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

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

[Out]

-1/800*FresnelC(10^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^5-1/96*FresnelC(6^(1/2)/Pi^(1/2)*arcc
os(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^5-1/16*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^5+1/5
*x^5*arccos(a*x)^(1/2)

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Rubi [A]  time = 0.28, antiderivative size = 121, normalized size of antiderivative = 1.00, 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 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 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 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]

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])

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*}

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Mathematica [C]  time = 0.32, size = 212, normalized size = 1.75 \[ -\frac {25 \sqrt {3} \left (-i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {3}{2},-3 i \cos ^{-1}(a x)\right )+3 \sqrt {5} \left (-i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {3}{2},-5 i \cos ^{-1}(a x)\right )-150 \sqrt {\cos ^{-1}(a x)^2} \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {3}{2},i \cos ^{-1}(a x)\right )-150 \sqrt {i \cos ^{-1}(a x)} \sqrt {\cos ^{-1}(a x)^2} \Gamma \left (\frac {3}{2},-i \cos ^{-1}(a x)\right )+25 \sqrt {3} \left (i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {3}{2},3 i \cos ^{-1}(a x)\right )+3 \sqrt {5} \left (i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {3}{2},5 i \cos ^{-1}(a x)\right )}{2400 a^5 \cos ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2400*(-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)*G
amma[3/2, (-5*I)*ArcCos[a*x]] + 3*Sqrt[5]*(I*ArcCos[a*x])^(3/2)*Gamma[3/2, (5*I)*ArcCos[a*x]])/(a^5*ArcCos[a*x
]^(3/2))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [B]  time = 0.54, size = 315, normalized size = 2.60 \[ \frac {\sqrt {10} \sqrt {\pi } i \operatorname {erf}\left (\frac {\sqrt {10} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{1600 \, a^{5} {\left (i - 1\right )}} + \frac {\sqrt {6} \sqrt {\pi } i \operatorname {erf}\left (\frac {\sqrt {6} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{192 \, a^{5} {\left (i - 1\right )}} + \frac {\sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{32 \, a^{5} {\left (i - 1\right )}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (5 \, i \arccos \left (a x\right )\right )}}{160 \, a^{5}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (3 \, i \arccos \left (a x\right )\right )}}{32 \, a^{5}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (i \arccos \left (a x\right )\right )}}{16 \, a^{5}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-i \arccos \left (a x\right )\right )}}{16 \, a^{5}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-3 \, i \arccos \left (a x\right )\right )}}{32 \, a^{5}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-5 \, i \arccos \left (a x\right )\right )}}{160 \, a^{5}} - \frac {\sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {10} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{1600 \, a^{5} {\left (i - 1\right )}} - \frac {\sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {6} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{192 \, a^{5} {\left (i - 1\right )}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {2} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{32 \, a^{5} {\left (i - 1\right )}} \]

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))

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maple [A]  time = 0.31, size = 143, normalized size = 1.18 \[ \frac {-3 \sqrt {5}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-25 \sqrt {3}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-150 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+300 a x \arccos \left (a x \right )+150 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )+30 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )}{2400 a^{5} \sqrt {\arccos \left (a x \right )}} \]

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)*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*
arccos(a*x)^(1/2))-25*3^(1/2)*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)
^(1/2))-150*2^(1/2)*Pi^(1/2)*arccos(a*x)^(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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,\sqrt {\mathrm {acos}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*acos(a*x)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \sqrt {\operatorname {acos}{\left (a x \right )}}\, dx \]

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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