\(\int \frac {x^4}{\arccos (a x)^{5/2}} \, dx\) [107]

Optimal result

Integrand size = 12, antiderivative size = 235 \[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\arccos (a x)}}+\frac {20 x^5}{3 \sqrt {\arccos (a x)}}+\frac {25 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a^5}-\frac {4 \sqrt {\frac {2 \pi }{3}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^5} \] Output:

2/3*x^4*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(3/2)-16/3*x^3/a^2/arccos(a*x)^(1 
/2)+20/3*x^5/arccos(a*x)^(1/2)+1/6*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1 
/2)*arccos(a*x)^(1/2))/a^5+3/4*6^(1/2)*Pi^(1/2)*FresnelS(6^(1/2)/Pi^(1/2)* 
arccos(a*x)^(1/2))/a^5+5/12*10^(1/2)*Pi^(1/2)*FresnelS(10^(1/2)/Pi^(1/2)*a 
rccos(a*x)^(1/2))/a^5
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.04 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.37 \[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=-\frac {2 \left (-\sqrt {1-a^2 x^2}-e^{-i \arccos (a x)} \arccos (a x)-e^{i \arccos (a x)} \arccos (a x)+\sqrt {-i \arccos (a x)} \arccos (a x) \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \arccos (a x) \Gamma \left (\frac {1}{2},i \arccos (a x)\right )\right )-5 \arccos (a x) \left (e^{-5 i \arccos (a x)}+e^{5 i \arccos (a x)}-\sqrt {5} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-5 i \arccos (a x)\right )-\sqrt {5} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},5 i \arccos (a x)\right )\right )-3 \left (3 \arccos (a x) \left (e^{-3 i \arccos (a x)}+e^{3 i \arccos (a x)}-\sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )-\sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )\right )+\sin (3 \arccos (a x))\right )-\sin (5 \arccos (a x))}{24 a^5 \arccos (a x)^{3/2}} \] Input:

Integrate[x^4/ArcCos[a*x]^(5/2),x]
 

Output:

-1/24*(2*(-Sqrt[1 - a^2*x^2] - ArcCos[a*x]/E^(I*ArcCos[a*x]) - E^(I*ArcCos 
[a*x])*ArcCos[a*x] + Sqrt[(-I)*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, (-I)*Ar 
cCos[a*x]] + Sqrt[I*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, I*ArcCos[a*x]]) - 
5*ArcCos[a*x]*(E^((-5*I)*ArcCos[a*x]) + E^((5*I)*ArcCos[a*x]) - Sqrt[5]*Sq 
rt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-5*I)*ArcCos[a*x]] - Sqrt[5]*Sqrt[I*ArcCo 
s[a*x]]*Gamma[1/2, (5*I)*ArcCos[a*x]]) - 3*(3*ArcCos[a*x]*(E^((-3*I)*ArcCo 
s[a*x]) + E^((3*I)*ArcCos[a*x]) - Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2 
, (-3*I)*ArcCos[a*x]] - Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCo 
s[a*x]]) + Sin[3*ArcCos[a*x]]) - Sin[5*ArcCos[a*x]])/(a^5*ArcCos[a*x]^(3/2 
))
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5145, 5223, 5147, 4906, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx\)

\(\Big \downarrow \) 5145

\(\displaystyle \frac {10}{3} a \int \frac {x^5}{\sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}dx-\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}dx}{3 a}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\)

\(\Big \downarrow \) 5223

\(\displaystyle \frac {10}{3} a \left (\frac {2 x^5}{a \sqrt {\arccos (a x)}}-\frac {10 \int \frac {x^4}{\sqrt {\arccos (a x)}}dx}{a}\right )-\frac {8 \left (\frac {2 x^3}{a \sqrt {\arccos (a x)}}-\frac {6 \int \frac {x^2}{\sqrt {\arccos (a x)}}dx}{a}\right )}{3 a}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\)

\(\Big \downarrow \) 5147

\(\displaystyle -\frac {8 \left (\frac {6 \int \frac {a^2 x^2 \sqrt {1-a^2 x^2}}{\sqrt {\arccos (a x)}}d\arccos (a x)}{a^4}+\frac {2 x^3}{a \sqrt {\arccos (a x)}}\right )}{3 a}+\frac {10}{3} a \left (\frac {10 \int \frac {a^4 x^4 \sqrt {1-a^2 x^2}}{\sqrt {\arccos (a x)}}d\arccos (a x)}{a^6}+\frac {2 x^5}{a \sqrt {\arccos (a x)}}\right )+\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {10}{3} a \left (\frac {10 \int \left (\frac {3 \sin (3 \arccos (a x))}{16 \sqrt {\arccos (a x)}}+\frac {\sin (5 \arccos (a x))}{16 \sqrt {\arccos (a x)}}+\frac {\sqrt {1-a^2 x^2}}{8 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a^6}+\frac {2 x^5}{a \sqrt {\arccos (a x)}}\right )-\frac {8 \left (\frac {6 \int \left (\frac {\sin (3 \arccos (a x))}{4 \sqrt {\arccos (a x)}}+\frac {\sqrt {1-a^2 x^2}}{4 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a^4}+\frac {2 x^3}{a \sqrt {\arccos (a x)}}\right )}{3 a}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {10}{3} a \left (\frac {10 \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )\right )}{a^6}+\frac {2 x^5}{a \sqrt {\arccos (a x)}}\right )-\frac {8 \left (\frac {6 \left (\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )\right )}{a^4}+\frac {2 x^3}{a \sqrt {\arccos (a x)}}\right )}{3 a}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\)

