3.1.0 ∫ (a+b arctan(c+dx))3 ______________________________

Integrand size = 20, antiderivative size = 372 \[ \int \frac {(a+b \arctan (c+d x))^3}{e+f x} \, dx=-\frac {(a+b \arctan (c+d x))^3 \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {(a+b \arctan (c+d x))^3 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}+\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{2 f}-\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}-\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-i (c+d x)}\right )}{4 f}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{4 f} \]

-(a+b*arctan(d*x+c))^3*ln(2/(1-I*(d*x+c)))/f+(a+b*arctan(d*x+c))^3*ln(2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))

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

(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/f-3/2*b^2*(a+b*arctan(d*x+c))*polylog(3,1-2/(1-I*(d*x+c)))/f+3/2*b^2*(a+b

*arctan(d*x+c))*polylog(3,1-2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/f-3/4*I*b^3*polylog(4,1-2/(1-I*(d*x+c)))/

f+3/4*I*b^3*polylog(4,1-2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/f

Rubi [A] (veriﬁed)

Time = 0.15 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5155, 4970} \[ \int \frac {(a+b \arctan (c+d x))^3}{e+f x} \, dx=\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 f}-\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right ) (a+b \arctan (c+d x))}{2 f}-\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 f}+\frac {(a+b \arctan (c+d x))^3 \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{f}+\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right ) (a+b \arctan (c+d x))^2}{2 f}-\frac {\log \left (\frac {2}{1-i (c+d x)}\right ) (a+b \arctan (c+d x))^3}{f}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{4 f}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-i (c+d x)}\right )}{4 f} \]

Int[(a + b*ArcTan[c + d*x])^3/(e + f*x),x]

-(((a + b*ArcTan[c + d*x])^3*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcTan[c + d*x])^3*Log[(2*d*(e + f*x))/((d

*e + I*f - c*f)*(1 - I*(c + d*x)))])/f + (((3*I)/2)*b*(a + b*ArcTan[c + d*x])^2*PolyLog[2, 1 - 2/(1 - I*(c + d

*x))])/f - (((3*I)/2)*b*(a + b*ArcTan[c + d*x])^2*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c

+ d*x)))])/f - (3*b^2*(a + b*ArcTan[c + d*x])*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*f) + (3*b^2*(a + b*ArcTa

n[c + d*x])*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f) - (((3*I)/4)*b^3*Poly

Log[4, 1 - 2/(1 - I*(c + d*x))])/f + (((3*I)/4)*b^3*PolyLog[4, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^3/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^3)*(Log[

2/(1 - I*c*x)]/e), x] + (Simp[(a + b*ArcTan[c*x])^3*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] + S

imp[3*I*b*(a + b*ArcTan[c*x])^2*(PolyLog[2, 1 - 2/(1 - I*c*x)]/(2*e)), x] - Simp[3*I*b*(a + b*ArcTan[c*x])^2*(

PolyLog[2, 1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(2*e)), x] - Simp[3*b^2*(a + b*ArcTan[c*x])*(PolyLog

[3, 1 - 2/(1 - I*c*x)]/(2*e)), x] + Simp[3*b^2*(a + b*ArcTan[c*x])*(PolyLog[3, 1 - 2*c*((d + e*x)/((c*d + I*e)

*(1 - I*c*x)))]/(2*e)), x] - Simp[3*I*b^3*(PolyLog[4, 1 - 2/(1 - I*c*x)]/(4*e)), x] + Simp[3*I*b^3*(PolyLog[4,

1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(4*e)), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2,

Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I

nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \arctan (x))^3}{\frac {d e-c f}{d}+\frac {f x}{d}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {(a+b \arctan (c+d x))^3 \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {(a+b \arctan (c+d x))^3 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}+\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{2 f}-\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}-\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-i (c+d x)}\right )}{4 f}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{4 f} \\ \end{align*}

Mathematica [F]

\[ \int \frac {(a+b \arctan (c+d x))^3}{e+f x} \, dx=\int \frac {(a+b \arctan (c+d x))^3}{e+f x} \, dx \]

Integrate[(a + b*ArcTan[c + d*x])^3/(e + f*x),x]

Integrate[(a + b*ArcTan[c + d*x])^3/(e + f*x), x]

Maple [C] (warning: unable to verify)

Time = 4.95 (sec) , antiderivative size = 3817, normalized size of antiderivative = 10.26


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(3817\)
default	\(\text {Expression too large to display}\)	\(3817\)
parts	\(\text {Expression too large to display}\)	\(4072\)

int((a+b*arctan(d*x+c))^3/(f*x+e),x,method=_RETURNVERBOSE)

1/d*(a^3*d*ln(c*f-d*e-f*(d*x+c))/f-b^3*d*(-ln(c*f-d*e-f*(d*x+c))/f*arctan(d*x+c)^3+3/f*(1/3*arctan(d*x+c)^3*ln

(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f+c*f

-d*e)-I*d*e*arctan(d*x+c)^2*polylog(2,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))/(2*I*f+2*c*f-

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

c))^2/(1+(d*x+c)^2))+1/2*I*c*f/(c*f-d*e+I*f)*arctan(d*x+c)^2*polylog(2,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c

)^2)/(d*e+I*f-c*f))+1/4*I*polylog(4,-(1+I*(d*x+c))^2/(1+(d*x+c)^2))-1/2*I*f/(c*f-d*e+I*f)*arctan(d*x+c)*polylo

g(3,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))-1/2*f/(c*f-d*e+I*f)*arctan(d*x+c)^2*polylog(2,(

c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))+1/4*f/(c*f-d*e+I*f)*polylog(4,(c*f-d*e+I*f)*(1+I*(d*

x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))-1/3*c*f/(c*f-d*e+I*f)*arctan(d*x+c)^3*ln(1-(c*f-d*e+I*f)*(1+I*(d*x+c))^2/

