\(\int \frac {x^4}{\arccos (a x)^{7/2}} \, dx\) [113]

Optimal result

Integrand size = 12, antiderivative size = 264 \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a^5}+\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}-\frac {8 \sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5} \] Output:

2/5*x^4*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(5/2)-16/15*x^3/a^2/arccos(a*x)^( 
3/2)+4/3*x^5/arccos(a*x)^(3/2)+32/5*x^2*(-a^2*x^2+1)^(1/2)/a^3/arccos(a*x) 
^(1/2)-40/3*x^4*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(1/2)+1/15*2^(1/2)*Pi^(1/ 
2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^5+9/10*6^(1/2)*Pi^(1/2)* 
FresnelC(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^5+5/6*10^(1/2)*Pi^(1/2)*Fre 
snelC(10^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^5
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.32 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.58 \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=-\frac {2 \left (-6 \sqrt {1-a^2 x^2}-2 i e^{i \arccos (a x)} \arccos (a x) (-i+2 \arccos (a x))-4 (-i \arccos (a x))^{3/2} \arccos (a x) \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+e^{-i \arccos (a x)} \arccos (a x) \left (-2+4 i \arccos (a x)-4 e^{i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )\right )\right )-5 \arccos (a x) \left (2 e^{5 i \arccos (a x)} (1+10 i \arccos (a x))+20 \sqrt {5} (-i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},-5 i \arccos (a x)\right )+e^{-5 i \arccos (a x)} \left (2-20 i \arccos (a x)+20 \sqrt {5} e^{5 i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},5 i \arccos (a x)\right )\right )\right )+9 \left (-2 \arccos (a x) \left (e^{3 i \arccos (a x)} (1+6 i \arccos (a x))+6 \sqrt {3} (-i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )+e^{-3 i \arccos (a x)} \left (1-6 i \arccos (a x)+6 \sqrt {3} e^{3 i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )\right )\right )-2 \sin (3 \arccos (a x))\right )-6 \sin (5 \arccos (a x))}{240 a^5 \arccos (a x)^{5/2}} \] Input:

Integrate[x^4/ArcCos[a*x]^(7/2),x]
 

Output:

-1/240*(2*(-6*Sqrt[1 - a^2*x^2] - (2*I)*E^(I*ArcCos[a*x])*ArcCos[a*x]*(-I 
+ 2*ArcCos[a*x]) - 4*((-I)*ArcCos[a*x])^(3/2)*ArcCos[a*x]*Gamma[1/2, (-I)* 
ArcCos[a*x]] + (ArcCos[a*x]*(-2 + (4*I)*ArcCos[a*x] - 4*E^(I*ArcCos[a*x])* 
(I*ArcCos[a*x])^(3/2)*Gamma[1/2, I*ArcCos[a*x]]))/E^(I*ArcCos[a*x])) - 5*A 
rcCos[a*x]*(2*E^((5*I)*ArcCos[a*x])*(1 + (10*I)*ArcCos[a*x]) + 20*Sqrt[5]* 
((-I)*ArcCos[a*x])^(3/2)*Gamma[1/2, (-5*I)*ArcCos[a*x]] + (2 - (20*I)*ArcC 
os[a*x] + 20*Sqrt[5]*E^((5*I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2 
, (5*I)*ArcCos[a*x]])/E^((5*I)*ArcCos[a*x])) + 9*(-2*ArcCos[a*x]*(E^((3*I) 
*ArcCos[a*x])*(1 + (6*I)*ArcCos[a*x]) + 6*Sqrt[3]*((-I)*ArcCos[a*x])^(3/2) 
*Gamma[1/2, (-3*I)*ArcCos[a*x]] + (1 - (6*I)*ArcCos[a*x] + 6*Sqrt[3]*E^((3 
*I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, (3*I)*ArcCos[a*x]])/E^(( 
3*I)*ArcCos[a*x])) - 2*Sin[3*ArcCos[a*x]]) - 6*Sin[5*ArcCos[a*x]])/(a^5*Ar 
cCos[a*x]^(5/2))
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5145, 5223, 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.

