\(\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) [239]

Optimal result

Integrand size = 26, antiderivative size = 49 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}} \]

[Out]

1/3*(a+b*arcsin(c*x))^3*(-c^2*x^2+1)^(1/2)/b/c/(-c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {4737} \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d - c^2*d*x^2])

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {1-c^2 x^2} \arcsin (c x) \left (3 a^2+3 a b \arcsin (c x)+b^2 \arcsin (c x)^2\right )}{3 c \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

(Sqrt[1 - c^2*x^2]*ArcSin[c*x]*(3*a^2 + 3*a*b*ArcSin[c*x] + b^2*ArcSin[c*x]^2))/(3*c*Sqrt[d - c^2*d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(142\) vs. \(2(43)=86\).

Time = 0.15 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.92

method	result	size
default	\(\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{3 c d \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{c d \left (c^{2} x^{2}-1\right )}\)	\(143\)
parts	\(\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{3 c d \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{c d \left (c^{2} x^{2}-1\right )}\)	\(143\)

[In]

int((a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

a^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/
2)/c/d/(c^2*x^2-1)*arcsin(c*x)^3-a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d/(c^2*x^2-1)*arcsin(c*x)^2

Fricas [F]

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

[In]

integrate((a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]

[In]

integrate((a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(1/2),x)

[Out]

Integral((a + b*asin(c*x))**2/sqrt(-d*(c*x - 1)*(c*x + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b^{2} \arcsin \left (c x\right )^{3}}{3 \, c \sqrt {d}} + \frac {a b \arcsin \left (c x\right )^{2}}{c \sqrt {d}} + \frac {a^{2} \arcsin \left (c x\right )}{c \sqrt {d}} \]

[In]

integrate((a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

1/3*b^2*arcsin(c*x)^3/(c*sqrt(d)) + a*b*arcsin(c*x)^2/(c*sqrt(d)) + a^2*arcsin(c*x)/(c*sqrt(d))

Giac [F]

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

[In]

integrate((a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^2/sqrt(-c^2*d*x^2 + d), x)

Mupad [F(-1)]

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

[In]

int((a + b*asin(c*x))^2/(d - c^2*d*x^2)^(1/2),x)

[Out]

int((a + b*asin(c*x))^2/(d - c^2*d*x^2)^(1/2), x)