Input:

Int[x^4/ArcCos[a*x]^(5/2),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG 
tQ[p, 0]
 

rule 5145

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[( 
-x^m)*Sqrt[1 - c^2*x^2]*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] + ( 
-Simp[c*((m + 1)/(b*(n + 1)))   Int[x^(m + 1)*((a + b*ArcCos[c*x])^(n + 1)/ 
Sqrt[1 - c^2*x^2]), x], x] + Simp[m/(b*c*(n + 1))   Int[x^(m - 1)*((a + b*A 
rcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && I 
GtQ[m, 0] && LtQ[n, -2]
 

rule 5147

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- 
(b*c^(m + 1))^(-1)   Subst[Int[x^n*Cos[-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]
 

rule 5223

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) 
+ (e_.)*(x_)^2], x_Symbol] :> Simp[(-(f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c 
^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1), x] + Simp[f*(m/(b*c*( 
n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[(f*x)^(m - 1)*(a + b 
*ArcCos[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2 
*d + e, 0] && LtQ[n, -1]
 

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.74

method	result	size
default	\(\frac {18 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+10 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+4 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+4 a x \arccos \left (a x \right )+18 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )+10 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )+2 \sqrt {-a^{2} x^{2}+1}+3 \sin \left (3 \arccos \left (a x \right )\right )+\sin \left (5 \arccos \left (a x \right )\right )}{24 a^{5} \arccos \left (a x \right )^{\frac {3}{2}}}\)	\(173\)

Input:

int(x^4/arccos(a*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/24/a^5*(18*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*ar 
ccos(a*x)^(1/2))*arccos(a*x)^(3/2)+10*2^(1/2)*Pi^(1/2)*5^(1/2)*FresnelS(2^ 
(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(3/2)+4*2^(1/2)*Pi^( 
1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(3/2)+4*a*x* 
arccos(a*x)+18*arccos(a*x)*cos(3*arccos(a*x))+10*arccos(a*x)*cos(5*arccos( 
a*x))+2*(-a^2*x^2+1)^(1/2)+3*sin(3*arccos(a*x))+sin(5*arccos(a*x)))/arccos 
(a*x)^(3/2)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4/arccos(a*x)^(5/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\int \frac {x^{4}}{\operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**4/acos(a*x)**(5/2),x)
 

Output:

Integral(x**4/acos(a*x)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4/arccos(a*x)^(5/2),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

\[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4/arccos(a*x)^(5/2),x, algorithm="giac")
 

Output:

integrate(x^4/arccos(a*x)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^{5/2}} \,d x \] Input:

int(x^4/acos(a*x)^(5/2),x)
 

Output:

int(x^4/acos(a*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\frac {-\frac {4 \mathit {acos} \left (a x \right )^{2} \left (\int \frac {\sqrt {\mathit {acos} \left (a x \right )}}{\mathit {acos} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{3}}d x \right ) a}{3}+\frac {4 \mathit {acos} \left (a x \right )^{2} \left (\int \frac {\sqrt {\mathit {acos} \left (a x \right )}\, x^{2}}{\mathit {acos} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{3}}d x \right ) a^{3}}{3}-\frac {10 \mathit {acos} \left (a x \right )^{2} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{5}}{\mathit {acos} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{6}}{3}+\frac {8 \mathit {acos} \left (a x \right )^{2} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{3}}{\mathit {acos} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{4}}{3}+\frac {8 \mathit {acos} \left (a x \right )^{2} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x}{\mathit {acos} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{2}}{9}+\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, a^{4} x^{4}}{3}-\frac {8 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}}{9}}{\mathit {acos} \left (a x \right )^{2} a^{5}} \] Input:

int(x^4/acos(a*x)^(5/2),x)
 

Output:

(2*( - 6*acos(a*x)**2*int(sqrt(acos(a*x))/(acos(a*x)**3*a**2*x**2 - acos(a 
*x)**3),x)*a + 6*acos(a*x)**2*int((sqrt(acos(a*x))*x**2)/(acos(a*x)**3*a** 
2*x**2 - acos(a*x)**3),x)*a**3 - 15*acos(a*x)**2*int((sqrt( - a**2*x**2 + 
1)*sqrt(acos(a*x))*x**5)/(acos(a*x)**2*a**2*x**2 - acos(a*x)**2),x)*a**6 + 
 12*acos(a*x)**2*int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x**3)/(acos(a 
*x)**2*a**2*x**2 - acos(a*x)**2),x)*a**4 + 4*acos(a*x)**2*int((sqrt( - a** 
2*x**2 + 1)*sqrt(acos(a*x))*x)/(acos(a*x)**2*a**2*x**2 - acos(a*x)**2),x)* 
a**2 + 3*sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*a**4*x**4 - 4*sqrt( - a**2 
*x**2 + 1)*sqrt(acos(a*x))))/(9*acos(a*x)**2*a**5)