3.2.81 \(\int (a+b \text {ArcSin}(c+d x)) \, dx\) [181]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4887, 4715, 267} \begin {gather*} a x+\frac {b (c+d x) \text {ArcSin}(c+d x)}{d}+\frac {b \sqrt {1-(c+d x)^2}}{d} \end {gather*}

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(159\) vs. \(2(40)=80\).
time = 0.25, size = 159, normalized size = 3.98 \begin {gather*} a x+b x \text {ArcSin}(c+d x)+\frac {b \left (2 d \sqrt {1-c^2-2 c d x-d^2 x^2}+2 c d \text {ArcTan}\left (\frac {\sqrt {-d^2} x-\sqrt {1-c^2-2 c d x-d^2 x^2}}{c}\right )+c \sqrt {-d^2} \log \left (-1+2 c d x+2 d^2 x^2+2 \sqrt {-d^2} x \sqrt {1-c^2-2 c d x-d^2 x^2}\right )\right )}{2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcSin[c + d*x],x]

[Out]

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

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Maple [A]
time = 0.06, size = 36, normalized size = 0.90

method	result	size
default	\(a x +\frac {b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\)	\(36\)
derivativedivides	\(\frac {\left (d x +c \right ) a +b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.47, size = 35, normalized size = 0.88 \begin {gather*} 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 {gather*}

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 = 3.07, size = 48, normalized size = 1.20 \begin {gather*} \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 {gather*}

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 = 0.07, size = 51, normalized size = 1.28 \begin {gather*} 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 {gather*}

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 = 0.40, size = 35, normalized size = 0.88 \begin {gather*} 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 {gather*}

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

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Mupad [B]
time = 0.61, size = 92, normalized size = 2.30 \begin {gather*} a\,x+b\,x\,\mathrm {asin}\left (c+d\,x\right )+\frac {b\,\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}}{d}+\frac {b\,c\,\ln \left (\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}-\frac {x\,d^2+c\,d}{\sqrt {-d^2}}\right )}{\sqrt {-d^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*asin(c + d*x),x)

[Out]

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

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