\(\int \frac {x^2}{(a+b \arcsin (c x))^3} \, dx\) [168]

Optimal result

Integrand size = 14, antiderivative size = 197 \[ \int \frac {x^2}{(a+b \arcsin (c x))^3} \, dx=-\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}-\frac {x}{b^2 c^2 (a+b \arcsin (c x))}+\frac {3 x^3}{2 b^2 (a+b \arcsin (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{8 b^3 c^3}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{8 b^3 c^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4729, 4807, 4731, 4491, 3384, 3380, 3383, 4719} \[ \int \frac {x^2}{(a+b \arcsin (c x))^3} \, dx=-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{8 b^3 c^3}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{8 b^3 c^3}-\frac {x}{b^2 c^2 (a+b \arcsin (c x))}+\frac {3 x^3}{2 b^2 (a+b \arcsin (c x))}-\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2} \]

[In]

Int[x^2/(a + b*ArcSin[c*x])^3,x]

[Out]

-1/2*(x^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])^2) - x/(b^2*c^2*(a + b*ArcSin[c*x])) + (3*x^3)/(2*b^2*(a
+ b*ArcSin[c*x])) - (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(8*b^3*c^3) + (9*Cos[(3*a)/b]*CosIntegral[(
3*(a + b*ArcSin[c*x]))/b])/(8*b^3*c^3) - (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b^3*c^3) + (9*Sin[(3
*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(8*b^3*c^3)

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 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rule 4719

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[-a/b + x/b], x], x,
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 4729

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[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] - Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2
]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[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]

Rule 4807

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] - Dist[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]

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{(a+b \arcsin (c x))^3} \, dx=-\frac {\frac {4 b^2 x^2 \sqrt {1-c^2 x^2}}{c (a+b \arcsin (c x))^2}+\frac {8 b x}{c^2 (a+b \arcsin (c x))}-\frac {12 b x^3}{a+b \arcsin (c x)}+\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{c^3}-\frac {9 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{c^3}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{c^3}-\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{c^3}}{8 b^3} \]

[In]

Integrate[x^2/(a + b*ArcSin[c*x])^3,x]

[Out]

-1/8*((4*b^2*x^2*Sqrt[1 - c^2*x^2])/(c*(a + b*ArcSin[c*x])^2) + (8*b*x)/(c^2*(a + b*ArcSin[c*x])) - (12*b*x^3)
/(a + b*ArcSin[c*x]) + (Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/c^3 - (9*Cos[(3*a)/b]*CosIntegral[3*(a/b + Ar
cSin[c*x])])/c^3 + (Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/c^3 - (9*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin
[c*x])])/c^3)/b^3

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-c^{2} x^{2}+1}}{8 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -x b c}{8 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}+\frac {\cos \left (3 \arcsin \left (c x \right )\right )}{8 \left (a +b \arcsin \left (c x \right )\right )^{2} b}+\frac {\frac {9 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a}{8}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a}{8}-\frac {3 \sin \left (3 \arcsin \left (c x \right )\right ) b}{8}}{\left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c^{3}}\)	\(290\)
default	\(\frac {-\frac {\sqrt {-c^{2} x^{2}+1}}{8 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -x b c}{8 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}+\frac {\cos \left (3 \arcsin \left (c x \right )\right )}{8 \left (a +b \arcsin \left (c x \right )\right )^{2} b}+\frac {\frac {9 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a}{8}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a}{8}-\frac {3 \sin \left (3 \arcsin \left (c x \right )\right ) b}{8}}{\left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c^{3}}\)	\(290\)

[In]

int(x^2/(a+b*arcsin(c*x))^3,x,method=_RETURNVERBOSE)

[Out]

1/c^3*(-1/8*(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x))^2/b-1/8*(arcsin(c*x)*Si(arcsin(c*x)+a/b)*sin(a/b)*b+arcsin(c*
x)*Ci(arcsin(c*x)+a/b)*cos(a/b)*b+Si(arcsin(c*x)+a/b)*sin(a/b)*a+Ci(arcsin(c*x)+a/b)*cos(a/b)*a-x*b*c)/(a+b*ar
csin(c*x))/b^3+1/8*cos(3*arcsin(c*x))/(a+b*arcsin(c*x))^2/b+3/8*(3*arcsin(c*x)*Si(3*arcsin(c*x)+3*a/b)*sin(3*a
/b)*b+3*arcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b+3*Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*a+3*Ci(3*arcsin(
c*x)+3*a/b)*cos(3*a/b)*a-sin(3*arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^3)

