\(\int \frac {x^4}{\arccos (a x)^{3/2}} \, dx\) [101]

Optimal result

Integrand size = 12, antiderivative size = 136 \[ \int \frac {x^4}{\arccos (a x)^{3/2}} \, dx=\frac {2 x^4 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a^5}-\frac {3 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{4 a^5}-\frac {\sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{4 a^5} \] Output:

2*x^4*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(1/2)-1/4*2^(1/2)*Pi^(1/2)*FresnelC 
(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^5-3/8*6^(1/2)*Pi^(1/2)*FresnelC(6^( 
1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^5-1/8*10^(1/2)*Pi^(1/2)*FresnelC(10^(1/ 
2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^5
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.71 \[ \int \frac {x^4}{\arccos (a x)^{3/2}} \, dx=\frac {i \left (-4 i \sqrt {1-a^2 x^2}+2 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )-2 \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )+3 \sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )-3 \sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )+\sqrt {5} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-5 i \arccos (a x)\right )-\sqrt {5} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},5 i \arccos (a x)\right )-6 i \sin (3 \arccos (a x))-2 i \sin (5 \arccos (a x))\right )}{16 a^5 \sqrt {\arccos (a x)}} \] Input:

Integrate[x^4/ArcCos[a*x]^(3/2),x]
 

Output:

((I/16)*((-4*I)*Sqrt[1 - a^2*x^2] + 2*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (- 
I)*ArcCos[a*x]] - 2*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]] + 3*Sqrt 
[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - 3*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]] - (6*I)*Sin[3*ArcCos[a*x]] - (2*I)*Sin[5*ArcCos[a*x] 
]))/(a^5*Sqrt[ArcCos[a*x]])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5143, 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.

\(\displaystyle \int \frac {x^4}{\arccos (a x)^{3/2}} \, dx\)

\(\Big \downarrow \) 5143

\(\displaystyle \frac {2 \int \left (-\frac {a x}{8 \sqrt {\arccos (a x)}}-\frac {9 \cos (3 \arccos (a x))}{16 \sqrt {\arccos (a x)}}-\frac {5 \cos (5 \arccos (a x))}{16 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a^5}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\)

\(\Big \downarrow \) 2009

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

Input:

Int[x^4/ArcCos[a*x]^(3/2),x]
 

Output:

(2*x^4*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) + (2*(-1/4*(Sqrt[Pi/2]*Fre 
snelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]]) - (3*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/P 
i]*Sqrt[ArcCos[a*x]]])/8 - (Sqrt[(5*Pi)/2]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCo 
s[a*x]]])/8))/a^5
 

Defintions of rubi rules used

rule 2009

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

rule 5143

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] - S 
imp[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 1), Cos[- 
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]
 

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.02

method	result	size
default	\(\frac {-\sqrt {5}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-3 \sqrt {3}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-2 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+2 \sqrt {-a^{2} x^{2}+1}+3 \sin \left (3 \arccos \left (a x \right )\right )+\sin \left (5 \arccos \left (a x \right )\right )}{8 a^{5} \sqrt {\arccos \left (a x \right )}}\)	\(139\)

Input:

int(x^4/arccos(a*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4/arccos(a*x)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^{4}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**4/acos(a*x)**(3/2),x)
 

Output:

Integral(x**4/acos(a*x)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4/arccos(a*x)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^4}{\arccos (a x)^{3/2}} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4/arccos(a*x)^(3/2),x, algorithm="giac")
 

Output:

integrate(x^4/arccos(a*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \] Input:

int(x^4/acos(a*x)^(3/2),x)
 

Output:

int(x^4/acos(a*x)^(3/2), x)
 

Reduce [F]

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

int(x^4/acos(a*x)^(3/2),x)
 

Output:

(2*( - 2*acos(a*x)*int(sqrt(acos(a*x))/(acos(a*x)**2*a**2*x**2 - acos(a*x) 
**2),x)*a + 2*acos(a*x)*int((sqrt(acos(a*x))*x**2)/(acos(a*x)**2*a**2*x**2 
 - acos(a*x)**2),x)*a**3 - 15*acos(a*x)*int((sqrt( - a**2*x**2 + 1)*sqrt(a 
cos(a*x))*x**5)/(acos(a*x)*a**2*x**2 - acos(a*x)),x)*a**6 + 12*acos(a*x)*i 
nt((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x**3)/(acos(a*x)*a**2*x**2 - ac 
os(a*x)),x)*a**4 + 4*acos(a*x)*int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x)) 
*x)/(acos(a*x)*a**2*x**2 - acos(a*x)),x)*a**2 + 3*sqrt( - a**2*x**2 + 1)*s 
qrt(acos(a*x))*a**4*x**4 - 4*sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))))/(3*a 
cos(a*x)*a**5)