3.77 \(\int x \sqrt{\cos ^{-1}(a x)} \, dx\)

Optimal. Leaf size=59 \[ -\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a^2}-\frac{\sqrt{\cos ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\cos ^{-1}(a x)} \]

[Out]

-Sqrt[ArcCos[a*x]]/(4*a^2) + (x^2*Sqrt[ArcCos[a*x]])/2 - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(
8*a^2)

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Rubi [A]  time = 0.150872, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4630, 4724, 3312, 3304, 3352} \[ -\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a^2}-\frac{\sqrt{\cos ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\cos ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[ArcCos[a*x]],x]

[Out]

-Sqrt[ArcCos[a*x]]/(4*a^2) + (x^2*Sqrt[ArcCos[a*x]])/2 - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(
8*a^2)

Rule 4630

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

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

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

Mathematica [A]  time = 0.0364452, size = 49, normalized size = 0.83 \[ \frac{\frac{1}{4} \sqrt{\cos ^{-1}(a x)} \cos \left (2 \cos ^{-1}(a x)\right )-\frac{1}{8} \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[ArcCos[a*x]],x]

[Out]

((Sqrt[ArcCos[a*x]]*Cos[2*ArcCos[a*x]])/4 - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/8)/a^2

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Maple [A]  time = 0.066, size = 42, normalized size = 0.7 \begin{align*} -{\frac{1}{8\,{a}^{2}\sqrt{\pi }} \left ( \pi \,{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -2\,\sqrt{\pi }\sqrt{\arccos \left ( ax \right ) }\cos \left ( 2\,\arccos \left ( ax \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arccos(a*x)^(1/2),x)

[Out]

-1/8/a^2/Pi^(1/2)*(Pi*FresnelC(2*arccos(a*x)^(1/2)/Pi^(1/2))-2*Pi^(1/2)*arccos(a*x)^(1/2)*cos(2*arccos(a*x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccos(a*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccos(a*x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\operatorname{acos}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*acos(a*x)**(1/2),x)

[Out]

Integral(x*sqrt(acos(a*x)), x)

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Giac [B]  time = 1.2481, size = 120, normalized size = 2.03 \begin{align*} \frac{\sqrt{\pi } i \operatorname{erf}\left (-{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{16 \, a^{2}{\left (i - 1\right )}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (2 \, i \arccos \left (a x\right )\right )}}{8 \, a^{2}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (-2 \, i \arccos \left (a x\right )\right )}}{8 \, a^{2}} - \frac{\sqrt{\pi } \operatorname{erf}\left ({\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{16 \, a^{2}{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccos(a*x)^(1/2),x, algorithm="giac")

[Out]

1/16*sqrt(pi)*i*erf(-(i + 1)*sqrt(arccos(a*x)))/(a^2*(i - 1)) + 1/8*sqrt(arccos(a*x))*e^(2*i*arccos(a*x))/a^2
+ 1/8*sqrt(arccos(a*x))*e^(-2*i*arccos(a*x))/a^2 - 1/16*sqrt(pi)*erf((i - 1)*sqrt(arccos(a*x)))/(a^2*(i - 1))