3.150 \(\int (a+b \cos ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=47 \[ -\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{c}+x \left (a+b \cos ^{-1}(c x)\right )^2-2 b^2 x \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4620, 4678, 8} \[ -\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{c}+x \left (a+b \cos ^{-1}(c x)\right )^2-2 b^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[c*x])^2,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4620

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 4678

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcCos[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 76, normalized size = 1.62 \[ x \left (a^2-2 b^2\right )-\frac {2 a b \sqrt {1-c^2 x^2}}{c}+\frac {2 b \cos ^{-1}(c x) \left (a c x-b \sqrt {1-c^2 x^2}\right )}{c}+b^2 x \cos ^{-1}(c x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCos[c*x])^2,x]

[Out]

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

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fricas [A]  time = 0.43, size = 65, normalized size = 1.38 \[ \frac {b^{2} c x \arccos \left (c x\right )^{2} + 2 \, a b c x \arccos \left (c x\right ) + {\left (a^{2} - 2 \, b^{2}\right )} c x - 2 \, \sqrt {-c^{2} x^{2} + 1} {\left (b^{2} \arccos \left (c x\right ) + a b\right )}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 75, normalized size = 1.60 \[ b^{2} x \arccos \left (c x\right )^{2} + 2 \, a b x \arccos \left (c x\right ) + a^{2} x - 2 \, b^{2} x - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} \arccos \left (c x\right )}{c} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 74, normalized size = 1.57 \[ \frac {c x \,a^{2}+b^{2} \left (c x \arccos \left (c x \right )^{2}-2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+2 a b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccos(c*x))^2,x)

[Out]

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

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maxima [A]  time = 0.42, size = 73, normalized size = 1.55 \[ b^{2} x \arccos \left (c x\right )^{2} - 2 \, b^{2} {\left (x + \frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )}{c}\right )} + a^{2} x + \frac {2 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} a b}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.46, size = 96, normalized size = 2.04 \[ \left \{\begin {array}{cl} x\,\left (a^2+\pi \,a\,b+\frac {\pi ^2\,b^2}{4}\right ) & \text {\ if\ \ }c=0\\ a^2\,x+b^2\,x\,\left ({\mathrm {acos}\left (c\,x\right )}^2-2\right )-\frac {2\,b^2\,\mathrm {acos}\left (c\,x\right )\,\sqrt {1-c^2\,x^2}}{c}-\frac {2\,a\,b\,\left (\sqrt {1-c^2\,x^2}-c\,x\,\mathrm {acos}\left (c\,x\right )\right )}{c} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acos(c*x))^2,x)

[Out]

piecewise(c == 0, x*(a^2 + (b^2*pi^2)/4 + a*b*pi), c ~= 0, a^2*x + b^2*x*(acos(c*x)^2 - 2) - (2*b^2*acos(c*x)*
(- c^2*x^2 + 1)^(1/2))/c - (2*a*b*((- c^2*x^2 + 1)^(1/2) - c*x*acos(c*x)))/c)

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sympy [A]  time = 0.26, size = 87, normalized size = 1.85 \[ \begin {cases} a^{2} x + 2 a b x \operatorname {acos}{\left (c x \right )} - \frac {2 a b \sqrt {- c^{2} x^{2} + 1}}{c} + b^{2} x \operatorname {acos}^{2}{\left (c x \right )} - 2 b^{2} x - \frac {2 b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{c} & \text {for}\: c \neq 0 \\x \left (a + \frac {\pi b}{2}\right )^{2} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acos(c*x))**2,x)

[Out]

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

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