\(\int \frac {x^5 \arctan (a x)^2}{(c+a^2 c x^2)^{5/2}} \, dx\) [347]

Optimal result

Integrand size = 24, antiderivative size = 400 \[ \int \frac {x^5 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2}{27 a^6 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {32}{9 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^3 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {10 x \arctan (a x)}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^2}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{a^6 c^3}+\frac {4 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}} \] Output:

2/27/a^6/c/(a^2*c*x^2+c)^(3/2)-32/9/a^6/c^2/(a^2*c*x^2+c)^(1/2)-2/9*x^3*ar 
ctan(a*x)/a^3/c/(a^2*c*x^2+c)^(3/2)-10/3*x*arctan(a*x)/a^5/c^2/(a^2*c*x^2+ 
c)^(1/2)+1/3*x^2*arctan(a*x)^2/a^4/c/(a^2*c*x^2+c)^(3/2)+5/3*arctan(a*x)^2 
/a^6/c^2/(a^2*c*x^2+c)^(1/2)+(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/a^6/c^3+4*I 
*(a^2*x^2+1)^(1/2)*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/a^6 
/c^2/(a^2*c*x^2+c)^(1/2)-2*I*(a^2*x^2+1)^(1/2)*polylog(2,-I*(1+I*a*x)^(1/2 
)/(1-I*a*x)^(1/2))/a^6/c^2/(a^2*c*x^2+c)^(1/2)+2*I*(a^2*x^2+1)^(1/2)*polyl 
og(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/a^6/c^2/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.05 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.57 \[ \int \frac {x^5 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {8 (-95+\cos (2 \arctan (a x)))-9 \left (1+a^2 x^2\right ) \arctan (a x)^2 (-45-20 \cos (2 \arctan (a x))+\cos (4 \arctan (a x)))-432 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+432 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+6 \arctan (a x) \left (-124 a x-72 \sqrt {1+a^2 x^2} \log \left (1-i e^{i \arctan (a x)}\right )+72 \sqrt {1+a^2 x^2} \log \left (1+i e^{i \arctan (a x)}\right )+\left (1+a^2 x^2\right ) \sin (4 \arctan (a x))\right )}{216 a^6 c^2 \sqrt {c+a^2 c x^2}} \] Input:

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

Output:

(8*(-95 + Cos[2*ArcTan[a*x]]) - 9*(1 + a^2*x^2)*ArcTan[a*x]^2*(-45 - 20*Co 
s[2*ArcTan[a*x]] + Cos[4*ArcTan[a*x]]) - (432*I)*Sqrt[1 + a^2*x^2]*PolyLog 
[2, (-I)*E^(I*ArcTan[a*x])] + (432*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, I*E^(I* 
ArcTan[a*x])] + 6*ArcTan[a*x]*(-124*a*x - 72*Sqrt[1 + a^2*x^2]*Log[1 - I*E 
^(I*ArcTan[a*x])] + 72*Sqrt[1 + a^2*x^2]*Log[1 + I*E^(I*ArcTan[a*x])] + (1 
 + a^2*x^2)*Sin[4*ArcTan[a*x]]))/(216*a^6*c^2*Sqrt[c + a^2*c*x^2])
 

Rubi [A] (verified)

Time = 2.54 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.20, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5499, 5475, 243, 53, 2009, 5465, 5429, 5499, 5465, 5425, 5421, 5429}

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.

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

\(\Big \downarrow \) 5499

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

\(\Big \downarrow \) 5475

\(\displaystyle \frac {\int \frac {x^3 \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {2}{9} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2}}dx-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\int \frac {x^3 \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {1}{9} \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2}}dx^2-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {\int \frac {x^3 \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {1}{9} \int \left (\frac {1}{a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {1}{a^2 \left (a^2 c x^2+c\right )^{5/2}}\right )dx^2-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\int \frac {x^3 \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a^2}\)

\(\Big \downarrow \) 5465

\(\displaystyle \frac {\int \frac {x^3 \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {2 \left (\frac {2 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a^2}\)

\(\Big \downarrow \) 5429

\(\displaystyle \frac {\int \frac {x^3 \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a^2}\)

\(\Big \downarrow \) 5499

\(\displaystyle \frac {\frac {\int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a^2 c}-\frac {\int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}}{a^2 c}-\frac {-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a^2}\)

\(\Big \downarrow \) 5465

\(\displaystyle \frac {\frac {\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}}{a^2 c}-\frac {\frac {2 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a^2}\)

\(\Big \downarrow \) 5425

\(\displaystyle \frac {\frac {\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {\frac {2 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a^2}\)