Fricas [F]

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

[In]

integrate(x^2/(a+b*arcsin(c*x))^3,x, algorithm="fricas")

[Out]

integral(x^2/(b^3*arcsin(c*x)^3 + 3*a*b^2*arcsin(c*x)^2 + 3*a^2*b*arcsin(c*x) + a^3), x)

Sympy [F]

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

[In]

integrate(x**2/(a+b*asin(c*x))**3,x)

[Out]

Integral(x**2/(a + b*asin(c*x))**3, x)

Maxima [F]

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

[In]

integrate(x^2/(a+b*arcsin(c*x))^3,x, algorithm="maxima")

[Out]

1/2*(3*a*c^2*x^3 - sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x^2 - 2*a*x + (3*b*c^2*x^3 - 2*b*x)*arctan2(c*x, sqrt(c*x
+ 1)*sqrt(-c*x + 1)) - 2*(b^4*c^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b^3*c^2*arctan2(c*x, sqrt
(c*x + 1)*sqrt(-c*x + 1)) + a^2*b^2*c^2)*integrate(1/2*(9*c^2*x^2 - 2)/(b^3*c^2*arctan2(c*x, sqrt(c*x + 1)*sqr
t(-c*x + 1)) + a*b^2*c^2), x))/(b^4*c^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b^3*c^2*arctan2(c*x
, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a^2*b^2*c^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1539 vs. \(2 (183) = 366\).

Time = 0.36 (sec) , antiderivative size = 1539, normalized size of antiderivative = 7.81 \[ \int \frac {x^2}{(a+b \arcsin (c x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(x^2/(a+b*arcsin(c*x))^3,x, algorithm="giac")

[Out]

9/2*b^2*arcsin(c*x)^2*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcs
in(c*x) + a^2*b^3*c^3) + 9/2*b^2*arcsin(c*x)^2*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^
3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) + 9*a*b*arcsin(c*x)*cos(a/b)^3*cos_integral(3*a/b + 3
*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) + 9*a*b*arcsin(c*x)*cos(a/b)^2*s
in(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) +
3/2*(c^2*x^2 - 1)*b^2*c*x*arcsin(c*x)/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) - 27/8*b
^2*arcsin(c*x)^2*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x)
+ a^2*b^3*c^3) + 9/2*a^2*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*
arcsin(c*x) + a^2*b^3*c^3) - 1/8*b^2*arcsin(c*x)^2*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^5*c^3*arcsin(c*
x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) - 9/8*b^2*arcsin(c*x)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c
*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) + 9/2*a^2*cos(a/b)^2*sin(a/b)*sin_integra
l(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) - 1/8*b^2*arcsin(c*x)
^2*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) +
3/2*(c^2*x^2 - 1)*a*b*c*x/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) + 1/2*b^2*c*x*arcsin
(c*x)/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) - 27/4*a*b*arcsin(c*x)*cos(a/b)*cos_inte
gral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) - 1/4*a*b*arcsin(c
*x)*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) -
9/4*a*b*arcsin(c*x)*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(
c*x) + a^2*b^3*c^3) - 1/4*a*b*arcsin(c*x)*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*
a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) + 1/2*a*b*c*x/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*
c^3) - 27/8*a^2*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x)
+ a^2*b^3*c^3) - 1/8*a^2*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(
c*x) + a^2*b^3*c^3) - 9/8*a^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^
3*arcsin(c*x) + a^2*b^3*c^3) - 1/8*a^2*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b
^4*c^3*arcsin(c*x) + a^2*b^3*c^3) + 1/2*(-c^2*x^2 + 1)^(3/2)*b^2/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c
*x) + a^2*b^3*c^3) - 1/2*sqrt(-c^2*x^2 + 1)*b^2/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3
)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{(a+b \arcsin (c x))^3} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3} \,d x \]

[In]

int(x^2/(a + b*asin(c*x))^3,x)

[Out]

int(x^2/(a + b*asin(c*x))^3, x)