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

Optimal. Leaf size=40 \[ a x+\frac{b \sqrt{1-(c+d x)^2}}{d}+\frac{b (c+d x) \sin ^{-1}(c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.024191, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4803, 4619, 261} \[ a x+\frac{b \sqrt{1-(c+d x)^2}}{d}+\frac{b (c+d x) \sin ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcSin[c + d*x],x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0399415, size = 51, normalized size = 1.27 \[ a x+\frac{b \left (\sqrt{-c^2-2 c d x-d^2 x^2+1}+c \sin ^{-1}(c+d x)\right )}{d}+b x \sin ^{-1}(c+d x) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcSin[c + d*x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 36, normalized size = 0.9 \begin{align*} ax+{\frac{b}{d} \left ( \left ( dx+c \right ) \arcsin \left ( dx+c \right ) +\sqrt{1- \left ( dx+c \right ) ^{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42312, size = 47, normalized size = 1.18 \begin{align*} a x + \frac{{\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt{-{\left (d x + c\right )}^{2} + 1}\right )} b}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.92261, size = 111, normalized size = 2.78 \begin{align*} \frac{a d x +{\left (b d x + b c\right )} \arcsin \left (d x + c\right ) + \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} b}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.220161, size = 51, normalized size = 1.27 \begin{align*} a x + b \left (\begin{cases} \frac{c \operatorname{asin}{\left (c + d x \right )}}{d} + x \operatorname{asin}{\left (c + d x \right )} + \frac{\sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} & \text{for}\: d \neq 0 \\x \operatorname{asin}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15692, size = 47, normalized size = 1.18 \begin{align*} a x + \frac{{\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt{-{\left (d x + c\right )}^{2} + 1}\right )} b}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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