\(\int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx\) [221]

Optimal result

Integrand size = 23, antiderivative size = 258 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^4 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{8 b^2 d}-\frac {9 e^4 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{16 b^2 d}+\frac {5 e^4 \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{16 b^2 d}-\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^2 d}+\frac {9 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}-\frac {5 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d} \]

[Out]

-1/8*e^4*cos(a/b)*Si((a+b*arcsin(d*x+c))/b)/b^2/d+9/16*e^4*cos(3*a/b)*Si(3*(a+b*arcsin(d*x+c))/b)/b^2/d-5/16*e
^4*cos(5*a/b)*Si(5*(a+b*arcsin(d*x+c))/b)/b^2/d+1/8*e^4*Ci((a+b*arcsin(d*x+c))/b)*sin(a/b)/b^2/d-9/16*e^4*Ci(3
*(a+b*arcsin(d*x+c))/b)*sin(3*a/b)/b^2/d+5/16*e^4*Ci(5*(a+b*arcsin(d*x+c))/b)*sin(5*a/b)/b^2/d-e^4*(d*x+c)^4*(
1-(d*x+c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4889, 12, 4727, 3384, 3380, 3383} \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\frac {e^4 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^2 d}-\frac {9 e^4 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}+\frac {5 e^4 \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}-\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^2 d}+\frac {9 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}-\frac {5 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))} \]

[In]

Int[(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x])^2,x]

[Out]

-((e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(b*d*(a + b*ArcSin[c + d*x]))) + (e^4*CosIntegral[(a + b*ArcSin[c +
d*x])/b]*Sin[a/b])/(8*b^2*d) - (9*e^4*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b]*Sin[(3*a)/b])/(16*b^2*d) + (5
*e^4*CosIntegral[(5*(a + b*ArcSin[c + d*x]))/b]*Sin[(5*a)/b])/(16*b^2*d) - (e^4*Cos[a/b]*SinIntegral[(a + b*Ar
cSin[c + d*x])/b])/(8*b^2*d) + (9*e^4*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(16*b^2*d) - (5
*e^4*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/(16*b^2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[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]

Rule 4727

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] - Dist[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]

Rule 4889

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^4 x^4}{(a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int \frac {x^4}{(a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^4 \text {Subst}\left (\int \left (\frac {5 \sin \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}-\frac {9 \sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}+\frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{b^2 d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^4 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{8 b^2 d}+\frac {\left (5 e^4\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d}-\frac {\left (9 e^4\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}-\frac {\left (e^4 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{8 b^2 d}+\frac {\left (9 e^4 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d}-\frac {\left (5 e^4 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d}+\frac {\left (e^4 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{8 b^2 d}-\frac {\left (9 e^4 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d}+\frac {\left (5 e^4 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b 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 )}{8 b^2 d}-\frac {9 e^4 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{16 b^2 d}+\frac {5 e^4 \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{16 b^2 d}-\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^2 d}+\frac {9 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}-\frac {5 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.61 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.10 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\frac {e^4 \left (-\frac {16 b (c+d x)^4 \sqrt {1-(c+d x)^2}}{a+b \arcsin (c+d x)}+16 \left (-3 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right ) \sin \left (\frac {a}{b}\right )+\operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )-\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )+5 \left (10 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right ) \sin \left (\frac {a}{b}\right )-5 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+\operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {5 a}{b}\right )-10 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )+5 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )-\cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )\right )}{16 b^2 d} \]

[In]

Integrate[(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x])^2,x]

[Out]

(e^4*((-16*b*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x]) + 16*(-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])]) + 5*(10*CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a
/b] - 5*CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] + CosIntegral[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])])))/(16*b^2*d)

