*sin_integral(5*a/b + 5*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) - 125/2*a*b^2*e^4*arcsin(d*x + c)^2*cos(a/b)^4*cos_integral(5*a/b + 5* arcsin(d*x + c))*sin(a/b)/(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) + 125/2*a*b^2*e^4*arcsin (d*x + c)^2*cos(a/b)^5*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^7*d*arcs in(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) + 125/8*b^3*e^4*arcsin(d*x + c)^3*cos(a/b)^2*cos_integral(5*a /b + 5*arcsin(d*x + c))*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcs in(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 125/2*a^2*b*e^4 *arcsin(d*x + c)*cos(a/b)^4*cos_integral(5*a/b + 5*arcsin(d*x + c))*sin(a/ b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*ar csin(d*x + c) + a^3*b^4*d) + 27/8*b^3*e^4*arcsin(d*x + c)^3*cos(a/b)^2*cos _integral(3*a/b + 3*arcsin(d*x + c))*sin(a/b)/(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) - 62 5/24*b^3*e^4*arcsin(d*x + c)^3*cos(a/b)^3*sin_integral(5*a/b + 5*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*...
Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^4} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4} \,d x \]