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

Optimal. Leaf size=59 \[ \frac{2 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-2 b^2 x \]

[Out]

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

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Rubi [A]  time = 0.0721905, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4803, 4619, 4677, 8} \[ \frac{2 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-2 b^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c + d*x])^2,x]

[Out]

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

Rule 4803

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

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 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 8

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

Rubi steps

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

Mathematica [A]  time = 0.082908, size = 63, normalized size = 1.07 \[ \frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac{2 b \left (b (c+d x)-\sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 92, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ( \left ( dx+c \right ){a}^{2}+{b}^{2} \left ( \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) -2\,dx-2\,c+2\,\sqrt{1- \left ( dx+c \right ) ^{2}}\arcsin \left ( dx+c \right ) \right ) +2\,ab \left ( \left ( dx+c \right ) \arcsin \left ( dx+c \right ) +\sqrt{1- \left ( dx+c \right ) ^{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.39363, size = 224, normalized size = 3.8 \begin{align*} \frac{{\left (a^{2} - 2 \, b^{2}\right )} d x +{\left (b^{2} d x + b^{2} c\right )} \arcsin \left (d x + c\right )^{2} + 2 \,{\left (a b d x + a b c\right )} \arcsin \left (d x + c\right ) + 2 \, \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (b^{2} \arcsin \left (d x + c\right ) + a b\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.462011, size = 143, normalized size = 2.42 \begin{align*} \begin{cases} a^{2} x + \frac{2 a b c \operatorname{asin}{\left (c + d x \right )}}{d} + 2 a b x \operatorname{asin}{\left (c + d x \right )} + \frac{2 a b \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac{b^{2} c \operatorname{asin}^{2}{\left (c + d x \right )}}{d} + b^{2} x \operatorname{asin}^{2}{\left (c + d x \right )} - 2 b^{2} x + \frac{2 b^{2} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname{asin}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \operatorname{asin}{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19288, size = 150, normalized size = 2.54 \begin{align*} \frac{{\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right )^{2}}{d} + \frac{2 \,{\left (d x + c\right )} a b \arcsin \left (d x + c\right )}{d} + \frac{2 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} b^{2} \arcsin \left (d x + c\right )}{d} + \frac{{\left (d x + c\right )} a^{2}}{d} - \frac{2 \,{\left (d x + c\right )} b^{2}}{d} + \frac{2 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} a b}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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