\(\int \frac {x^6}{\arccos (a x)^{3/2}} \, dx\) [99]

Optimal result

Integrand size = 12, antiderivative size = 171 \[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {5 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {9 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {\sqrt {\frac {7 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {14}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7} \] Output:

2*x^6*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(1/2)-5/32*2^(1/2)*Pi^(1/2)*Fresnel 
C(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^7-9/32*6^(1/2)*Pi^(1/2)*FresnelC(6 
^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^7-5/32*10^(1/2)*Pi^(1/2)*FresnelC(10^ 
(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^7-1/32*14^(1/2)*Pi^(1/2)*FresnelC(14^( 
1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^7
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.79 \[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\frac {i \left (-10 i \sqrt {1-a^2 x^2}+5 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )-5 \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )+9 \sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )-9 \sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )+5 \sqrt {5} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-5 i \arccos (a x)\right )-5 \sqrt {5} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},5 i \arccos (a x)\right )+\sqrt {7} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-7 i \arccos (a x)\right )-\sqrt {7} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},7 i \arccos (a x)\right )-18 i \sin (3 \arccos (a x))-10 i \sin (5 \arccos (a x))-2 i \sin (7 \arccos (a x))\right )}{64 a^7 \sqrt {\arccos (a x)}} \] Input:

Integrate[x^6/ArcCos[a*x]^(3/2),x]
 

Output:

((I/64)*((-10*I)*Sqrt[1 - a^2*x^2] + 5*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, ( 
-I)*ArcCos[a*x]] - 5*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]] + 9*Sqr 
t[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - 9*Sqrt[3]*Sqr 
t[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]] + 5*Sqrt[5]*Sqrt[(-I)*ArcCo 
s[a*x]]*Gamma[1/2, (-5*I)*ArcCos[a*x]] - 5*Sqrt[5]*Sqrt[I*ArcCos[a*x]]*Gam 
ma[1/2, (5*I)*ArcCos[a*x]] + Sqrt[7]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-7 
*I)*ArcCos[a*x]] - Sqrt[7]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (7*I)*ArcCos[a*x 
]] - (18*I)*Sin[3*ArcCos[a*x]] - (10*I)*Sin[5*ArcCos[a*x]] - (2*I)*Sin[7*A 
rcCos[a*x]]))/(a^7*Sqrt[ArcCos[a*x]])
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.96, 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.

\(\displaystyle \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx\)

\(\Big \downarrow \) 5143

\(\displaystyle \frac {2 \int \left (-\frac {5 a x}{64 \sqrt {\arccos (a x)}}-\frac {27 \cos (3 \arccos (a x))}{64 \sqrt {\arccos (a x)}}-\frac {25 \cos (5 \arccos (a x))}{64 \sqrt {\arccos (a x)}}-\frac {7 \cos (7 \arccos (a x))}{64 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a^7}+\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {5}{32} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )-\frac {9}{32} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )-\frac {5}{32} \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )-\frac {1}{32} \sqrt {\frac {7 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {14}{\pi }} \sqrt {\arccos (a x)}\right )\right )}{a^7}+\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\)

Input:

Int[x^6/ArcCos[a*x]^(3/2),x]
 

Output:

(2*x^6*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) + (2*((-5*Sqrt[Pi/2]*Fresn 
elC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/32 - (9*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/ 
Pi]*Sqrt[ArcCos[a*x]]])/32 - (5*Sqrt[(5*Pi)/2]*FresnelC[Sqrt[10/Pi]*Sqrt[A 
rcCos[a*x]]])/32 - (Sqrt[(7*Pi)/2]*FresnelC[Sqrt[14/Pi]*Sqrt[ArcCos[a*x]]] 
)/32))/a^7
 

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 = 0.21 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.07

method	result	size
default	\(-\frac {9 \sqrt {3}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+5 \sqrt {5}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+\sqrt {2}\, \sqrt {\pi }\, \sqrt {7}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {7}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arccos \left (a x \right )}+5 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-5 \sqrt {-a^{2} x^{2}+1}-9 \sin \left (3 \arccos \left (a x \right )\right )-5 \sin \left (5 \arccos \left (a x \right )\right )-\sin \left (7 \arccos \left (a x \right )\right )}{32 a^{7} \sqrt {\arccos \left (a x \right )}}\)	\(183\)

