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

Optimal. Leaf size=130 \[ -\frac {a^2 \sin ^{-1}(a+b x)^2}{2 b^2}-\frac {(a+b x)^2}{4 b^2}+\frac {\sqrt {1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{2 b^2}-\frac {\sin ^{-1}(a+b x)^2}{4 b^2}-\frac {2 a \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^2}+\frac {1}{2} x^2 \sin ^{-1}(a+b x)^2+\frac {2 a x}{b} \]

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4805, 4743, 4763, 4641, 4677, 8, 4707, 30} \[ -\frac {a^2 \sin ^{-1}(a+b x)^2}{2 b^2}-\frac {(a+b x)^2}{4 b^2}+\frac {\sqrt {1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{2 b^2}-\frac {\sin ^{-1}(a+b x)^2}{4 b^2}-\frac {2 a \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^2}+\frac {1}{2} x^2 \sin ^{-1}(a+b x)^2+\frac {2 a x}{b} \]

Antiderivative was successfully verified.

[In]

Int[x*ArcSin[a + b*x]^2,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4641

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

Rule 4677

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

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4743

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

Rule 4763

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g},
x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 ||
(n == 1 && p > -1) || (m == 2 && p < -2))

Rule 4805

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

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Mathematica [A]  time = 0.10, size = 83, normalized size = 0.64 \[ \frac {\left (-2 a^2+2 b^2 x^2-1\right ) \sin ^{-1}(a+b x)^2-2 (3 a-b x) \sqrt {-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)+b x (6 a-b x)}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*ArcSin[a + b*x]^2,x]

[Out]

(b*x*(6*a - b*x) - 2*(3*a - b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x] + (-1 - 2*a^2 + 2*b^2*x^2)*
ArcSin[a + b*x]^2)/(4*b^2)

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fricas [A]  time = 0.49, size = 80, normalized size = 0.62 \[ -\frac {b^{2} x^{2} - 6 \, a b x - {\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \arcsin \left (b x + a\right )^{2} - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x - 3 \, a\right )} \arcsin \left (b x + a\right )}{4 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b^2*x^2 - 6*a*b*x - (2*b^2*x^2 - 2*a^2 - 1)*arcsin(b*x + a)^2 - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*
x - 3*a)*arcsin(b*x + a))/b^2

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giac [A]  time = 1.93, size = 139, normalized size = 1.07 \[ -\frac {{\left (b x + a\right )} a \arcsin \left (b x + a\right )^{2}}{b^{2}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )^{2}}{2 \, b^{2}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{2 \, b^{2}} - \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a \arcsin \left (b x + a\right )}{b^{2}} + \frac {2 \, {\left (b x + a\right )} a}{b^{2}} + \frac {\arcsin \left (b x + a\right )^{2}}{4 \, b^{2}} - \frac {{\left (b x + a\right )}^{2} - 1}{4 \, b^{2}} - \frac {1}{8 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x + a)*a*arcsin(b*x + a)^2/b^2 + 1/2*((b*x + a)^2 - 1)*arcsin(b*x + a)^2/b^2 + 1/2*sqrt(-(b*x + a)^2 + 1)*
(b*x + a)*arcsin(b*x + a)/b^2 - 2*sqrt(-(b*x + a)^2 + 1)*a*arcsin(b*x + a)/b^2 + 2*(b*x + a)*a/b^2 + 1/4*arcsi
n(b*x + a)^2/b^2 - 1/4*((b*x + a)^2 - 1)/b^2 - 1/8/b^2

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maple [A]  time = 0.08, size = 124, normalized size = 0.95 \[ \frac {\frac {\arcsin \left (b x +a \right )^{2} \left (-1+\left (b x +a \right )^{2}\right )}{2}+\frac {\arcsin \left (b x +a \right ) \left (\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\arcsin \left (b x +a \right )\right )}{2}-\frac {\arcsin \left (b x +a \right )^{2}}{4}-\frac {\left (b x +a \right )^{2}}{4}-a \left (\arcsin \left (b x +a \right )^{2} \left (b x +a \right )-2 b x -2 a +2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2} + b \int \frac {\sqrt {b x + a + 1} \sqrt {-b x - a + 1} x^{2} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {asin}\left (a+b\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*asin(a + b*x)^2,x)

[Out]

int(x*asin(a + b*x)^2, x)

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sympy [A]  time = 0.59, size = 138, normalized size = 1.06 \[ \begin {cases} - \frac {a^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{2 b^{2}} + \frac {3 a x}{2 b} - \frac {3 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{2 b^{2}} + \frac {x^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{2} - \frac {x^{2}}{4} + \frac {x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{2 b} - \frac {\operatorname {asin}^{2}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {asin}^{2}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*asin(a + b*x)**2/(2*b**2) + 3*a*x/(2*b) - 3*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a +
b*x)/(2*b**2) + x**2*asin(a + b*x)**2/2 - x**2/4 + x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(2*b)
- asin(a + b*x)**2/(4*b**2), Ne(b, 0)), (x**2*asin(a)**2/2, True))

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