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

Optimal. Leaf size=35 \[ \frac{\sqrt{1-(a+b x)^2}}{b}+\frac{(a+b x) \sin ^{-1}(a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.0155002, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4803, 4619, 261} \[ \frac{\sqrt{1-(a+b x)^2}}{b}+\frac{(a+b x) \sin ^{-1}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a + b*x],x]

[Out]

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

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

Mathematica [A]  time = 0.03416, size = 41, normalized size = 1.17 \[ \frac{\sqrt{-a^2-2 a b x-b^2 x^2+1}+(a+b x) \sin ^{-1}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.001, size = 31, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ( bx+a \right ) \arcsin \left ( bx+a \right ) +\sqrt{1- \left ( bx+a \right ) ^{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.46266, size = 41, normalized size = 1.17 \begin{align*} \frac{{\left (b x + a\right )} \arcsin \left (b x + a\right ) + \sqrt{-{\left (b x + a\right )}^{2} + 1}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.88263, size = 92, normalized size = 2.63 \begin{align*} \frac{{\left (b x + a\right )} \arcsin \left (b x + a\right ) + \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.186078, size = 46, normalized size = 1.31 \begin{align*} \begin{cases} \frac{a \operatorname{asin}{\left (a + b x \right )}}{b} + x \operatorname{asin}{\left (a + b x \right )} + \frac{\sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{b} & \text{for}\: b \neq 0 \\x \operatorname{asin}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17596, size = 41, normalized size = 1.17 \begin{align*} \frac{{\left (b x + a\right )} \arcsin \left (b x + a\right ) + \sqrt{-{\left (b x + a\right )}^{2} + 1}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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