\(\int \frac {(c+a^2 c x^2) \arctan (a x)^3}{x} \, dx\) [367]

Optimal result

Integrand size = 20, antiderivative size = 276 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x} \, dx=-\frac {3}{2} i c \arctan (a x)^2-\frac {3}{2} a c x \arctan (a x)^2+\frac {1}{2} c \arctan (a x)^3+\frac {1}{2} a^2 c x^2 \arctan (a x)^3+2 c \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x} \, dx=\frac {1}{2} c \left (1+a^2 x^2\right ) \arctan (a x)^3-\frac {3}{2} c \left (-i \arctan (a x)^2+a x \arctan (a x)^2+2 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )-\frac {1}{64} i c \left (\pi ^4-32 \arctan (a x)^4+64 i \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )-64 i \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )-96 \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-96 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+96 i \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-96 i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )\right ) \] Input:

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

Output:

(c*(1 + a^2*x^2)*ArcTan[a*x]^3)/2 - (3*c*((-I)*ArcTan[a*x]^2 + a*x*ArcTan[ 
a*x]^2 + 2*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - I*PolyLog[2, -E^(( 
2*I)*ArcTan[a*x])]))/2 - (I/64)*c*(Pi^4 - 32*ArcTan[a*x]^4 + (64*I)*ArcTan 
[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] - (64*I)*ArcTan[a*x]^3*Log[1 + E^( 
(2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] 
- 96*ArcTan[a*x]^2*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x] 
*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - (96*I)*ArcTan[a*x]*PolyLog[3, -E^((2 
*I)*ArcTan[a*x])] + 48*PolyLog[4, E^((-2*I)*ArcTan[a*x])] + 48*PolyLog[4, 
-E^((2*I)*ArcTan[a*x])])
 

Rubi [A] (verified)

Time = 2.33 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.23, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5485, 5357, 5361, 5451, 5345, 5419, 5455, 5379, 2849, 2752, 5523, 5529, 5533, 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.

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5357

\(\displaystyle a^2 c \int x \arctan (a x)^3dx+c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\)

\(\Big \downarrow \) 5361

\(\displaystyle a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx\right )+c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\)

\(\Big \downarrow \) 5451

\(\displaystyle a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \left (\frac {\int \arctan (a x)^2dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\)

\(\Big \downarrow \) 5345

\(\displaystyle a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \left (\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\)

\(\Big \downarrow \) 5419

\(\displaystyle a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \left (\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\arctan (a x)^3}{3 a^3}\right )\right )+c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\)

\(\Big \downarrow \) 5455

\(\displaystyle c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )+a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \left (-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {\int \frac {\arctan (a x)}{i-a x}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )}{a^2}\right )\right )\)

\(\Big \downarrow \) 5379

\(\displaystyle c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )+a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \left (-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}-\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )}{a^2}\right )\right )\)

\(\Big \downarrow \) 2849

\(\displaystyle c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )+a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \left (-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )}{a^2}\right )\right )\)

\(\Big \downarrow \) 2752

\(\displaystyle c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )+a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \left (-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )}{a^2}\right )\right )\)

\(\Big \downarrow \) 5523

\(\displaystyle c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \left (\frac {1}{2} \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {1}{2} \int \frac {\arctan (a x)^2 \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\right )+a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \left (-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )}{a^2}\right )\right )\)

\(\Big \downarrow \) 5529

\(\displaystyle c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \left (\frac {1}{2} \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-i \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )+\frac {1}{2} \left (i \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )}{2 a}\right )\right )\right )+a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \left (-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )}{a^2}\right )\right )\)

\(\Big \downarrow \) 5533

\(\displaystyle c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \left (\frac {1}{2} \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-i \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\right )+\frac {1}{2} \left (i \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )}{a^2 x^2+1}dx\right )-\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )}{2 a}\right )\right )\right )+a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \left (-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )}{a^2}\right )\right )\)

\(\Big \downarrow \) 7164

\(\displaystyle a^2 c \left (\frac {1}{2} x^2 \arctan (a x)^3-\frac {3}{2} a \left (-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )}{a^2}\right )\right )+c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \left (\frac {1}{2} \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-i \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,1-\frac {2}{i a x+1}\right )}{4 a}\right )\right )+\frac {1}{2} \left (i \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,\frac {2}{i a x+1}-1\right )}{4 a}\right )-\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )}{2 a}\right )\right )\right )\)

Input:

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

Output:

a^2*c*((x^2*ArcTan[a*x]^3)/2 - (3*a*(-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))/2) + c*(2*ArcTan 
[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 6*a*((((I/2)*ArcTan[a*x]^2*PolyLog[2, 
 1 - 2/(1 + I*a*x)])/a - I*(((I/2)*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x 
)])/a + PolyLog[4, 1 - 2/(1 + I*a*x)]/(4*a)))/2 + (((-1/2*I)*ArcTan[a*x]^2 
*PolyLog[2, -1 + 2/(1 + I*a*x)])/a + I*(((I/2)*ArcTan[a*x]*PolyLog[3, -1 + 
 2/(1 + I*a*x)])/a + PolyLog[4, -1 + 2/(1 + I*a*x)]/(4*a)))/2))
 

Defintions of rubi rules used

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 5357

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

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 5523

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x 
_)^2), x_Symbol] :> Simp[1/2   Int[Log[1 + u]*((a + b*ArcTan[c*x])^p/(d + e 
*x^2)), x], x] - Simp[1/2   Int[Log[1 - u]*((a + b*ArcTan[c*x])^p/(d + e*x^ 
2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && 
EqQ[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 5533

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

Maple [A] (verified)

Time = 21.72 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.67

method	result	size
derivativedivides	\(\frac {c \arctan \left (a x \right )^{2} \left (-3-i \arctan \left (a x \right )+\arctan \left (a x \right ) a x \right ) \left (a x +i\right )}{2}+6 i c \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 c \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+\frac {3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-c \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {3 i c \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+c \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {3 i c \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+c \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 i c \arctan \left (a x \right )^{2}\)	\(460\)
default	\(\frac {c \arctan \left (a x \right )^{2} \left (-3-i \arctan \left (a x \right )+\arctan \left (a x \right ) a x \right ) \left (a x +i\right )}{2}+6 i c \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 c \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+\frac {3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-c \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {3 i c \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+c \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {3 i c \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+c \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 i c \arctan \left (a x \right )^{2}\)	\(460\)

Input:

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

Output:

1/2*c*arctan(a*x)^2*(-3-I*arctan(a*x)+arctan(a*x)*a*x)*(a*x+I)+6*I*c*polyl 
og(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*c*arctan(a*x)*ln((1+I*a*x)^2/(a^2*x^2 
+1)+1)+3/2*I*c*arctan(a*x)^2*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-c*arctan( 
a*x)^3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)-3*I*c*arctan(a*x)^2*polylog(2,-(1+I*a 
*x)/(a^2*x^2+1)^(1/2))-3/2*c*arctan(a*x)*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1 
))-3/4*I*c*polylog(4,-(1+I*a*x)^2/(a^2*x^2+1))+c*arctan(a*x)^3*ln(1-(1+I*a 
*x)/(a^2*x^2+1)^(1/2))+6*I*c*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*c*ar 
ctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+3/2*I*c*polylog(2,-(1+I*a 
*x)^2/(a^2*x^2+1))+c*arctan(a*x)^3*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*c 
*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*c*arctan(a*x)*poly 
log(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*I*c*arctan(a*x)^2
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/16*a^2*c*x^2*arctan(a*x)^3 - 3/64*a^2*c*x^2*arctan(a*x)*log(a^2*x^2 + 1) 
^2 + integrate(1/64*(12*a^4*c*x^4*arctan(a*x)*log(a^2*x^2 + 1) - 12*a^3*c* 
x^3*arctan(a*x)^2 + 56*(a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x)^3 + 3*(a^ 
3*c*x^3 + 2*(a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x))*log(a^2*x^2 + 1)^2) 
/(a^2*x^3 + x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x} \, dx=\frac {c \left (\mathit {atan} \left (a x \right )^{3} a^{2} x^{2}+\mathit {atan} \left (a x \right )^{3}-3 \mathit {atan} \left (a x \right )^{2} a x +3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+3 \mathit {atan} \left (a x \right )+2 \left (\int \frac {\mathit {atan} \left (a x \right )^{3}}{x}d x \right )-6 \left (\int \frac {\mathit {atan} \left (a x \right ) x^{3}}{a^{2} x^{2}+1}d x \right ) a^{4}-3 a x \right )}{2} \] Input:

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

Output:

(c*(atan(a*x)**3*a**2*x**2 + atan(a*x)**3 - 3*atan(a*x)**2*a*x + 3*atan(a* 
x)*a**2*x**2 + 3*atan(a*x) + 2*int(atan(a*x)**3/x,x) - 6*int((atan(a*x)*x* 
*3)/(a**2*x**2 + 1),x)*a**4 - 3*a*x))/2