3.1.92 \(\int \frac {x^4}{\sqrt {\text {ArcCos}(a x)}} \, dx\) [92]

Optimal. Leaf size=106 \[ -\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{4 a^5}-\frac {\sqrt {\frac {3 \pi }{2}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{8 a^5}-\frac {\sqrt {\frac {\pi }{10}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{8 a^5} \]

[Out]

-1/80*FresnelS(10^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^5-1/8*FresnelS(2^(1/2)/Pi^(1/2)*arccos
(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^5-1/16*FresnelS(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^5

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Rubi [A]
time = 0.07, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4732, 4491, 3386, 3432} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{4 a^5}-\frac {\sqrt {\frac {3 \pi }{2}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{8 a^5}-\frac {\sqrt {\frac {\pi }{10}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{8 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^5 - (Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos
[a*x]]])/(8*a^5) - (Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^5)

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[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 3432

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

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4732

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(b*c^(m + 1))^(-1), Subst[Int[x^n*C
os[-a/b + x/b]^m*Sin[-a/b + x/b], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^4}{\sqrt {\cos ^{-1}(a x)}} \, dx &=-\frac {\text {Subst}\left (\int \frac {\cos ^4(x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^5}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {\sin (x)}{8 \sqrt {x}}+\frac {3 \sin (3 x)}{16 \sqrt {x}}+\frac {\sin (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^5}\\ &=-\frac {\text {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^5}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^5}-\frac {3 \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac {\text {Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{8 a^5}-\frac {\text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{4 a^5}-\frac {3 \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{8 a^5}\\ &=-\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{4 a^5}-\frac {\sqrt {\frac {3 \pi }{2}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^5}-\frac {\sqrt {\frac {\pi }{10}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^5}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 192, normalized size = 1.81 \begin {gather*} -\frac {-10 \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-i \text {ArcCos}(a x)\right )-10 \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},i \text {ArcCos}(a x)\right )-5 \sqrt {3} \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-3 i \text {ArcCos}(a x)\right )-5 \sqrt {3} \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},3 i \text {ArcCos}(a x)\right )-\sqrt {5} \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-5 i \text {ArcCos}(a x)\right )-\sqrt {5} \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},5 i \text {ArcCos}(a x)\right )}{160 a^5 \sqrt {\text {ArcCos}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/160*(-10*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] - 10*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a
*x]] - 5*Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - 5*Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gamma[1
/2, (3*I)*ArcCos[a*x]] - Sqrt[5]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-5*I)*ArcCos[a*x]] - Sqrt[5]*Sqrt[I*ArcCos
[a*x]]*Gamma[1/2, (5*I)*ArcCos[a*x]])/(a^5*Sqrt[ArcCos[a*x]])

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Maple [A]
time = 0.22, size = 72, normalized size = 0.68

method	result	size
default	\(-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\sqrt {5}\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+5 \sqrt {3}\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+10 \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{80 a^{5}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/80/a^5*2^(1/2)*Pi^(1/2)*(5^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/2))+5*3^(1/2)*FresnelS(2^
(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))+10*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2)))

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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 \frac {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 = 0.47, size = 139, normalized size = 1.31 \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 )}{320 \, 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 )}{320 \, 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 )}{64 \, 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 )}{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 )}{32 \, 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 )}{32 \, 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]

-(1/320*I - 1/320)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(10)*sqrt(arccos(a*x)))/a^5 + (1/320*I + 1/320)*sqr
t(10)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(10)*sqrt(arccos(a*x)))/a^5 - (1/64*I - 1/64)*sqrt(6)*sqrt(pi)*erf((1/2*
I - 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^5 + (1/64*I + 1/64)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arc
cos(a*x)))/a^5 - (1/32*I - 1/32)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^5 + (1/32*I +
1/32)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^5

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {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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