\(\int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx\) [282]

Optimal result

Integrand size = 22, antiderivative size = 166 \[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {x}{3 a^4 c}-\frac {\arctan (a x)}{3 a^5 c}-\frac {x^2 \arctan (a x)}{3 a^3 c}-\frac {4 i \arctan (a x)^2}{3 a^5 c}-\frac {x \arctan (a x)^2}{a^4 c}+\frac {x^3 \arctan (a x)^2}{3 a^2 c}+\frac {\arctan (a x)^3}{3 a^5 c}-\frac {8 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^5 c}-\frac {4 i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{3 a^5 c} \] Output:

1/3*x/a^4/c-1/3*arctan(a*x)/a^5/c-1/3*x^2*arctan(a*x)/a^3/c-4/3*I*arctan(a 
*x)^2/a^5/c-x*arctan(a*x)^2/a^4/c+1/3*x^3*arctan(a*x)^2/a^2/c+1/3*arctan(a 
*x)^3/a^5/c-8/3*arctan(a*x)*ln(2/(1+I*a*x))/a^5/c-4/3*I*polylog(2,1-2/(1+I 
*a*x))/a^5/c
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.54 \[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {a x+\left (4 i-3 a x+a^3 x^3\right ) \arctan (a x)^2+\arctan (a x)^3-\arctan (a x) \left (1+a^2 x^2+8 \log \left (1+e^{2 i \arctan (a x)}\right )\right )+4 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )}{3 a^5 c} \] Input:

Integrate[(x^4*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]
 

Output:

(a*x + (4*I - 3*a*x + a^3*x^3)*ArcTan[a*x]^2 + ArcTan[a*x]^3 - ArcTan[a*x] 
*(1 + a^2*x^2 + 8*Log[1 + E^((2*I)*ArcTan[a*x])]) + (4*I)*PolyLog[2, -E^(( 
2*I)*ArcTan[a*x])])/(3*a^5*c)
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.46, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5451, 27, 5361, 5451, 5345, 5361, 262, 216, 5419, 5455, 5379, 2849, 2752}

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.

\(\displaystyle \int \frac {x^4 \arctan (a x)^2}{a^2 c x^2+c} \, dx\)

\(\Big \downarrow \) 5451

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int x^2 \arctan (a x)^2dx}{a^2 c}-\frac {\int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2 c}\)

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5451

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

\(\Big \downarrow \) 5345

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 5419

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

\(\Big \downarrow \) 5455

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

\(\Big \downarrow \) 5379

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

\(\Big \downarrow \) 2849

\(\displaystyle \frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2}\right )}{a^2 c}-\frac {-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )}{a^2}}{a^2 c}\)

\(\Big \downarrow \) 2752

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

Input:

Int[(x^4*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 5345

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
+ b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcTan[c*x^n])^(p 
 - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && 
 (EqQ[n, 1] || EqQ[p, 1])
 

rule 5361

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> 
 Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 
1))   Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], 
x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & 
& IntegerQ[m])) && NeQ[m, -1]
 

rule 5379

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

rule 5419

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

rule 5451

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

rule 5455

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

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.35

method	result	size
derivativedivides	\(\frac {\frac {\arctan \left (a x \right )^{2} a^{3} x^{3}}{3 c}-\frac {\arctan \left (a x \right )^{2} a x}{c}+\frac {\arctan \left (a x \right )^{3}}{c}-\frac {2 \left (\frac {x^{2} a^{2} \arctan \left (a x \right )}{2}-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a x}{2}+\frac {\arctan \left (a x \right )}{2}-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )+\arctan \left (a x \right )^{3}\right )}{3 c}}{a^{5}}\)	\(224\)
default	\(\frac {\frac {\arctan \left (a x \right )^{2} a^{3} x^{3}}{3 c}-\frac {\arctan \left (a x \right )^{2} a x}{c}+\frac {\arctan \left (a x \right )^{3}}{c}-\frac {2 \left (\frac {x^{2} a^{2} \arctan \left (a x \right )}{2}-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a x}{2}+\frac {\arctan \left (a x \right )}{2}-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )+\arctan \left (a x \right )^{3}\right )}{3 c}}{a^{5}}\)	\(224\)
parts	\(\frac {x^{3} \arctan \left (a x \right )^{2}}{3 a^{2} c}-\frac {x \arctan \left (a x \right )^{2}}{a^{4} c}+\frac {\arctan \left (a x \right )^{3}}{a^{5} c}-\frac {2 \left (\frac {\arctan \left (a x \right )^{3}}{3 a^{5}}+\frac {\frac {x^{2} a^{2} \arctan \left (a x \right )}{2}-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a x}{2}+\frac {\arctan \left (a x \right )}{2}-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{3 a^{5}}\right )}{c}\)	\(236\)

