3.74 \(\int x^4 \sqrt{\cos ^{-1}(a x)} \, dx\)

Optimal. Leaf size=121 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^5}-\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^5}-\frac{\sqrt{\frac{\pi }{10}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{80 a^5}+\frac{1}{5} x^5 \sqrt{\cos ^{-1}(a x)} \]

[Out]

(x^5*Sqrt[ArcCos[a*x]])/5 - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^5) - (Sqrt[Pi/6]*FresnelC
[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^5) - (Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(80*a^5)

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Rubi [A]  time = 0.281232, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4630, 4724, 3312, 3304, 3352} \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^5}-\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^5}-\frac{\sqrt{\frac{\pi }{10}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{80 a^5}+\frac{1}{5} x^5 \sqrt{\cos ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^4*Sqrt[ArcCos[a*x]],x]

[Out]

(x^5*Sqrt[ArcCos[a*x]])/5 - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^5) - (Sqrt[Pi/6]*FresnelC
[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^5) - (Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(80*a^5)

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^4 \sqrt{\cos ^{-1}(a x)} \, dx &=\frac{1}{5} x^5 \sqrt{\cos ^{-1}(a x)}+\frac{1}{10} a \int \frac{x^5}{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}} \, dx\\ &=\frac{1}{5} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos ^5(x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{10 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{5 \cos (x)}{8 \sqrt{x}}+\frac{5 \cos (3 x)}{16 \sqrt{x}}+\frac{\cos (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{10 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (5 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{160 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{80 a^5}-\frac{\operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{16 a^5}-\frac{\operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{8 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\cos ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^5}-\frac{\sqrt{\frac{\pi }{6}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^5}-\frac{\sqrt{\frac{\pi }{10}} C\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{80 a^5}\\ \end{align*}

Mathematica [C]  time = 0.266099, size = 212, normalized size = 1.75 \[ -\frac{25 \sqrt{3} \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{3}{2},-3 i \cos ^{-1}(a x)\right )+3 \sqrt{5} \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{3}{2},-5 i \cos ^{-1}(a x)\right )-150 \sqrt{\cos ^{-1}(a x)^2} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},i \cos ^{-1}(a x)\right )-150 \sqrt{i \cos ^{-1}(a x)} \sqrt{\cos ^{-1}(a x)^2} \text{Gamma}\left (\frac{3}{2},-i \cos ^{-1}(a x)\right )+25 \sqrt{3} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{3}{2},3 i \cos ^{-1}(a x)\right )+3 \sqrt{5} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{3}{2},5 i \cos ^{-1}(a x)\right )}{2400 a^5 \cos ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4*Sqrt[ArcCos[a*x]],x]

[Out]

-(-150*Sqrt[I*ArcCos[a*x]]*Sqrt[ArcCos[a*x]^2]*Gamma[3/2, (-I)*ArcCos[a*x]] - 150*Sqrt[(-I)*ArcCos[a*x]]*Sqrt[
ArcCos[a*x]^2]*Gamma[3/2, I*ArcCos[a*x]] + 25*Sqrt[3]*((-I)*ArcCos[a*x])^(3/2)*Gamma[3/2, (-3*I)*ArcCos[a*x]]
+ 25*Sqrt[3]*(I*ArcCos[a*x])^(3/2)*Gamma[3/2, (3*I)*ArcCos[a*x]] + 3*Sqrt[5]*((-I)*ArcCos[a*x])^(3/2)*Gamma[3/
2, (-5*I)*ArcCos[a*x]] + 3*Sqrt[5]*(I*ArcCos[a*x])^(3/2)*Gamma[3/2, (5*I)*ArcCos[a*x]])/(2400*a^5*ArcCos[a*x]^
(3/2))

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Maple [A]  time = 0.096, size = 143, normalized size = 1.2 \begin{align*}{\frac{1}{2400\,{a}^{5}} \left ( -3\,\sqrt{5}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{5}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -25\,\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -150\,\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +300\,ax\arccos \left ( ax \right ) +150\,\arccos \left ( ax \right ) \cos \left ( 3\,\arccos \left ( ax \right ) \right ) +30\,\arccos \left ( ax \right ) \cos \left ( 5\,\arccos \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arccos \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccos(a*x)^(1/2),x)

[Out]

1/2400/a^5/arccos(a*x)^(1/2)*(-3*5^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*
arccos(a*x)^(1/2))-25*3^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)
^(1/2))-150*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))+300*a*x*arccos(a*x
)+150*arccos(a*x)*cos(3*arccos(a*x))+30*arccos(a*x)*cos(5*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^4*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^4*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^{4} \sqrt{\operatorname{acos}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acos(a*x)**(1/2),x)

[Out]

Integral(x**4*sqrt(acos(a*x)), x)

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Giac [B]  time = 1.32936, size = 425, normalized size = 3.51 \begin{align*} \frac{\sqrt{10} \sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{10} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{1600 \, a^{5}{\left (i - 1\right )}} + \frac{\sqrt{6} \sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{6} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{192 \, a^{5}{\left (i - 1\right )}} + \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{32 \, a^{5}{\left (i - 1\right )}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (5 \, i \arccos \left (a x\right )\right )}}{160 \, a^{5}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (3 \, i \arccos \left (a x\right )\right )}}{32 \, a^{5}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (i \arccos \left (a x\right )\right )}}{16 \, a^{5}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (-i \arccos \left (a x\right )\right )}}{16 \, a^{5}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (-3 \, i \arccos \left (a x\right )\right )}}{32 \, a^{5}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (-5 \, i \arccos \left (a x\right )\right )}}{160 \, a^{5}} - \frac{\sqrt{10} \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{10} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{1600 \, a^{5}{\left (i - 1\right )}} - \frac{\sqrt{6} \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{6} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{192 \, a^{5}{\left (i - 1\right )}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{32 \, a^{5}{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccos(a*x)^(1/2),x, algorithm="giac")

[Out]

1/1600*sqrt(10)*sqrt(pi)*i*erf(sqrt(10)*sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1)) + 1/192*sqrt(6)*sqrt(pi)*i*er
f(sqrt(6)*sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1)) + 1/32*sqrt(2)*sqrt(pi)*i*erf(sqrt(2)*sqrt(arccos(a*x))/(i
- 1))/(a^5*(i - 1)) + 1/160*sqrt(arccos(a*x))*e^(5*i*arccos(a*x))/a^5 + 1/32*sqrt(arccos(a*x))*e^(3*i*arccos(a
*x))/a^5 + 1/16*sqrt(arccos(a*x))*e^(i*arccos(a*x))/a^5 + 1/16*sqrt(arccos(a*x))*e^(-i*arccos(a*x))/a^5 + 1/32
*sqrt(arccos(a*x))*e^(-3*i*arccos(a*x))/a^5 + 1/160*sqrt(arccos(a*x))*e^(-5*i*arccos(a*x))/a^5 - 1/1600*sqrt(1
0)*sqrt(pi)*erf(-sqrt(10)*i*sqrt(arccos(a*x))/(i - 1))/(a^5*(i - 1)) - 1/192*sqrt(6)*sqrt(pi)*erf(-sqrt(6)*i*s
qrt(arccos(a*x))/(i - 1))/(a^5*(i - 1)) - 1/32*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*i*sqrt(arccos(a*x))/(i - 1))/(a^5
*(i - 1))