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.53

method	result	size
derivativedivides	\(-\frac {e^{4} \left (5 \arcsin \left (d x +c \right ) \cos \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b +9 \arcsin \left (d x +c \right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b -5 \arcsin \left (d x +c \right ) \sin \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b +2 \arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b -2 \arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b -9 \arcsin \left (d x +c \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b +5 \cos \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a +9 \,\operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a -5 \sin \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a +2 \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -2 \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -9 \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a +\cos \left (5 \arcsin \left (d x +c \right )\right ) b -3 \cos \left (3 \arcsin \left (d x +c \right )\right ) b +2 \sqrt {1-\left (d x +c \right )^{2}}\, b \right )}{16 d \left (a +b \arcsin \left (d x +c \right )\right ) b^{2}}\)	\(396\)
default	\(-\frac {e^{4} \left (5 \arcsin \left (d x +c \right ) \cos \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b +9 \arcsin \left (d x +c \right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b -5 \arcsin \left (d x +c \right ) \sin \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b +2 \arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b -2 \arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b -9 \arcsin \left (d x +c \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b +5 \cos \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a +9 \,\operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a -5 \sin \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a +2 \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -2 \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -9 \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a +\cos \left (5 \arcsin \left (d x +c \right )\right ) b -3 \cos \left (3 \arcsin \left (d x +c \right )\right ) b +2 \sqrt {1-\left (d x +c \right )^{2}}\, b \right )}{16 d \left (a +b \arcsin \left (d x +c \right )\right ) b^{2}}\)	\(396\)

[In]

int((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/16/d*e^4*(5*arcsin(d*x+c)*cos(5*a/b)*Si(5*arcsin(d*x+c)+5*a/b)*b+9*arcsin(d*x+c)*Ci(3*arcsin(d*x+c)+3*a/b)*
sin(3*a/b)*b-5*arcsin(d*x+c)*sin(5*a/b)*Ci(5*arcsin(d*x+c)+5*a/b)*b+2*arcsin(d*x+c)*cos(a/b)*Si(arcsin(d*x+c)+
a/b)*b-2*arcsin(d*x+c)*sin(a/b)*Ci(arcsin(d*x+c)+a/b)*b-9*arcsin(d*x+c)*cos(3*a/b)*Si(3*arcsin(d*x+c)+3*a/b)*b
+5*cos(5*a/b)*Si(5*arcsin(d*x+c)+5*a/b)*a+9*Ci(3*arcsin(d*x+c)+3*a/b)*sin(3*a/b)*a-5*sin(5*a/b)*Ci(5*arcsin(d*
x+c)+5*a/b)*a+2*cos(a/b)*Si(arcsin(d*x+c)+a/b)*a-2*sin(a/b)*Ci(arcsin(d*x+c)+a/b)*a-9*cos(3*a/b)*Si(3*arcsin(d
*x+c)+3*a/b)*a+cos(5*arcsin(d*x+c))*b-3*cos(3*arcsin(d*x+c))*b+2*(1-(d*x+c)^2)^(1/2)*b)/(a+b*arcsin(d*x+c))/b^
2

Fricas [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^2,x, algorithm="fricas")

[Out]

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^2*arcsin(d*x + c)^2
+ 2*a*b*arcsin(d*x + c) + a^2), x)

Sympy [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=e^{4} \left (\int \frac {c^{4}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx\right ) \]

[In]

integrate((d*e*x+c*e)**4/(a+b*asin(d*x+c))**2,x)

[Out]

e**4*(Integral(c**4/(a**2 + 2*a*b*asin(c + d*x) + b**2*asin(c + d*x)**2), x) + Integral(d**4*x**4/(a**2 + 2*a*
b*asin(c + d*x) + b**2*asin(c + d*x)**2), x) + Integral(4*c*d**3*x**3/(a**2 + 2*a*b*asin(c + d*x) + b**2*asin(
c + d*x)**2), x) + Integral(6*c**2*d**2*x**2/(a**2 + 2*a*b*asin(c + d*x) + b**2*asin(c + d*x)**2), x) + Integr
al(4*c**3*d*x/(a**2 + 2*a*b*asin(c + d*x) + b**2*asin(c + d*x)**2), x))

Maxima [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^2,x, algorithm="maxima")

[Out]

