\(\int x (c+a^2 c x^2) \arctan (a x)^3 \, dx\) [365]

Optimal result

Integrand size = 18, antiderivative size = 160 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=-\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \arctan (a x)}{4 a^2}-\frac {i c \arctan (a x)^2}{2 a^2}-\frac {c x \arctan (a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \arctan (a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{4 a^2}-\frac {c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^2}-\frac {i c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.63 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {c \left (-a x-\left (-2 i+3 a x+a^3 x^3\right ) \arctan (a x)^2+\left (1+a^2 x^2\right )^2 \arctan (a x)^3+\arctan (a x) \left (1+a^2 x^2-4 \log \left (1+e^{2 i \arctan (a x)}\right )\right )+2 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{4 a^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5465, 27, 5415, 24, 5345, 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.

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5415

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 5345

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

\(\Big \downarrow \) 5455

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

\(\Big \downarrow \) 5379

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

(c*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/(4*a^2) - (3*c*(x/3 - ((1 + a^2*x^2)*Arc 
Tan[a*x])/(3*a) + (x*(1 + a^2*x^2)*ArcTan[a*x]^2)/3 + (2*(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)))/3))/(4*a)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 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 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 5415

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

Maple [A] (verified)

Time = 4.04 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.48

method	result	size
parts	\(\frac {c \arctan \left (a x \right )^{3} a^{2} x^{4}}{4}+\frac {c \arctan \left (a x \right )^{3} x^{2}}{2}+\frac {c \arctan \left (a x \right )^{3}}{4 a^{2}}-\frac {3 c \left (\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}+a \arctan \left (a x \right )^{2} x -\frac {x^{2} a^{2} \arctan \left (a x \right )}{3}-\frac {2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{3}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{3}\right )}{4 a^{2}}\)	\(237\)
derivativedivides	\(\frac {\frac {c \arctan \left (a x \right )^{3} a^{4} x^{4}}{4}+\frac {a^{2} c \,x^{2} \arctan \left (a x \right )^{3}}{2}+\frac {c \arctan \left (a x \right )^{3}}{4}-\frac {3 c \left (\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}+a \arctan \left (a x \right )^{2} x -\frac {x^{2} a^{2} \arctan \left (a x \right )}{3}-\frac {2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{3}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{3}\right )}{4}}{a^{2}}\)	\(238\)
default	\(\frac {\frac {c \arctan \left (a x \right )^{3} a^{4} x^{4}}{4}+\frac {a^{2} c \,x^{2} \arctan \left (a x \right )^{3}}{2}+\frac {c \arctan \left (a x \right )^{3}}{4}-\frac {3 c \left (\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}+a \arctan \left (a x \right )^{2} x -\frac {x^{2} a^{2} \arctan \left (a x \right )}{3}-\frac {2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{3}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{3}\right )}{4}}{a^{2}}\)	\(238\)

Input:

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

Output:

1/4*c*arctan(a*x)^3*a^2*x^4+1/2*c*arctan(a*x)^3*x^2+1/4*c*arctan(a*x)^3/a^ 
2-3/4/a^2*c*(1/3*a^3*arctan(a*x)^2*x^3+a*arctan(a*x)^2*x-1/3*x^2*a^2*arcta 
n(a*x)-2/3*arctan(a*x)*ln(a^2*x^2+1)+1/3*a*x-1/3*arctan(a*x)-1/3*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)))+1/3*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))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/64*(8*(a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x)^3 + 4*(512*a^5*c*integra 
te(1/64*x^5*arctan(a*x)^3/(a^3*x^2 + a), x) - 192*a^4*c*integrate(1/64*x^4 
*arctan(a*x)^2/(a^3*x^2 + a), x) - 48*a^4*c*integrate(1/64*x^4*log(a^2*x^2 
 + 1)^2/(a^3*x^2 + a), x) - 64*a^4*c*integrate(1/64*x^4*log(a^2*x^2 + 1)/( 
a^3*x^2 + a), x) + 1024*a^3*c*integrate(1/64*x^3*arctan(a*x)^3/(a^3*x^2 + 
a), x) + 128*a^3*c*integrate(1/64*x^3*arctan(a*x)/(a^3*x^2 + a), x) - 384* 
a^2*c*integrate(1/64*x^2*arctan(a*x)^2/(a^3*x^2 + a), x) - 96*a^2*c*integr 
ate(1/64*x^2*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x) - 192*a^2*c*integrate(1/ 
64*x^2*log(a^2*x^2 + 1)/(a^3*x^2 + a), x) + 512*a*c*integrate(1/64*x*arcta 
n(a*x)^3/(a^3*x^2 + a), x) + 384*a*c*integrate(1/64*x*arctan(a*x)/(a^3*x^2 
 + a), x) - c*arctan(a*x)^3/a^2 - 48*c*integrate(1/64*log(a^2*x^2 + 1)^2/( 
a^3*x^2 + a), x))*a^2 - 4*(a^3*c*x^3 + 3*a*c*x)*arctan(a*x)^2 + (a^3*c*x^3 
 + 3*a*c*x)*log(a^2*x^2 + 1)^2)/a^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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