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

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Rubi [A]  time = 0.0578785, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 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]

Rule 8

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

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*}

Mathematica [A]  time = 0.0615706, 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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Maple [A]  time = 0.046, size = 74, normalized size = 1.6 \begin{align*}{\frac{1}{c} \left ( cx{a}^{2}+{b}^{2} \left ( cx \left ( \arccos \left ( cx \right ) \right ) ^{2}-2\,cx-2\,\arccos \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1} \right ) +2\,ab \left ( cx\arccos \left ( cx \right ) -\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) } \end{align*}

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 = 1.43825, size = 99, normalized size = 2.11 \begin{align*} 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} \end{align*}

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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Fricas [A]  time = 2.41873, size = 159, normalized size = 3.38 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 0.331679, size = 87, normalized size = 1.85 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 1.14457, size = 101, normalized size = 2.15 \begin{align*} 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} \end{align*}

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