-((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)*sqrt(d*x + c + 1)*sqrt(-d*x -
c + 1) - (b^2*d*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + a*b*d)*integrate((5*d^5*e^4*x^5 + 25*
c*d^4*e^4*x^4 + 2*(25*c^2 - 2)*d^3*e^4*x^3 + 2*(25*c^3 - 6*c)*d^2*e^4*x^2 + (25*c^4 - 12*c^2)*d*e^4*x + (5*c^5
- 4*c^3)*e^4)*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)/(a*b*d^2*x^2 + 2*a*b*c*d*x + a*b*c^2 - a*b + (b^2*d^2*x^2
+ 2*b^2*c*d*x + b^2*c^2 - b^2)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))), x))/(b^2*d*arctan2(d*x
+ c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + a*b*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1401 vs. \(2 (244) = 488\).

Time = 0.42 (sec) , antiderivative size = 1401, normalized size of antiderivative = 5.43 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^2,x, algorithm="giac")

[Out]

5*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^3*d*arcsin(d*x + c) + a
*b^2*d) - 5*b*e^4*arcsin(d*x + c)*cos(a/b)^5*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) +
a*b^2*d) + 5*a*e^4*cos(a/b)^4*cos_integral(5*a/b + 5*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2*
d) - 5*a*e^4*cos(a/b)^5*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2*d) - 15/4*b*e^4
*arcsin(d*x + c)*cos(a/b)^2*cos_integral(5*a/b + 5*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2*d)
- 9/4*b*e^4*arcsin(d*x + c)*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c
) + a*b^2*d) + 25/4*b*e^4*arcsin(d*x + c)*cos(a/b)^3*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^3*d*arcsin(d*x
+ c) + a*b^2*d) + 9/4*b*e^4*arcsin(d*x + c)*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^3*d*arcsin(
d*x + c) + a*b^2*d) - 15/4*a*e^4*cos(a/b)^2*cos_integral(5*a/b + 5*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x
+ c) + a*b^2*d) - 9/4*a*e^4*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c
) + a*b^2*d) + 25/4*a*e^4*cos(a/b)^3*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2*d)
+ 9/4*a*e^4*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2*d) + 5/16*b*e^4
*arcsin(d*x + c)*cos_integral(5*a/b + 5*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2*d) + 9/16*b*e
^4*arcsin(d*x + c)*cos_integral(3*a/b + 3*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2*d) + 1/8*b*
e^4*arcsin(d*x + c)*cos_integral(a/b + arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2*d) - 25/16*b*e
^4*arcsin(d*x + c)*cos(a/b)*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2*d) - 27/16*
b*e^4*arcsin(d*x + c)*cos(a/b)*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2*d) - 1/8
*b*e^4*arcsin(d*x + c)*cos(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2*d) - ((d*x
+ c)^2 - 1)^2*sqrt(-(d*x + c)^2 + 1)*b*e^4/(b^3*d*arcsin(d*x + c) + a*b^2*d) + 5/16*a*e^4*cos_integral(5*a/b +
5*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2*d) + 9/16*a*e^4*cos_integral(3*a/b + 3*arcsin(d*x
+ c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2*d) + 1/8*a*e^4*cos_integral(a/b + arcsin(d*x + c))*sin(a/b)/(b^3
*d*arcsin(d*x + c) + a*b^2*d) - 25/16*a*e^4*cos(a/b)*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^3*d*arcsin(d*x
+ c) + a*b^2*d) - 27/16*a*e^4*cos(a/b)*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2
*d) - 1/8*a*e^4*cos(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2*d) + 2*(-(d*x + c)
^2 + 1)^(3/2)*b*e^4/(b^3*d*arcsin(d*x + c) + a*b^2*d) - sqrt(-(d*x + c)^2 + 1)*b*e^4/(b^3*d*arcsin(d*x + c) +
a*b^2*d)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2} \,d x \]

[In]

int((c*e + d*e*x)^4/(a + b*asin(c + d*x))^2,x)

[Out]

int((c*e + d*e*x)^4/(a + b*asin(c + d*x))^2, x)