3.4.68 \(\int \frac {(c+a^2 c x^2) \arctan (a x)^3}{x^2} \, dx\) [368]

3.4.68.1 Optimal result

Integrand size = 20, antiderivative size = 169 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=-\frac {c \arctan (a x)^3}{x}+a^2 c x \arctan (a x)^3+3 a c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \]

output

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

3.4.68.2 Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=-i a c \arctan (a x)^3+a^2 c x \arctan (a x)^3+3 a c \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+a c \left (-\frac {i \pi ^3}{8}+i \arctan (a x)^3-\frac {\arctan (a x)^3}{a x}+3 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+3 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right ) \]

input

Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^2,x]
 

output

(-I)*a*c*ArcTan[a*x]^3 + a^2*c*x*ArcTan[a*x]^3 + 3*a*c*ArcTan[a*x]^2*Log[1 
 + E^((2*I)*ArcTan[a*x])] - (3*I)*a*c*ArcTan[a*x]*PolyLog[2, -E^((2*I)*Arc 
Tan[a*x])] + a*c*((-1/8*I)*Pi^3 + I*ArcTan[a*x]^3 - ArcTan[a*x]^3/(a*x) + 
3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLo 
g[2, E^((-2*I)*ArcTan[a*x])] + (3*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2) + 
 (3*a*c*PolyLog[3, -E^((2*I)*ArcTan[a*x])])/2
 

3.4.68.3 Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.40, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5485, 5345, 5361, 5455, 5379, 5459, 5403, 5527, 5529, 7164}

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.

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5345

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5455

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

\(\Big \downarrow \) 5379

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 5527

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

\(\Big \downarrow \) 5529

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

\(\Big \downarrow \) 7164

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

input

Int[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^2,x]
 

output

c*(-(ArcTan[a*x]^3/x) + 3*a*((-1/3*I)*ArcTan[a*x]^3 + I*((-I)*ArcTan[a*x]^ 
2*Log[2 - 2/(1 - I*a*x)] + (2*I)*a*(((I/2)*ArcTan[a*x]*PolyLog[2, -1 + 2/( 
1 - I*a*x)])/a - PolyLog[3, -1 + 2/(1 - I*a*x)]/(4*a))))) + a^2*c*(x*ArcTa 
n[a*x]^3 - 3*a*(((-1/3*I)*ArcTan[a*x]^3)/a^2 - ((ArcTan[a*x]^2*Log[2/(1 + 
I*a*x)])/a - 2*(((-1/2*I)*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a - P 
olyLog[3, 1 - 2/(1 + I*a*x)]/(4*a)))/a))
 

3.4.68.3.1 Defintions of rubi rules used

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 5403

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ 
Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si 
mp[b*c*(p/d)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 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 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]
 

rule 5459

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

rule 5485

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

rule 5527

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

rule 5529

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

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

3.4.68.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 14.21 (sec) , antiderivative size = 1653, normalized size of antiderivative = 9.78

method	result	size
parts	\(\text {Expression too large to display}\)	\(1653\)
derivativedivides	\(\text {Expression too large to display}\)	\(1654\)
default	\(\text {Expression too large to display}\)	\(1654\)

input

int((a^2*c*x^2+c)*arctan(a*x)^3/x^2,x,method=_RETURNVERBOSE)
 

output

a^2*c*x*arctan(a*x)^3-c*arctan(a*x)^3/x-3*c*(a*arctan(a*x)^2*ln(a^2*x^2+1) 
-a*arctan(a*x)^2*ln(a*x)-2*a*(arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2) 
)-1/3*I*arctan(a*x)^3+1/4*(I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a 
*x)^2/(a^2*x^2+1)+1))^3-I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3+I*Pi*csgn(I 
*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)-I*Pi 
*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/(( 
1+I*a*x)^2/(a^2*x^2+1)+1))^2-I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x 
)^2/(a^2*x^2+1)+1)^2)^3+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*csgn(I/(( 
1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2 
/(a^2*x^2+1)+1))-I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a* 
x)^2/(a^2*x^2+1))+I*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2* 
x^2+1)+1))^3-I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2 
/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2 
)-I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1) 
-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1) 
^2)^3+I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/ 
((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2-I*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1 
+I*a*x)^2/(a^2*x^2+1)+1))^2-I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I* 
a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2* 
x^2+1)+1))^2+I*Pi+I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+...
 

3.4.68.5 Fricas [F]

\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \]

input

integrate((a^2*c*x^2+c)*arctan(a*x)^3/x^2,x, algorithm="fricas")
 

output

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

3.4.68.6 Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=c \left (\int a^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx\right ) \]

input

integrate((a**2*c*x**2+c)*atan(a*x)**3/x**2,x)
 

output

c*(Integral(a**2*atan(a*x)**3, x) + Integral(atan(a*x)**3/x**2, x))
 

3.4.68.7 Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \]

input

integrate((a^2*c*x^2+c)*arctan(a*x)^3/x^2,x, algorithm="maxima")
 

output

1/64*(8*(a^2*c*x^2 - c)*arctan(a*x)^3 - 6*(a^2*c*x^2 - c)*arctan(a*x)*log( 
a^2*x^2 + 1)^2 + (28*a*c*arctan(a*x)^4 + 1792*a^4*c*integrate(1/32*x^4*arc 
tan(a*x)^3/(a^2*x^4 + x^2), x) + 192*a^4*c*integrate(1/32*x^4*arctan(a*x)* 
log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 768*a^4*c*integrate(1/32*x^4*arct 
an(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 768*a^3*c*integrate(1/32*x^ 
3*arctan(a*x)^2/(a^2*x^4 + x^2), x) + a*c*log(a^2*x^2 + 1)^3 + 384*a^2*c*i 
ntegrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 768 
*a^2*c*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) 
 + 768*a*c*integrate(1/32*x*arctan(a*x)^2/(a^2*x^4 + x^2), x) - 192*a*c*in 
tegrate(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 1792*c*integrate(1 
/32*arctan(a*x)^3/(a^2*x^4 + x^2), x) + 192*c*integrate(1/32*arctan(a*x)*l 
og(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x))*x)/x
 

3.4.68.8 Giac [F]

\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \]

input

integrate((a^2*c*x^2+c)*arctan(a*x)^3/x^2,x, algorithm="giac")
 

output

sage0*x
 

3.4.68.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )}{x^2} \,d x \]

input

int((atan(a*x)^3*(c + a^2*c*x^2))/x^2,x)
 

output

int((atan(a*x)^3*(c + a^2*c*x^2))/x^2, x)