3.1.100 \(\int \frac {x^5}{\text {ArcCos}(a x)^{3/2}} \, dx\) [100]

Optimal. Leaf size=127 \[ \frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\text {ArcCos}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a^6}-\frac {\sqrt {3 \pi } \text {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{8 a^6}-\frac {5 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{8 a^6} \]

[Out]

-1/2*FresnelC(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^6-5/8*FresnelC(2*arccos(a*x)^(1/2)/Pi^(
1/2))*Pi^(1/2)/a^6-1/8*FresnelC(2*3^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^6+2*x^5*(-a^2*x^2+1)^
(1/2)/a/arccos(a*x)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4728, 3385, 3433} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a^6}-\frac {\sqrt {3 \pi } \text {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{8 a^6}-\frac {5 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{8 a^6}+\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\text {ArcCos}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/ArcCos[a*x]^(3/2),x]

[Out]

(2*x^5*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^6 -
(Sqrt[3*Pi]*FresnelC[2*Sqrt[3/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^6) - (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqr
t[Pi]])/(8*a^6)

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 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 4728

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(-x^m)*Sqrt[1 - c^2*x^2]*((a + b*Arc
Cos[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), C
os[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cos[-a/b + x/b]^2), x], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c},
x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.35, size = 226, normalized size = 1.78 \begin {gather*} \frac {i \left (5 \sqrt {2} \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcCos}(a x)\right )-5 \sqrt {2} \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},2 i \text {ArcCos}(a x)\right )+8 \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-4 i \text {ArcCos}(a x)\right )-8 \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},4 i \text {ArcCos}(a x)\right )+\sqrt {6} \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-6 i \text {ArcCos}(a x)\right )-\sqrt {6} \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},6 i \text {ArcCos}(a x)\right )-10 i \sin (2 \text {ArcCos}(a x))-8 i \sin (4 \text {ArcCos}(a x))-2 i \sin (6 \text {ArcCos}(a x))\right )}{32 a^6 \sqrt {\text {ArcCos}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/ArcCos[a*x]^(3/2),x]

[Out]

((I/32)*(5*Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-2*I)*ArcCos[a*x]] - 5*Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma
[1/2, (2*I)*ArcCos[a*x]] + 8*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-4*I)*ArcCos[a*x]] - 8*Sqrt[I*ArcCos[a*x]]*Gam
ma[1/2, (4*I)*ArcCos[a*x]] + Sqrt[6]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-6*I)*ArcCos[a*x]] - Sqrt[6]*Sqrt[I*Ar
cCos[a*x]]*Gamma[1/2, (6*I)*ArcCos[a*x]] - (10*I)*Sin[2*ArcCos[a*x]] - (8*I)*Sin[4*ArcCos[a*x]] - (2*I)*Sin[6*
ArcCos[a*x]]))/(a^6*Sqrt[ArcCos[a*x]])

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Maple [A]
time = 0.27, size = 121, normalized size = 0.95

method	result	size
default	\(\frac {-2 \sqrt {\pi }\, \sqrt {3}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {6}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arccos \left (a x \right )}-8 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-10 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+\sin \left (6 \arccos \left (a x \right )\right )+5 \sin \left (2 \arccos \left (a x \right )\right )+4 \sin \left (4 \arccos \left (a x \right )\right )}{16 a^{6} \sqrt {\arccos \left (a x \right )}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/arccos(a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/16/a^6/arccos(a*x)^(1/2)*(-2*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*6^(1/2)*arccos(a*x)^(1/2))*arccos(a*
x)^(1/2)-8*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))-10*arccos(a*x)^(1
/2)*Pi^(1/2)*FresnelC(2*arccos(a*x)^(1/2)/Pi^(1/2))+sin(6*arccos(a*x))+5*sin(2*arccos(a*x))+4*sin(4*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^5/arccos(a*x)^(3/2),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^5/arccos(a*x)^(3/2),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^{5}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/acos(a*x)**(3/2),x)

[Out]

Integral(x**5/acos(a*x)**(3/2), x)

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Giac [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^5/arccos(a*x)^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^5}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/acos(a*x)^(3/2),x)

[Out]

int(x^5/acos(a*x)^(3/2), x)

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