\(\int \frac {x^2}{(a+b \arcsin (c x))^2} \, dx\) [163]

Optimal result

Integrand size = 14, antiderivative size = 156 \[ \int \frac {x^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {\operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 c^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}+\frac {3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3} \]

[Out]

-1/4*cos(a/b)*Si((a+b*arcsin(c*x))/b)/b^2/c^3+3/4*cos(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b^2/c^3+1/4*Ci((a+b*arc
sin(c*x))/b)*sin(a/b)/b^2/c^3-3/4*Ci(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b^2/c^3-x^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b
*arcsin(c*x))

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4727, 3384, 3380, 3383} \[ \int \frac {x^2}{(a+b \arcsin (c x))^2} \, dx=\frac {\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}-\frac {3 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}+\frac {3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3}-\frac {x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))} \]

[In]

Int[x^2/(a + b*ArcSin[c*x])^2,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{(a+b \arcsin (c x))^2} \, dx=\frac {-\frac {4 b c^2 x^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )-3 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{4 b^2 c^3} \]

[In]

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

[Out]

((-4*b*c^2*x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - 3*CosIntegra
l[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] - Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 3*Cos[(3*a)/b]*SinIntegral[3
*(a/b + ArcSin[c*x])])/(4*b^2*c^3)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-c^{2} x^{2}+1}}{4 \left (a +b \arcsin \left (c x \right )\right ) b}-\frac {\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )-\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^{2}}+\frac {\cos \left (3 \arcsin \left (c x \right )\right )}{4 \left (a +b \arcsin \left (c x \right )\right ) b}+\frac {\frac {3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )}{4}-\frac {3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4}}{b^{2}}}{c^{3}}\)	\(149\)
default	\(\frac {-\frac {\sqrt {-c^{2} x^{2}+1}}{4 \left (a +b \arcsin \left (c x \right )\right ) b}-\frac {\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )-\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^{2}}+\frac {\cos \left (3 \arcsin \left (c x \right )\right )}{4 \left (a +b \arcsin \left (c x \right )\right ) b}+\frac {\frac {3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )}{4}-\frac {3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4}}{b^{2}}}{c^{3}}\)	\(149\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (146) = 292\).

Time = 0.31 (sec) , antiderivative size = 646, normalized size of antiderivative = 4.14 \[ \int \frac {x^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {3 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {3 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac {3 \, a \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {3 \, a \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} + \frac {3 \, b \arcsin \left (c x\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} + \frac {b \arcsin \left (c x\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} - \frac {9 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} - \frac {b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} + \frac {3 \, a \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} + \frac {a \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} - \frac {9 \, a \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} - \frac {a \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}\right )}} + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} - \frac {\sqrt {-c^{2} x^{2} + 1} b}{b^{3} c^{3} \arcsin \left (c x\right ) + a b^{2} c^{3}} \]

[In]

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

[Out]

-3*b*arcsin(c*x)*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 3
*b*arcsin(c*x)*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 3*a*cos(a/b)
^2*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 3*a*cos(a/b)^3*sin_integra
l(3*a/b + 3*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 3/4*b*arcsin(c*x)*cos_integral(3*a/b + 3*arcsin(c
*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/4*b*arcsin(c*x)*cos_integral(a/b + arcsin(c*x))*sin(a/b)/(
b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 9/4*b*arcsin(c*x)*cos(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^3*arc
sin(c*x) + a*b^2*c^3) - 1/4*b*arcsin(c*x)*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^
2*c^3) + 3/4*a*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/4*a*cos_inte
gral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 9/4*a*cos(a/b)*sin_integral(3*a/b + 3*arc
sin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 1/4*a*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^3*arcsin(c
*x) + a*b^2*c^3) + (-c^2*x^2 + 1)^(3/2)*b/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - sqrt(-c^2*x^2 + 1)*b/(b^3*c^3*ar
csin(c*x) + a*b^2*c^3)

Mupad [F(-1)]

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

[In]

int(x^2/(a + b*asin(c*x))^2,x)

[Out]

int(x^2/(a + b*asin(c*x))^2, x)