\(\int \frac {x^3}{\arccos (a x)^{5/2}} \, dx\) [108]

Optimal result

Integrand size = 12, antiderivative size = 126 \[ \int \frac {x^3}{\arccos (a x)^{5/2}} \, dx=\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\arccos (a x)}}+\frac {16 x^4}{3 \sqrt {\arccos (a x)}}+\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^4}+\frac {4 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{3 a^4} \] Output:

2/3*x^3*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(3/2)-4*x^2/a^2/arccos(a*x)^(1/2) 
+16/3*x^4/arccos(a*x)^(1/2)+4/3*2^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/ 
2)*arccos(a*x)^(1/2))/a^4+4/3*Pi^(1/2)*FresnelS(2*arccos(a*x)^(1/2)/Pi^(1/ 
2))/a^4
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.61 \[ \int \frac {x^3}{\arccos (a x)^{5/2}} \, dx=-\frac {-4 \arccos (a x) \left (e^{-4 i \arccos (a x)}+e^{4 i \arccos (a x)}-2 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-4 i \arccos (a x)\right )-2 \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},4 i \arccos (a x)\right )\right )-2 \left (2 \arccos (a x) \left (e^{-2 i \arccos (a x)}+e^{2 i \arccos (a x)}-\sqrt {2} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-2 i \arccos (a x)\right )-\sqrt {2} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},2 i \arccos (a x)\right )\right )+\sin (2 \arccos (a x))\right )-\sin (4 \arccos (a x))}{12 a^4 \arccos (a x)^{3/2}} \] Input:

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

Output:

-1/12*(-4*ArcCos[a*x]*(E^((-4*I)*ArcCos[a*x]) + E^((4*I)*ArcCos[a*x]) - 2* 
Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-4*I)*ArcCos[a*x]] - 2*Sqrt[I*ArcCos[a* 
x]]*Gamma[1/2, (4*I)*ArcCos[a*x]]) - 2*(2*ArcCos[a*x]*(E^((-2*I)*ArcCos[a* 
x]) + E^((2*I)*ArcCos[a*x]) - Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (- 
2*I)*ArcCos[a*x]] - Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (2*I)*ArcCos[a* 
x]]) + Sin[2*ArcCos[a*x]]) - Sin[4*ArcCos[a*x]])/(a^4*ArcCos[a*x]^(3/2))
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.33, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5145, 5223, 5147, 4906, 27, 2009, 3042, 3786, 3832}

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.

\(\displaystyle \int \frac {x^3}{\arccos (a x)^{5/2}} \, dx\)

\(\Big \downarrow \) 5145

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

\(\Big \downarrow \) 5223

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

\(\Big \downarrow \) 5147

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

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

Input:

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

Output:

(2*x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) - (2*((2*x^2)/(a*Sqrt[Ar 
cCos[a*x]]) + (2*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/a^3))/ 
a + (8*a*((2*x^4)/(a*Sqrt[ArcCos[a*x]]) + (8*((Sqrt[Pi/2]*FresnelS[2*Sqrt[ 
2/Pi]*Sqrt[ArcCos[a*x]]])/8 + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqr 
t[Pi]])/4))/a^5))/3
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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 3832

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 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 = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {16 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+16 \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+8 \cos \left (2 \arccos \left (a x \right )\right ) \arccos \left (a x \right )+8 \arccos \left (a x \right ) \cos \left (4 \arccos \left (a x \right )\right )+\sin \left (4 \arccos \left (a x \right )\right )+2 \sin \left (2 \arccos \left (a x \right )\right )}{12 a^{4} \arccos \left (a x \right )^{\frac {3}{2}}}\)	\(107\)

Input:

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

Output:

1/12/a^4*(16*2^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2 
))*arccos(a*x)^(3/2)+16*Pi^(1/2)*FresnelS(2*arccos(a*x)^(1/2)/Pi^(1/2))*ar 
ccos(a*x)^(3/2)+8*cos(2*arccos(a*x))*arccos(a*x)+8*arccos(a*x)*cos(4*arcco 
s(a*x))+sin(4*arccos(a*x))+2*sin(2*arccos(a*x)))/arccos(a*x)^(3/2)
                                                                                    
                                                                                    
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3/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]

\[ \int \frac {x^3}{\arccos (a x)^{5/2}} \, dx=\int \frac {x^{3}}{\operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^3/arccos(a*x)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\arccos (a x)^{5/2}} \, dx=\int \frac {x^3}{{\mathrm {acos}\left (a\,x\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^3}{\arccos (a x)^{5/2}} \, dx=\frac {-9 \mathit {acos} \left (a x \right )^{2} \left (\int \frac {\sqrt {\mathit {acos} \left (a x \right )}\, x^{3}}{\mathit {acos} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{3}}d x \right ) a^{4}+9 \mathit {acos} \left (a x \right )^{2} \left (\int \frac {\sqrt {\mathit {acos} \left (a x \right )}\, x}{\mathit {acos} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{3}}d x \right ) a^{2}-16 \mathit {acos} \left (a x \right )^{2} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{4}}{\mathit {acos} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{5}-12 \sqrt {\mathit {acos} \left (a x \right )}\, \mathit {acos} \left (a x \right )+4 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, a^{3} x^{3}+6 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, a x}{6 \mathit {acos} \left (a x \right )^{2} a^{4}} \] Input:

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

Output:

( - 9*acos(a*x)**2*int((sqrt(acos(a*x))*x**3)/(acos(a*x)**3*a**2*x**2 - ac 
os(a*x)**3),x)*a**4 + 9*acos(a*x)**2*int((sqrt(acos(a*x))*x)/(acos(a*x)**3 
*a**2*x**2 - acos(a*x)**3),x)*a**2 - 16*acos(a*x)**2*int((sqrt( - a**2*x** 
2 + 1)*sqrt(acos(a*x))*x**4)/(acos(a*x)**2*a**2*x**2 - acos(a*x)**2),x)*a* 
*5 - 12*sqrt(acos(a*x))*acos(a*x) + 4*sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x 
))*a**3*x**3 + 6*sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*a*x)/(6*acos(a*x)* 
*2*a**4)