3.71 \(\int x^3 \cos ^{-1}(a+b x^4) \, dx\)

Optimal. Leaf size=47 \[ \frac{\left (a+b x^4\right ) \cos ^{-1}\left (a+b x^4\right )}{4 b}-\frac{\sqrt{1-\left (a+b x^4\right )^2}}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0514806, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6715, 4804, 4620, 261} \[ \frac{\left (a+b x^4\right ) \cos ^{-1}\left (a+b x^4\right )}{4 b}-\frac{\sqrt{1-\left (a+b x^4\right )^2}}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCos[a + b*x^4],x]

[Out]

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

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 4804

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]

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 261

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]

Rubi steps

\begin{align*} \int x^3 \cos ^{-1}\left (a+b x^4\right ) \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \cos ^{-1}(a+b x) \, dx,x,x^4\right )\\ &=\frac{\operatorname{Subst}\left (\int \cos ^{-1}(x) \, dx,x,a+b x^4\right )}{4 b}\\ &=\frac{\left (a+b x^4\right ) \cos ^{-1}\left (a+b x^4\right )}{4 b}+\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2}} \, dx,x,a+b x^4\right )}{4 b}\\ &=-\frac{\sqrt{1-\left (a+b x^4\right )^2}}{4 b}+\frac{\left (a+b x^4\right ) \cos ^{-1}\left (a+b x^4\right )}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0276867, size = 43, normalized size = 0.91 \[ \frac{\left (a+b x^4\right ) \cos ^{-1}\left (a+b x^4\right )-\sqrt{1-\left (a+b x^4\right )^2}}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcCos[a + b*x^4],x]

[Out]

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

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Maple [A]  time = 0.003, size = 40, normalized size = 0.9 \begin{align*}{\frac{1}{4\,b} \left ( \left ( b{x}^{4}+a \right ) \arccos \left ( b{x}^{4}+a \right ) -\sqrt{1- \left ( b{x}^{4}+a \right ) ^{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccos(b*x^4+a),x)

[Out]

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

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Maxima [A]  time = 1.61154, size = 53, normalized size = 1.13 \begin{align*} \frac{{\left (b x^{4} + a\right )} \arccos \left (b x^{4} + a\right ) - \sqrt{-{\left (b x^{4} + a\right )}^{2} + 1}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(b*x^4+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.43861, size = 105, normalized size = 2.23 \begin{align*} \frac{{\left (b x^{4} + a\right )} \arccos \left (b x^{4} + a\right ) - \sqrt{-b^{2} x^{8} - 2 \, a b x^{4} - a^{2} + 1}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(b*x^4+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.14307, size = 61, normalized size = 1.3 \begin{align*} \begin{cases} \frac{a \operatorname{acos}{\left (a + b x^{4} \right )}}{4 b} + \frac{x^{4} \operatorname{acos}{\left (a + b x^{4} \right )}}{4} - \frac{\sqrt{- a^{2} - 2 a b x^{4} - b^{2} x^{8} + 1}}{4 b} & \text{for}\: b \neq 0 \\\frac{x^{4} \operatorname{acos}{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acos(b*x**4+a),x)

[Out]

Piecewise((a*acos(a + b*x**4)/(4*b) + x**4*acos(a + b*x**4)/4 - sqrt(-a**2 - 2*a*b*x**4 - b**2*x**8 + 1)/(4*b)
, Ne(b, 0)), (x**4*acos(a)/4, True))

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Giac [A]  time = 1.26132, size = 53, normalized size = 1.13 \begin{align*} \frac{{\left (b x^{4} + a\right )} \arccos \left (b x^{4} + a\right ) - \sqrt{-{\left (b x^{4} + a\right )}^{2} + 1}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(b*x^4+a),x, algorithm="giac")

[Out]

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