3.2.25 \(\int \text {ArcSin}(a+b x) \, dx\) [125]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4887, 4715, 267} \begin {gather*} \frac {(a+b x) \text {ArcSin}(a+b x)}{b}+\frac {\sqrt {1-(a+b x)^2}}{b} \end {gather*}

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 267

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 4715

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 4887

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]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(154\) vs. \(2(35)=70\).
time = 0.23, size = 154, normalized size = 4.40 \begin {gather*} x \text {ArcSin}(a+b x)+\frac {2 b \sqrt {1-a^2-2 a b x-b^2 x^2}+2 a b \text {ArcTan}\left (\frac {\sqrt {-b^2} x-\sqrt {1-a^2-2 a b x-b^2 x^2}}{a}\right )+a \sqrt {-b^2} \log \left (-1+2 a b x+2 b^2 x^2+2 \sqrt {-b^2} x \sqrt {1-a^2-2 a b x-b^2 x^2}\right )}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a + b*x],x]

[Out]

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

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Maple [A]
time = 0.01, size = 31, normalized size = 0.89

method	result	size
derivativedivides	\(\frac {\left (b x +a \right ) \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}}{b}\)	\(31\)
default	\(\frac {\left (b x +a \right ) \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}}{b}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.47, size = 30, normalized size = 0.86 \begin {gather*} \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right ) + \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \end {gather*}

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 = 2.86, size = 39, normalized size = 1.11 \begin {gather*} \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 {gather*}

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 = 0.07, size = 46, normalized size = 1.31 \begin {gather*} \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 {gather*}

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 = 0.40, size = 30, normalized size = 0.86 \begin {gather*} \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right ) + \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \end {gather*}

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

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Mupad [B]
time = 0.56, size = 86, normalized size = 2.46 \begin {gather*} x\,\mathrm {asin}\left (a+b\,x\right )+\frac {\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}}{b}+\frac {a\,\ln \left (\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}-\frac {x\,b^2+a\,b}{\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a + b*x),x)

[Out]

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

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