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

Optimal. Leaf size=63 \[ -\frac{4 b \sqrt{2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )}{d x}+x \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^2-8 b^2 x \]

[Out]

-8*b^2*x - (4*b*Sqrt[2*d*x^2 - d^2*x^4]*(a + b*ArcCos[-1 + d*x^2]))/(d*x) + x*(a + b*ArcCos[-1 + d*x^2])^2

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Rubi [A]  time = 0.0115222, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4815, 8} \[ -\frac{4 b \sqrt{2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )}{d x}+x \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^2-8 b^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[-1 + d*x^2])^2,x]

[Out]

-8*b^2*x - (4*b*Sqrt[2*d*x^2 - d^2*x^4]*(a + b*ArcCos[-1 + d*x^2]))/(d*x) + x*(a + b*ArcCos[-1 + d*x^2])^2

Rule 4815

Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[x*(a + b*ArcCos[c + d*x^2])^n, x] + (-
Dist[4*b^2*n*(n - 1), Int[(a + b*ArcCos[c + d*x^2])^(n - 2), x], x] - Simp[(2*b*n*Sqrt[-2*c*d*x^2 - d^2*x^4]*(
a + b*ArcCos[c + d*x^2])^(n - 1))/(d*x), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]

Rule 8

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

Rubi steps

\begin{align*} \int \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^2 \, dx &=-\frac{4 b \sqrt{2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )}{d x}+x \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^2-\left (8 b^2\right ) \int 1 \, dx\\ &=-8 b^2 x-\frac{4 b \sqrt{2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )}{d x}+x \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^2\\ \end{align*}

Mathematica [A]  time = 0.0658086, size = 98, normalized size = 1.56 \[ x \left (a^2-8 b^2\right )-\frac{4 a b \sqrt{-d x^2 \left (d x^2-2\right )}}{d x}+\frac{2 b \cos ^{-1}\left (d x^2-1\right ) \left (a d x^2-2 b \sqrt{-d x^2 \left (d x^2-2\right )}\right )}{d x}+b^2 x \cos ^{-1}\left (d x^2-1\right )^2 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCos[-1 + d*x^2])^2,x]

[Out]

(a^2 - 8*b^2)*x - (4*a*b*Sqrt[-(d*x^2*(-2 + d*x^2))])/(d*x) + (2*b*(a*d*x^2 - 2*b*Sqrt[-(d*x^2*(-2 + d*x^2))])
*ArcCos[-1 + d*x^2])/(d*x) + b^2*x*ArcCos[-1 + d*x^2]^2

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Maple [F]  time = 0.118, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\arccos \left ( d{x}^{2}-1 \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccos(d*x^2-1))^2,x)

[Out]

int((a+b*arccos(d*x^2-1))^2,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(d*x^2-1))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.50507, size = 205, normalized size = 3.25 \begin{align*} \frac{b^{2} d x^{2} \arccos \left (d x^{2} - 1\right )^{2} + 2 \, a b d x^{2} \arccos \left (d x^{2} - 1\right ) +{\left (a^{2} - 8 \, b^{2}\right )} d x^{2} - 4 \, \sqrt{-d^{2} x^{4} + 2 \, d x^{2}}{\left (b^{2} \arccos \left (d x^{2} - 1\right ) + a b\right )}}{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(d*x^2-1))^2,x, algorithm="fricas")

[Out]

(b^2*d*x^2*arccos(d*x^2 - 1)^2 + 2*a*b*d*x^2*arccos(d*x^2 - 1) + (a^2 - 8*b^2)*d*x^2 - 4*sqrt(-d^2*x^4 + 2*d*x
^2)*(b^2*arccos(d*x^2 - 1) + a*b))/(d*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acos}{\left (d x^{2} - 1 \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acos(d*x**2-1))**2,x)

[Out]

Integral((a + b*acos(d*x**2 - 1))**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arccos \left (d x^{2} - 1\right ) + a\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(d*x^2-1))^2,x, algorithm="giac")

[Out]

integrate((b*arccos(d*x^2 - 1) + a)^2, x)