3.1.74 \(\int x^4 \sqrt {\text {ArcCos}(a x)} \, dx\) [74]

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

[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 = 0.18, 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 = {4726, 4810, 3393, 3385, 3433} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{8 a^5}-\frac {\sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{16 a^5}-\frac {\sqrt {\frac {\pi }{10}} \text {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{80 a^5}+\frac {1}{5} x^5 \sqrt {\text {ArcCos}(a x)} \end {gather*}

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 3385

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 3393

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 3433

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

Rule 4726

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 4810

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(-(b*c^
(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1),
x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGt
Q[m, 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 {\text {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 {\text {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 {\text {Subst}\left (\int \frac {\cos (5 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{160 a^5}-\frac {\text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^5}-\frac {\text {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 {\text {Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{80 a^5}-\frac {\text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{16 a^5}-\frac {\text {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] Result contains complex when optimal does not.
time = 0.20, size = 212, normalized size = 1.75 \begin {gather*} -\frac {-150 \sqrt {i \text {ArcCos}(a x)} \sqrt {\text {ArcCos}(a x)^2} \text {Gamma}\left (\frac {3}{2},-i \text {ArcCos}(a x)\right )-150 \sqrt {-i \text {ArcCos}(a x)} \sqrt {\text {ArcCos}(a x)^2} \text {Gamma}\left (\frac {3}{2},i \text {ArcCos}(a x)\right )+25 \sqrt {3} (-i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {3}{2},-3 i \text {ArcCos}(a x)\right )+25 \sqrt {3} (i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {3}{2},3 i \text {ArcCos}(a x)\right )+3 \sqrt {5} (-i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {3}{2},-5 i \text {ArcCos}(a x)\right )+3 \sqrt {5} (i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {3}{2},5 i \text {ArcCos}(a x)\right )}{2400 a^5 \text {ArcCos}(a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[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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Maple [A]
time = 0.42, size = 143, normalized size = 1.18

method	result	size
default	\(\frac {-3 \sqrt {5}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-25 \sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-150 \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }+300 a x \arccos \left (a x \right )+150 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )+30 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )}{2400 a^{5} \sqrt {\arccos \left (a x \right )}}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*arccos(a*x)^(1/2)*Pi^(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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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 [C] Result contains complex when optimal does not.
time = 0.49, size = 247, normalized size = 2.04 \begin {gather*} \frac {\left (i + 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arccos \left (a x\right )}\right )}{3200 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arccos \left (a x\right )}\right )}{3200 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{384 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{384 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{5}} + \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/3200*I + 1/3200)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(10)*sqrt(arccos(a*x)))/a^5 - (1/3200*I - 1/3200)*
sqrt(10)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(10)*sqrt(arccos(a*x)))/a^5 + (1/384*I + 1/384)*sqrt(6)*sqrt(pi)*erf(
(1/2*I - 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^5 - (1/384*I - 1/384)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*s
qrt(arccos(a*x)))/a^5 + (1/64*I + 1/64)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^5 - (1
/64*I - 1/64)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^5 + 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 + 1/1
60*sqrt(arccos(a*x))*e^(-5*I*arccos(a*x))/a^5

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

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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