Input:

int(x^6/arccos(a*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/32/a^7*(9*3^(1/2)*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2^(1/2)/P 
i^(1/2)*3^(1/2)*arccos(a*x)^(1/2))+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))+2^(1/2)*Pi^(1/2 
)*7^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*7^(1/2)*arccos(a*x)^(1/2))*arccos(a*x) 
^(1/2)+5*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arcc 
os(a*x)^(1/2))-5*(-a^2*x^2+1)^(1/2)-9*sin(3*arccos(a*x))-5*sin(5*arccos(a* 
x))-sin(7*arccos(a*x)))/arccos(a*x)^(1/2)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^6/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]

\[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^{6}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**6/acos(a*x)**(3/2),x)
 

Output:

Integral(x**6/acos(a*x)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^6/arccos(a*x)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\int { \frac {x^{6}}{\arccos \left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^6/arccos(a*x)^(3/2),x, algorithm="giac")
 

Output:

integrate(x^6/arccos(a*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^6}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \] Input:

int(x^6/acos(a*x)^(3/2),x)
 

Output:

int(x^6/acos(a*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\frac {-\frac {12 \mathit {acos} \left (a x \right ) \left (\int \frac {\sqrt {\mathit {acos} \left (a x \right )}}{\mathit {acos} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{2}}d x \right ) a}{5}+\frac {2 \mathit {acos} \left (a x \right ) \left (\int \frac {\sqrt {\mathit {acos} \left (a x \right )}\, x^{4}}{\mathit {acos} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{5}}{5}+2 \mathit {acos} \left (a x \right ) \left (\int \frac {\sqrt {\mathit {acos} \left (a x \right )}\, x^{2}}{\mathit {acos} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{3}-14 \mathit {acos} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{7}}{\mathit {acos} \left (a x \right ) a^{2} x^{2}-\mathit {acos} \left (a x \right )}d x \right ) a^{8}+12 \mathit {acos} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{5}}{\mathit {acos} \left (a x \right ) a^{2} x^{2}-\mathit {acos} \left (a x \right )}d x \right ) a^{6}+\frac {12 \mathit {acos} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{3}}{\mathit {acos} \left (a x \right ) a^{2} x^{2}-\mathit {acos} \left (a x \right )}d x \right ) a^{4}}{5}+\frac {16 \mathit {acos} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x}{\mathit {acos} \left (a x \right ) a^{2} x^{2}-\mathit {acos} \left (a x \right )}d x \right ) a^{2}}{5}+2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, a^{6} x^{6}-\frac {4 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, a^{2} x^{2}}{5}-\frac {24 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}}{5}}{\mathit {acos} \left (a x \right ) a^{7}} \] Input:

int(x^6/acos(a*x)^(3/2),x)
 

Output:

(2*( - 6*acos(a*x)*int(sqrt(acos(a*x))/(acos(a*x)**2*a**2*x**2 - acos(a*x) 
**2),x)*a + acos(a*x)*int((sqrt(acos(a*x))*x**4)/(acos(a*x)**2*a**2*x**2 - 
 acos(a*x)**2),x)*a**5 + 5*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 - 35*acos(a*x)*int((sqrt( - a**2*x** 
2 + 1)*sqrt(acos(a*x))*x**7)/(acos(a*x)*a**2*x**2 - acos(a*x)),x)*a**8 + 3 
0*acos(a*x)*int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x**5)/(acos(a*x)*a 
**2*x**2 - acos(a*x)),x)*a**6 + 6*acos(a*x)*int((sqrt( - a**2*x**2 + 1)*sq 
rt(acos(a*x))*x**3)/(acos(a*x)*a**2*x**2 - acos(a*x)),x)*a**4 + 8*acos(a*x 
)*int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x)/(acos(a*x)*a**2*x**2 - ac 
os(a*x)),x)*a**2 + 5*sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*a**6*x**6 - 2* 
sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*a**2*x**2 - 12*sqrt( - a**2*x**2 + 
1)*sqrt(acos(a*x))))/(5*acos(a*x)*a**7)