3.344 \(\int x (a+b \sin ^{-1}(c x^2)) \, dx\)

Optimal. Leaf size=45 \[ \frac {a x^2}{2}+\frac {b \sqrt {1-c^2 x^4}}{2 c}+\frac {1}{2} b x^2 \sin ^{-1}\left (c x^2\right ) \]

[Out]

1/2*a*x^2+1/2*b*x^2*arcsin(c*x^2)+1/2*b*(-c^2*x^4+1)^(1/2)/c

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6715, 4619, 261} \[ \frac {a x^2}{2}+\frac {b \sqrt {1-c^2 x^4}}{2 c}+\frac {1}{2} b x^2 \sin ^{-1}\left (c x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSin[c*x^2]),x]

[Out]

(a*x^2)/2 + (b*Sqrt[1 - c^2*x^4])/(2*c) + (b*x^2*ArcSin[c*x^2])/2

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {align*} \int x \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(c x)\right ) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {1}{2} b \operatorname {Subst}\left (\int \sin ^{-1}(c x) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {1}{2} b x^2 \sin ^{-1}\left (c x^2\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x^2}} \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}+\frac {b \sqrt {1-c^2 x^4}}{2 c}+\frac {1}{2} b x^2 \sin ^{-1}\left (c x^2\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 43, normalized size = 0.96 \[ \frac {a x^2}{2}+\frac {1}{2} b \left (\frac {\sqrt {1-c^2 x^4}}{c}+x^2 \sin ^{-1}\left (c x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcSin[c*x^2]),x]

[Out]

(a*x^2)/2 + (b*(Sqrt[1 - c^2*x^4]/c + x^2*ArcSin[c*x^2]))/2

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 38, normalized size = 0.84 \[ \frac {b c x^{2} \arcsin \left (c x^{2}\right ) + a c x^{2} + \sqrt {-c^{2} x^{4} + 1} b}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*c*x^2*arcsin(c*x^2) + a*c*x^2 + sqrt(-c^2*x^4 + 1)*b)/c

________________________________________________________________________________________

giac [A]  time = 0.23, size = 38, normalized size = 0.84 \[ \frac {a c x^{2} + {\left (c x^{2} \arcsin \left (c x^{2}\right ) + \sqrt {-c^{2} x^{4} + 1}\right )} b}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*c*x^2 + (c*x^2*arcsin(c*x^2) + sqrt(-c^2*x^4 + 1))*b)/c

________________________________________________________________________________________

maple [A]  time = 0.00, size = 39, normalized size = 0.87 \[ \frac {a \,x^{2} c +b \left (x^{2} c \arcsin \left (c \,x^{2}\right )+\sqrt {-c^{2} x^{4}+1}\right )}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/c*(a*x^2*c+b*(x^2*c*arcsin(c*x^2)+(-c^2*x^4+1)^(1/2)))

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 37, normalized size = 0.82 \[ \frac {1}{2} \, a x^{2} + \frac {{\left (c x^{2} \arcsin \left (c x^{2}\right ) + \sqrt {-c^{2} x^{4} + 1}\right )} b}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 + 1/2*(c*x^2*arcsin(c*x^2) + sqrt(-c^2*x^4 + 1))*b/c

________________________________________________________________________________________

mupad [B]  time = 0.37, size = 37, normalized size = 0.82 \[ \frac {a\,x^2}{2}+\frac {b\,\sqrt {1-c^2\,x^4}}{2\,c}+\frac {b\,x^2\,\mathrm {asin}\left (c\,x^2\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asin(c*x^2)),x)

[Out]

(a*x^2)/2 + (b*(1 - c^2*x^4)^(1/2))/(2*c) + (b*x^2*asin(c*x^2))/2

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 42, normalized size = 0.93 \[ \begin {cases} \frac {a x^{2}}{2} + \frac {b x^{2} \operatorname {asin}{\left (c x^{2} \right )}}{2} + \frac {b \sqrt {- c^{2} x^{4} + 1}}{2 c} & \text {for}\: c \neq 0 \\\frac {a x^{2}}{2} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*x**2/2 + b*x**2*asin(c*x**2)/2 + b*sqrt(-c**2*x**4 + 1)/(2*c), Ne(c, 0)), (a*x**2/2, True))

________________________________________________________________________________________