3.6.91 \(\int \frac {x}{(c+a^2 c x^2)^{5/2} \arctan (a x)^2} \, dx\) [591]

3.6.91.1 Optimal result

Integrand size = 22, antiderivative size = 116 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=-\frac {x}{a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{4 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{4 a^2 c^2 \sqrt {c+a^2 c x^2}} \]

output

-x/a/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)+1/4*Ci(arctan(a*x))*(a^2*x^2+1)^(1/ 
2)/a^2/c^2/(a^2*c*x^2+c)^(1/2)+3/4*Ci(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/a^2 
/c^2/(a^2*c*x^2+c)^(1/2)
 

3.6.91.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\frac {-4 a x+\left (1+a^2 x^2\right )^{3/2} \arctan (a x) \operatorname {CosIntegral}(\arctan (a x))+3 \left (1+a^2 x^2\right )^{3/2} \arctan (a x) \operatorname {CosIntegral}(3 \arctan (a x))}{4 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)} \]

input

Integrate[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]
 

output

(-4*a*x + (1 + a^2*x^2)^(3/2)*ArcTan[a*x]*CosIntegral[ArcTan[a*x]] + 3*(1 
+ a^2*x^2)^(3/2)*ArcTan[a*x]*CosIntegral[3*ArcTan[a*x]])/(4*a^2*c^2*(1 + a 
^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])
 

3.6.91.3 Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.22, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5503, 5440, 5439, 3042, 3793, 2009, 5506, 5505, 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.

\(\displaystyle \int \frac {x}{\arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5503

\(\displaystyle \frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx}{a}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5440

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{a c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5439

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^3}{\arctan (a x)}d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \left (\frac {\cos (3 \arctan (a x))}{4 \arctan (a x)}+\frac {3}{4 \sqrt {a^2 x^2+1} \arctan (a x)}\right )d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5506

\(\displaystyle -\frac {2 a \sqrt {a^2 x^2+1} \int \frac {x^2}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5505

\(\displaystyle -\frac {2 \sqrt {a^2 x^2+1} \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {2 \sqrt {a^2 x^2+1} \int \left (\frac {1}{4 \sqrt {a^2 x^2+1} \arctan (a x)}-\frac {\cos (3 \arctan (a x))}{4 \arctan (a x)}\right )d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \operatorname {CosIntegral}(\arctan (a x))-\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\)

input

Int[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]
 

output

-(x/(a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (2*Sqrt[1 + a^2*x^2]*(CosIn 
tegral[ArcTan[a*x]]/4 - CosIntegral[3*ArcTan[a*x]]/4))/(a^2*c^2*Sqrt[c + a 
^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*((3*CosIntegral[ArcTan[a*x]])/4 + CosInteg 
ral[3*ArcTan[a*x]]/4))/(a^2*c^2*Sqrt[c + a^2*c*x^2])
 

3.6.91.3.1 Defintions of rubi rules used

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 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

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 5439

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ 
Symbol] :> Simp[d^q/c   Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar 
cTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*( 
q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
 

rule 5440

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ 
Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2])   Int[(1 + 
c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && 
 EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] &&  !(IntegerQ[q] || GtQ[d, 0])
 

rule 5503

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[x^m*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p 
+ 1)/(b*c*d*(p + 1))), x] + (-Simp[c*((m + 2*q + 2)/(b*(p + 1)))   Int[x^(m 
 + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] - Simp[m/(b*c*(p + 
1))   Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x]) /; F 
reeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] & 
& LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
 

rule 5505

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1)   Subst[Int[(a + b*x)^p*(Sin[x]^m/ 
Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p 
}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q 
] || GtQ[d, 0])
 

rule 5506

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]) 
  Int[x^m*(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, 
d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] &&  !(I 
ntegerQ[q] || GtQ[d, 0])
 

3.6.91.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 7.33 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.59

method	result	size
default	\(-\frac {\left (3 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+\arctan \left (a x \right ) \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{4} x^{4}+3 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+\arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{4} x^{4}+6 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{2} x^{2}+2 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}+6 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{2} x^{2}+2 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}+8 \sqrt {a^{2} x^{2}+1}\, a x +3 \,\operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+3 \,\operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) a^{2} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\)	\(301\)

input

int(x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x,method=_RETURNVERBOSE)
 

output

-1/8*(3*arctan(a*x)*Ei(1,3*I*arctan(a*x))*a^4*x^4+arctan(a*x)*Ei(1,I*arcta 
n(a*x))*a^4*x^4+3*arctan(a*x)*Ei(1,-3*I*arctan(a*x))*a^4*x^4+arctan(a*x)*E 
i(1,-I*arctan(a*x))*a^4*x^4+6*arctan(a*x)*Ei(1,3*I*arctan(a*x))*a^2*x^2+2* 
arctan(a*x)*Ei(1,I*arctan(a*x))*a^2*x^2+6*arctan(a*x)*Ei(1,-3*I*arctan(a*x 
))*a^2*x^2+2*arctan(a*x)*Ei(1,-I*arctan(a*x))*a^2*x^2+8*(a^2*x^2+1)^(1/2)* 
a*x+3*Ei(1,3*I*arctan(a*x))*arctan(a*x)+Ei(1,I*arctan(a*x))*arctan(a*x)+3* 
Ei(1,-3*I*arctan(a*x))*arctan(a*x)+Ei(1,-I*arctan(a*x))*arctan(a*x))/(a^2* 
x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)/arctan(a*x)/a^2/c^3/(a^4*x^4+2*a^2* 
x^2+1)
 

3.6.91.5 Fricas [F]

\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]

input

integrate(x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="fricas")
 

output

integral(sqrt(a^2*c*x^2 + c)*x/((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x 
^2 + c^3)*arctan(a*x)^2), x)
 

3.6.91.6 Sympy [F]

\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]

input

integrate(x/(a**2*c*x**2+c)**(5/2)/atan(a*x)**2,x)
 

output

Integral(x/((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**2), x)
 

3.6.91.7 Maxima [F]

\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]

input

integrate(x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="maxima")
 

output

integrate(x/((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^2), x)
 

3.6.91.8 Giac [F(-2)]

Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\text {Exception raised: TypeError} \]

input

integrate(x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.6.91.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

input

int(x/(atan(a*x)^2*(c + a^2*c*x^2)^(5/2)),x)
 

output

int(x/(atan(a*x)^2*(c + a^2*c*x^2)^(5/2)), x)