(1+(d*x+c)^2)/(d*e+I*f-c*f))-1/2*c*f/(c*f-d*e+I*f)*arctan(d*x+c)*polylog(3,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d

*x+c)^2)/(d*e+I*f-c*f))-1/6*I*Pi*csgn(I*(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d

*e*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f+c*f-d*e)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))*(csgn(I*(I*f*(1+I*(d*x+c))^2/

(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f+c*f-d*e))*csgn(I/(1+(1+I

*(d*x+c))^2/(1+(d*x+c)^2)))-csgn(I*(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1

+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f+c*f-d*e)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))*csgn(I/(1+(1+I*(d*x+c))^2/(1+(d*x+

c)^2)))-csgn(I*(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d*

x+c)^2)-I*f+c*f-d*e))*csgn(I*(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*

x+c))^2/(1+(d*x+c)^2)-I*f+c*f-d*e)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))+csgn(I*(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2

)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f+c*f-d*e)/(1+(1+I*(d*x+c))^2/(1+(d*x+

c)^2)))^2)*arctan(d*x+c)^3-1/3*I*f/(c*f-d*e+I*f)*arctan(d*x+c)^3*ln(1-(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)

^2)/(d*e+I*f-c*f))+1/3*d*e/(c*f-d*e+I*f)*arctan(d*x+c)^3*ln(1-(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e

+I*f-c*f))+1/2*d*e/(c*f-d*e+I*f)*arctan(d*x+c)*polylog(3,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-

c*f))+I*d*e*polylog(4,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))/(4*I*f+4*c*f-4*d*e)-1/4*I*c*f

/(c*f-d*e+I*f)*polylog(4,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))))-3*a*b^2*d*(-ln(c*f-d*e-f

*(d*x+c))/f*arctan(d*x+c)^2+2/f*(1/2*arctan(d*x+c)^2*ln(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/

(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f+c*f-d*e)-1/4*I*Pi*csgn(I*(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2

)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f+c*f-d*e)/(1+(1+I*(d*x+c))^2/(1+(d*x+

c)^2)))*(csgn(I*(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d

*x+c)^2)-I*f+c*f-d*e))*csgn(I/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))-csgn(I*(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f

*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f+c*f-d*e)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)

))*csgn(I/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))-csgn(I*(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+

(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f+c*f-d*e))*csgn(I*(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f*(1+I*

(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f+c*f-d*e)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))+csg

n(I*(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f

+c*f-d*e)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))^2)*arctan(d*x+c)^2-1/2*I*arctan(d*x+c)*polylog(2,-(1+I*(d*x+c))^2

/(1+(d*x+c)^2))+1/4*polylog(3,-(1+I*(d*x+c))^2/(1+(d*x+c)^2))-1/2*f/(c*f-d*e+I*f)*arctan(d*x+c)*polylog(2,(c*f

-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))-1/2*I*f/(c*f-d*e+I*f)*arctan(d*x+c)^2*ln(1-(c*f-d*e+I*f

)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))-1/4*I*f/(c*f-d*e+I*f)*polylog(3,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(

1+(d*x+c)^2)/(d*e+I*f-c*f))+1/2*I*c*f/(c*f-d*e+I*f)*arctan(d*x+c)*polylog(2,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(

d*x+c)^2)/(d*e+I*f-c*f))-1/2*c*f/(c*f-d*e+I*f)*arctan(d*x+c)^2*ln(1-(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2

)/(d*e+I*f-c*f))-1/4*c*f/(c*f-d*e+I*f)*polylog(3,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))-I*

d*e*arctan(d*x+c)*polylog(2,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))/(2*I*f+2*c*f-2*d*e)+1/2

*d*e/(c*f-d*e+I*f)*arctan(d*x+c)^2*ln(1-(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))+1/4*d*e/(c*

f-d*e+I*f)*polylog(3,(c*f-d*e+I*f)*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(d*e+I*f-c*f))))-3*a^2*b*d*(-ln(c*f-d*e-f*(d*

x+c))/f*arctan(d*x+c)+1/2*I*ln(c*f-d*e-f*(d*x+c))*(ln((I*f+f*(d*x+c))/(c*f-d*e+I*f))-ln((I*f-f*(d*x+c))/(d*e+I

*f-c*f)))/f+1/2*I*(dilog((I*f+f*(d*x+c))/(c*f-d*e+I*f))-dilog((I*f-f*(d*x+c))/(d*e+I*f-c*f)))/f))

Fricas [F]

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

integrate((a+b*arctan(d*x+c))^3/(f*x+e),x, algorithm="fricas")

integral((b^3*arctan(d*x + c)^3 + 3*a*b^2*arctan(d*x + c)^2 + 3*a^2*b*arctan(d*x + c) + a^3)/(f*x + e), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{e+f x} \, dx=\text {Timed out} \]

integrate((a+b*atan(d*x+c))**3/(f*x+e),x)

Maxima [F]

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

integrate((a+b*arctan(d*x+c))^3/(f*x+e),x, algorithm="maxima")

a^3*log(f*x + e)/f + integrate(1/32*(28*b^3*arctan(d*x + c)^3 + 3*b^3*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x +

c^2 + 1)^2 + 96*a*b^2*arctan(d*x + c)^2 + 96*a^2*b*arctan(d*x + c))/(f*x + e), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{e+f x} \, dx=\text {Timed out} \]

integrate((a+b*arctan(d*x+c))^3/(f*x+e),x, algorithm="giac")

Mupad [F(-1)]

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

int((a + b*atan(c + d*x))^3/(e + f*x), x)

\(\int \frac {(a+b \arctan (c+d x))^3}{e+f x} \, dx\) [39]

Optimal result