\(\int \arccos (a+b x) \, dx\) [27]

Optimal result

Integrand size = 6, antiderivative size = 36 \[ \int \arccos (a+b x) \, dx=-\frac {\sqrt {1-(a+b x)^2}}{b}+\frac {(a+b x) \arccos (a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4888, 4716, 267} \[ \int \arccos (a+b x) \, dx=\frac {(a+b x) \arccos (a+b x)}{b}-\frac {\sqrt {1-(a+b x)^2}}{b} \]

[In]

Int[ArcCos[a + b*x],x]

[Out]

-(Sqrt[1 - (a + b*x)^2]/b) + ((a + b*x)*ArcCos[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 4716

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[
x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4888

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int \arccos (x) \, dx,x,a+b x)}{b} \\ & = \frac {(a+b x) \arccos (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) \arccos (a+b x)}{b} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(154\) vs. \(2(36)=72\).

Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 4.28 \[ \int \arccos (a+b x) \, dx=x \arccos (a+b x)-\frac {2 b \sqrt {1-a^2-2 a b x-b^2 x^2}+2 a b \arctan \left (\frac {\sqrt {-b^2} x-\sqrt {1-a^2-2 a b x-b^2 x^2}}{a}\right )+a \sqrt {-b^2} \log \left (-1+2 a b x+2 b^2 x^2+2 \sqrt {-b^2} x \sqrt {1-a^2-2 a b x-b^2 x^2}\right )}{2 b^2} \]

[In]

Integrate[ArcCos[a + b*x],x]

[Out]

x*ArcCos[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)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\left (b x +a \right ) \arccos \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{b}\)	\(33\)
default	\(\frac {\left (b x +a \right ) \arccos \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{b}\)	\(33\)
parts	\(x \arccos \left (b x +a \right )+b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )\)	\(87\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \arccos (a+b x) \, dx=\frac {{\left (b x + a\right )} \arccos \left (b x + a\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.28 \[ \int \arccos (a+b x) \, dx=\begin {cases} \frac {a \operatorname {acos}{\left (a + b x \right )}}{b} + x \operatorname {acos}{\left (a + b x \right )} - \frac {\sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {acos}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \arccos (a+b x) \, dx=\frac {{\left (b x + a\right )} \arccos \left (b x + a\right ) - \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \arccos (a+b x) \, dx=\frac {{\left (b x + a\right )} \arccos \left (b x + a\right ) - \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.44 \[ \int \arccos (a+b x) \, dx=x\,\mathrm {acos}\left (a+b\,x\right )-\frac {\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}}{b}-\frac {a\,\ln \left (\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}-\frac {x\,b^2+a\,b}{\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]

[In]

int(acos(a + b*x),x)

[Out]

x*acos(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)