Input:

int(x^4*arctan(a*x)^2/(a^2*c*x^2+c),x,method=_RETURNVERBOSE)
 

Output:

1/a^5*(1/3/c*arctan(a*x)^2*a^3*x^3-1/c*arctan(a*x)^2*a*x+1/c*arctan(a*x)^3 
-2/3/c*(1/2*x^2*a^2*arctan(a*x)-2*arctan(a*x)*ln(a^2*x^2+1)-1/2*a*x+1/2*ar 
ctan(a*x)-I*(ln(a*x-I)*ln(a^2*x^2+1)-1/2*ln(a*x-I)^2-dilog(-1/2*I*(a*x+I)) 
-ln(a*x-I)*ln(-1/2*I*(a*x+I)))+I*(ln(a*x+I)*ln(a^2*x^2+1)-1/2*ln(a*x+I)^2- 
dilog(1/2*I*(a*x-I))-ln(a*x+I)*ln(1/2*I*(a*x-I)))+arctan(a*x)^3))
 

Fricas [F]

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

integrate(x^4*arctan(a*x)^2/(a^2*c*x^2+c),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(x**4*atan(a*x)**2/(a**2*c*x**2+c),x)
                                                                                    
                                                                                    
 

Output:

Integral(x**4*atan(a*x)**2/(a**2*x**2 + 1), x)/c
 

Maxima [F]

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

integrate(x^4*arctan(a*x)^2/(a^2*c*x^2+c),x, algorithm="maxima")
 

Output:

1/48*(4*(432*a^4*integrate(1/48*x^4*arctan(a*x)^2/(a^6*c*x^2 + a^4*c), x) 
+ 36*a^4*integrate(1/48*x^4*log(a^2*x^2 + 1)^2/(a^6*c*x^2 + a^4*c), x) + 4 
8*a^4*integrate(1/48*x^4*log(a^2*x^2 + 1)/(a^6*c*x^2 + a^4*c), x) - 96*a^3 
*integrate(1/48*x^3*arctan(a*x)/(a^6*c*x^2 + a^4*c), x) - 144*a^2*integrat 
e(1/48*x^2*log(a^2*x^2 + 1)/(a^6*c*x^2 + a^4*c), x) + 288*a*integrate(1/48 
*x*arctan(a*x)/(a^6*c*x^2 + a^4*c), x) - arctan(a*x)^3/(a^5*c) - 36*integr 
ate(1/48*log(a^2*x^2 + 1)^2/(a^6*c*x^2 + a^4*c), x))*a^5*c + 4*(a^3*x^3 - 
3*a*x)*arctan(a*x)^2 + 8*arctan(a*x)^3 - (a^3*x^3 - 3*a*x)*log(a^2*x^2 + 1 
)^2)/(a^5*c)
 

Giac [F]

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

integrate(x^4*arctan(a*x)^2/(a^2*c*x^2+c),x, algorithm="giac")
 

Output:

integrate(x^4*arctan(a*x)^2/(a^2*c*x^2 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\int \frac {x^4\,{\mathrm {atan}\left (a\,x\right )}^2}{c\,a^2\,x^2+c} \,d x \] Input:

int((x^4*atan(a*x)^2)/(c + a^2*c*x^2),x)
 

Output:

int((x^4*atan(a*x)^2)/(c + a^2*c*x^2), x)
 

Reduce [F]

\[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {\mathit {atan} \left (a x \right )^{3}+\mathit {atan} \left (a x \right )^{2} a^{3} x^{3}-3 \mathit {atan} \left (a x \right )^{2} a x -\mathit {atan} \left (a x \right ) a^{2} x^{2}-\mathit {atan} \left (a x \right )+8 \left (\int \frac {\mathit {atan} \left (a x \right ) x}{a^{2} x^{2}+1}d x \right ) a^{2}+a x}{3 a^{5} c} \] Input:

int(x^4*atan(a*x)^2/(a^2*c*x^2+c),x)
 

Output:

(atan(a*x)**3 + atan(a*x)**2*a**3*x**3 - 3*atan(a*x)**2*a*x - atan(a*x)*a* 
*2*x**2 - atan(a*x) + 8*int((atan(a*x)*x)/(a**2*x**2 + 1),x)*a**2 + a*x)/( 
3*a**5*c)