\(\displaystyle \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx\)

\(\Big \downarrow \) 5145

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

\(\Big \downarrow \) 5223

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

\(\Big \downarrow \) 5143

\(\displaystyle 2 a \left (\frac {2 x^5}{3 a \arccos (a x)^{3/2}}-\frac {10 \left (\frac {2 \int \left (-\frac {a x}{8 \sqrt {\arccos (a x)}}-\frac {9 \cos (3 \arccos (a x))}{16 \sqrt {\arccos (a x)}}-\frac {5 \cos (5 \arccos (a x))}{16 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a^5}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\right )}{3 a}\right )-\frac {8 \left (\frac {2 x^3}{3 a \arccos (a x)^{3/2}}-\frac {2 \left (\frac {2 \int \left (-\frac {a x}{4 \sqrt {\arccos (a x)}}-\frac {3 \cos (3 \arccos (a x))}{4 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a^3}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\right )}{a}\right )}{5 a}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}\)

\(\Big \downarrow \) 2009

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

Input:

Int[x^4/ArcCos[a*x]^(7/2),x]
 

Output:

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

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]
 

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 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 = 0.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {-100 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-108 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-8 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}+8 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+108 \arccos \left (a x \right )^{2} \sin \left (3 \arccos \left (a x \right )\right )+100 \arccos \left (a x \right )^{2} \sin \left (5 \arccos \left (a x \right )\right )-4 a x \arccos \left (a x \right )-18 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )-10 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )-6 \sqrt {-a^{2} x^{2}+1}-9 \sin \left (3 \arccos \left (a x \right )\right )-3 \sin \left (5 \arccos \left (a x \right )\right )}{120 a^{5} \arccos \left (a x \right )^{\frac {5}{2}}}\)	\(225\)

Input:

int(x^4/arccos(a*x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/120/a^5*(-100*2^(1/2)*Pi^(1/2)*5^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2 
)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)-108*2^(1/2)*Pi^(1/2)*3^(1/2)*Fresne 
lC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)-8*2^(1/2) 
*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)+8 
*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)+108*arccos(a*x)^2*sin(3*arccos(a*x))+100 
*arccos(a*x)^2*sin(5*arccos(a*x))-4*a*x*arccos(a*x)-18*arccos(a*x)*cos(3*a 
rccos(a*x))-10*arccos(a*x)*cos(5*arccos(a*x))-6*(-a^2*x^2+1)^(1/2)-9*sin(3 
*arccos(a*x))-3*sin(5*arccos(a*x)))/arccos(a*x)^(5/2)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4/arccos(a*x)^(7/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int \frac {x^{4}}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**4/acos(a*x)**(7/2),x)
 

Output:

Integral(x**4/acos(a*x)**(7/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4/arccos(a*x)^(7/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(x^4/arccos(a*x)^(7/2),x, algorithm="giac")
 

Output:

integrate(x^4/arccos(a*x)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \] Input:

int(x^4/acos(a*x)^(7/2),x)
 

Output:

int(x^4/acos(a*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\frac {-\frac {4 \mathit {acos} \left (a x \right )^{3} \left (\int \frac {\sqrt {\mathit {acos} \left (a x \right )}}{\mathit {acos} \left (a x \right )^{4} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{4}}d x \right ) a}{3}+\frac {4 \mathit {acos} \left (a x \right )^{3} \left (\int \frac {\sqrt {\mathit {acos} \left (a x \right )}\, x^{2}}{\mathit {acos} \left (a x \right )^{4} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{4}}d x \right ) a^{3}}{3}-2 \mathit {acos} \left (a x \right )^{3} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{5}}{\mathit {acos} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{3}}d x \right ) a^{6}+\frac {8 \mathit {acos} \left (a x \right )^{3} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{3}}{\mathit {acos} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{3}}d x \right ) a^{4}}{5}+\frac {8 \mathit {acos} \left (a x \right )^{3} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x}{\mathit {acos} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{3}}d x \right ) a^{2}}{15}+\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, a^{4} x^{4}}{5}-\frac {8 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}}{15}}{\mathit {acos} \left (a x \right )^{3} a^{5}} \] Input:

int(x^4/acos(a*x)^(7/2),x)
 

Output:

(2*( - 10*acos(a*x)**3*int(sqrt(acos(a*x))/(acos(a*x)**4*a**2*x**2 - acos( 
a*x)**4),x)*a + 10*acos(a*x)**3*int((sqrt(acos(a*x))*x**2)/(acos(a*x)**4*a 
**2*x**2 - acos(a*x)**4),x)*a**3 - 15*acos(a*x)**3*int((sqrt( - a**2*x**2 
+ 1)*sqrt(acos(a*x))*x**5)/(acos(a*x)**3*a**2*x**2 - acos(a*x)**3),x)*a**6 
 + 12*acos(a*x)**3*int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x**3)/(acos 
(a*x)**3*a**2*x**2 - acos(a*x)**3),x)*a**4 + 4*acos(a*x)**3*int((sqrt( - a 
**2*x**2 + 1)*sqrt(acos(a*x))*x)/(acos(a*x)**3*a**2*x**2 - acos(a*x)**3),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))))/(15*acos(a*x)**3*a**5)