\(\Big \downarrow \) 5421

\(\displaystyle -\frac {-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a^2}+\frac {-\frac {\frac {2 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}+\frac {\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}}{a^2 c}}{a^2 c}\)

\(\Big \downarrow \) 5429

\(\displaystyle -\frac {-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a^2}+\frac {-\frac {\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}}{a^2}+\frac {\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}}{a^2 c}}{a^2 c}\)

Input:

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

Output:

-(((-2/(3*a^4*c*(c + a^2*c*x^2)^(3/2)) + 2/(a^4*c^2*Sqrt[c + a^2*c*x^2]))/ 
9 + (2*x^3*ArcTan[a*x])/(9*a*c*(c + a^2*c*x^2)^(3/2)) - (x^2*ArcTan[a*x]^2 
)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) + (2*(-(ArcTan[a*x]^2/(a^2*c*Sqrt[c + a^ 
2*c*x^2])) + (2*(1/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(c*Sqrt[c + 
 a^2*c*x^2])))/a))/(3*a^2*c))/a^2) + (-((-(ArcTan[a*x]^2/(a^2*c*Sqrt[c + a 
^2*c*x^2])) + (2*(1/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(c*Sqrt[c 
+ a^2*c*x^2])))/a)/a^2) + ((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^2*c) - ( 
2*Sqrt[1 + a^2*x^2]*(((-2*I)*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I 
*a*x]])/a + (I*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a - (I* 
PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a))/(a*Sqrt[c + a^2*c*x^2 
]))/(a^2*c))/(a^2*c)
 

Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

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

rule 5421

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] 
 :> Simp[-2*I*(a + b*ArcTan[c*x])*(ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/ 
(c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c 
*x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I 
*c*x])]/(c*Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 
GtQ[d, 0]
 

rule 5425

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

rule 5429

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbo 
l] :> Simp[b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTan[c*x])/(d*Sqr 
t[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
 

rule 5465

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

rule 5475

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

rule 5499

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

Maple [A] (verified)

Time = 2.33 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.14

method	result	size
default	\(\frac {\left (6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right ) \left (i a^{3} x^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} a^{6} c^{3}}+\frac {7 \left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} a^{6} \left (a^{2} x^{2}+1\right )}-\frac {7 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{8 c^{3} a^{6} \left (a^{2} x^{2}+1\right )}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{3} x^{3}-3 a^{2} x^{2}-3 i a x +1\right ) \left (-6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right )}{216 c^{3} a^{6} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{a^{6} c^{3}}+\frac {2 \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{3} a^{6}}\)	\(454\)

Input:

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

Output:

1/216*(6*I*arctan(a*x)+9*arctan(a*x)^2-2)*(I*a^3*x^3+3*a^2*x^2-3*I*a*x-1)* 
(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)^2/a^6/c^3+7/8*(arctan(a*x)^2-2+2*I*a 
rctan(a*x))*(1+I*a*x)*(c*(a*x-I)*(a*x+I))^(1/2)/c^3/a^6/(a^2*x^2+1)-7/8*(c 
*(a*x-I)*(a*x+I))^(1/2)*(I*a*x-1)*(arctan(a*x)^2-2-2*I*arctan(a*x))/c^3/a^ 
6/(a^2*x^2+1)-1/216*(c*(a*x-I)*(a*x+I))^(1/2)*(I*a^3*x^3-3*a^2*x^2-3*I*a*x 
+1)*(-6*I*arctan(a*x)+9*arctan(a*x)^2-2)/c^3/a^6/(a^4*x^4+2*a^2*x^2+1)+arc 
tan(a*x)^2*(c*(a*x-I)*(a*x+I))^(1/2)/a^6/c^3+2*(arctan(a*x)*ln(1+I*(1+I*a* 
x)/(a^2*x^2+1)^(1/2))-arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*di 
log(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/ 
2)))*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)^(1/2)/c^3/a^6
                                                                                    
                                                                                    
 

Fricas [F]

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

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

Output:

integral(sqrt(a^2*c*x^2 + c)*x^5*arctan(a*x)^2/(a^6*c^3*x^6 + 3*a^4*c^3*x^ 
4 + 3*a^2*c^3*x^2 + c^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^5 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\int \frac {\mathit {atan} \left (a x \right )^{2} x^{5}}{\sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}+2 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+\sqrt {a^{2} x^{2}+1}}d x}{\sqrt {c}\, c^{2}} \] Input:

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

Output:

int((atan(a*x)**2*x**5)/(sqrt(a**2*x**2 + 1)*a**4*x**4 + 2*sqrt(a**2*x**2 
+ 1)*a**2*x**2 + sqrt(a**2*x**2 + 1)),x)/(sqrt(c)*c**2)