3.4.70 \(\int \frac {(c+a^2 c x^2) \arctan (a x)^3}{x^4} \, dx\) [370]

3.4.70.1 Optimal result

Integrand size = 20, antiderivative size = 189 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^4} \, dx=-\frac {a^2 c \arctan (a x)}{x}-\frac {1}{2} a^3 c \arctan (a x)^2-\frac {a c \arctan (a x)^2}{2 x^2}-\frac {2}{3} i a^3 c \arctan (a x)^3-\frac {c \arctan (a x)^3}{3 x^3}-\frac {a^2 c \arctan (a x)^3}{x}+a^3 c \log (x)-\frac {1}{2} a^3 c \log \left (1+a^2 x^2\right )+2 a^3 c \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-2 i a^3 c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+a^3 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right ) \]

output

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

3.4.70.2 Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^4} \, dx=\frac {1}{12} c \left (-i a^3 \pi ^3-\frac {12 a^2 \arctan (a x)}{x}-6 a^3 \arctan (a x)^2-\frac {6 a \arctan (a x)^2}{x^2}+8 i a^3 \arctan (a x)^3-\frac {4 \arctan (a x)^3}{x^3}-\frac {12 a^2 \arctan (a x)^3}{x}+24 a^3 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+12 a^3 \log (a x)-6 a^3 \log \left (1+a^2 x^2\right )+24 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+12 a^3 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )\right ) \]

input

Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^4,x]
 

output

(c*((-I)*a^3*Pi^3 - (12*a^2*ArcTan[a*x])/x - 6*a^3*ArcTan[a*x]^2 - (6*a*Ar 
cTan[a*x]^2)/x^2 + (8*I)*a^3*ArcTan[a*x]^3 - (4*ArcTan[a*x]^3)/x^3 - (12*a 
^2*ArcTan[a*x]^3)/x + 24*a^3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] 
 + 12*a^3*Log[a*x] - 6*a^3*Log[1 + a^2*x^2] + (24*I)*a^3*ArcTan[a*x]*PolyL 
og[2, E^((-2*I)*ArcTan[a*x])] + 12*a^3*PolyLog[3, E^((-2*I)*ArcTan[a*x])]) 
)/12
 

3.4.70.3 Rubi [A] (verified)

Time = 2.00 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.61, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5485, 5361, 5453, 5361, 5453, 5361, 243, 47, 14, 16, 5419, 5459, 5403, 5527, 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.

\(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )}{x^4} \, dx\)

\(\Big \downarrow \) 5485

\(\displaystyle a^2 c \int \frac {\arctan (a x)^3}{x^2}dx+c \int \frac {\arctan (a x)^3}{x^4}dx\)

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 5419

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 5527

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

\(\Big \downarrow \) 7164

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

input

Int[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^4,x]
 

output

a^2*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))))) + c*(-1/3*Ar 
cTan[a*x]^3/x^3 + a*(-1/2*ArcTan[a*x]^2/x^2 + a*(-(ArcTan[a*x]/x) - (a*Arc 
Tan[a*x]^2)/2 + (a*(Log[x^2] - Log[1 + a^2*x^2]))/2) - a^2*((-1/3*I)*ArcTa 
n[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))))))
 

3.4.70.3.1 Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

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 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 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 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 5453

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

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 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]
 

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

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

Time = 38.36 (sec) , antiderivative size = 1807, normalized size of antiderivative = 9.56

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1807\)
default	\(\text {Expression too large to display}\)	\(1807\)
parts	\(\text {Expression too large to display}\)	\(1809\)

input

int((a^2*c*x^2+c)*arctan(a*x)^3/x^4,x,method=_RETURNVERBOSE)
 

output

a^3*(-c*arctan(a*x)^3/a/x-1/3*c*arctan(a*x)^3/a^3/x^3-c*(arctan(a*x)^2*ln( 
a^2*x^2+1)+1/2*arctan(a*x)^2/a^2/x^2-2*arctan(a*x)^2*ln(a*x)-2*arctan(a*x) 
^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+2*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2 
+1)-1)+1/6*arctan(a*x)*(-6*I*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*Pi*arctan 
(a*x)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2*a*x-3*I*Pi*arctan(a*x)*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)*a*x+3*I*P 
i*arctan(a*x)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*a*x-3*I*Pi*arctan(a*x)*csg 
n(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I*(1+I*a 
*x)^2/(a^2*x^2+1))*a*x+6*I*a*x+6*I*Pi*arctan(a*x)*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*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)- 
1))*a*x+6*I*Pi*arctan(a*x)*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*a*x+3*I*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*Pi*arctan(a*x)*csgn(I 
*(1+I*a*x)^2/(a^2*x^2+1))*a*x-6*I*Pi*arctan(a*x)*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*a*x-6*I*Pi*arctan(a*x)*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))*a*x-6*I*Pi*arctan(a*x)*csgn(((1+ 
I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*a*x+3*I*Pi*arctan(a 
*x)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((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)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*a*x+3*I*P...
 

3.4.70.5 Fricas [F]

\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate((a^2*c*x^2+c)*arctan(a*x)^3/x^4,x, algorithm="fricas")
 

output

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

3.4.70.6 Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^4} \, dx=c \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{4}}\, dx + \int \frac {a^{2} \operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx\right ) \]

input

integrate((a**2*c*x**2+c)*atan(a*x)**3/x**4,x)
 

output

c*(Integral(atan(a*x)**3/x**4, x) + Integral(a**2*atan(a*x)**3/x**2, x))
 

3.4.70.7 Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate((a^2*c*x^2+c)*arctan(a*x)^3/x^4,x, algorithm="maxima")
 

output

1/96*(3*(7*a^3*c*arctan(a*x)^4 + 96*a^4*c*integrate(1/32*x^4*arctan(a*x)*l 
og(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x) - 384*a^4*c*integrate(1/32*x^4*arcta 
n(a*x)*log(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) + 384*a^3*c*integrate(1/32*x^3 
*arctan(a*x)^2/(a^2*x^6 + x^4), x) - 96*a^3*c*integrate(1/32*x^3*log(a^2*x 
^2 + 1)^2/(a^2*x^6 + x^4), x) + 1792*a^2*c*integrate(1/32*x^2*arctan(a*x)^ 
3/(a^2*x^6 + x^4), x) + 192*a^2*c*integrate(1/32*x^2*arctan(a*x)*log(a^2*x 
^2 + 1)^2/(a^2*x^6 + x^4), x) - 128*a^2*c*integrate(1/32*x^2*arctan(a*x)*l 
og(a^2*x^2 + 1)/(a^2*x^6 + x^4), x) + 128*a*c*integrate(1/32*x*arctan(a*x) 
^2/(a^2*x^6 + x^4), x) - 32*a*c*integrate(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x 
^6 + x^4), x) + 896*c*integrate(1/32*arctan(a*x)^3/(a^2*x^6 + x^4), x) + 9 
6*c*integrate(1/32*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^6 + x^4), x))*x^3 
 - 4*(3*a^2*c*x^2 + c)*arctan(a*x)^3 + 3*(3*a^2*c*x^2 + c)*arctan(a*x)*log 
(a^2*x^2 + 1)^2)/x^3
 

3.4.70.8 Giac [F]

\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate((a^2*c*x^2+c)*arctan(a*x)^3/x^4,x, algorithm="giac")
 

output

sage0*x
 

3.4.70.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )}{x^4} \,d x \]

input

int((atan(a*x)^3*(c + a^2*c*x^2))/x^4,x)
 

output

int((atan(a*x)^3*(c + a^2*c*x^2))/x^4, x)