3.1.40 \(\int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx\) [40]

3.1.40.1 Optimal result

Integrand size = 20, antiderivative size = 1233 \[ \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx=\frac {3 a^2 b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \arctan (c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \arctan (c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \arctan (c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \arctan (c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {(a+b \arctan (c+d x))^3}{f (e+f x)}+\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {6 a b^2 d \arctan (c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \arctan (c+d x)^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \arctan (c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \]

output

3*a^2*b*d*(-c*f+d*e)*arctan(d*x+c)/f/(f^2+(-c*f+d*e)^2)+3*I*b^3*d*arctan(d 
*x+c)*polylog(2,1-2/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*a*b^2 
*d*(-c*f+d*e)*arctan(d*x+c)^2/f/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*a*b^2* 
d*polylog(2,1-2/(1+I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+b^3*d*(-c*f 
+d*e)*arctan(d*x+c)^3/f/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-(a+b*arctan(d*x+c) 
)^3/f/(f*x+e)+3*a^2*b*d*ln(f*x+e)/(f^2+(-c*f+d*e)^2)-6*a*b^2*d*arctan(d*x+ 
c)*ln(2/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3*b^3*d*arctan(d*x+ 
c)^2*ln(2/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+6*a*b^2*d*arctan( 
d*x+c)*ln(2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2 
+1)*f^2)+3*b^3*d*arctan(d*x+c)^2*ln(2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c) 
))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+6*a*b^2*d*arctan(d*x+c)*ln(2/(1+I*(d*x+ 
c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*b^3*d*arctan(d*x+c)^2*ln(2/(1+I*(d* 
x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3/2*a^2*b*d*ln(1+(d*x+c)^2)/(f^2+(- 
c*f+d*e)^2)+I*b^3*d*arctan(d*x+c)^3/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3*I*b^ 
3*d*arctan(d*x+c)*polylog(2,1-2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/(d^ 
2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*b^3*d*arctan(d*x+c)*polylog(2,1-2/(1+I*(d 
*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3*I*a*b^2*d*polylog(2,1-2*d*(f*x+e 
)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*a*b^2*d 
*polylog(2,1-2/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*a*b^2*d* 
arctan(d*x+c)^2/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3/2*b^3*d*polylog(3,1-2...
 

3.1.40.2 Mathematica [F]

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

input

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

output

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

3.1.40.3 Rubi [A] (verified)

Time = 2.24 (sec) , antiderivative size = 1265, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5568, 7292, 5580, 27, 7276, 2009}

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.

\(\displaystyle \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx\)

\(\Big \downarrow \) 5568

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 5580

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.1.40.3.1 Defintions of rubi rules used

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5568

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

rule 5580

Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m 
_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/d   Subs 
t[Int[((d*e - c*f)/d + f*(x/d))^m*(C/d^2 + (C/d^2)*x^2)^q*(a + b*ArcTan[x]) 
^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] & 
& EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

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

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

Time = 6.29 (sec) , antiderivative size = 3708, normalized size of antiderivative = 3.01

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3708\)
default	\(\text {Expression too large to display}\)	\(3708\)
parts	\(\text {Expression too large to display}\)	\(3834\)

input

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

output

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

3.1.40.5 Fricas [F]

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

input

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

output

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

3.1.40.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx=\text {Timed out} \]

input

integrate((a+b*atan(d*x+c))**3/(f*x+e)**2,x)
 

output

Timed out
 

3.1.40.7 Maxima [F]

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

input

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

output

3/2*(d*(2*(d^2*e - c*d*f)*arctan((d^2*x + c*d)/d)/((d^2*e^2*f - 2*c*d*e*f^ 
2 + (c^2 + 1)*f^3)*d) - log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*e^2 - 2*c*d* 
e*f + (c^2 + 1)*f^2) + 2*log(f*x + e)/(d^2*e^2 - 2*c*d*e*f + (c^2 + 1)*f^2 
)) - 2*arctan(d*x + c)/(f^2*x + e*f))*a^2*b - a^3/(f^2*x + e*f) - 1/32*(4* 
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 - 32*(f^2*x + e*f)*integrate(1/32*(28*(b^3*d^2*f*x^2 + 2*b^3*c*d*f* 
x + (b^3*c^2 + b^3)*f)*arctan(d*x + c)^3 + 12*(8*a*b^2*d^2*f*x^2 + b^3*d*e 
 + (16*a*b^2*c + b^3)*d*f*x + 8*(a*b^2*c^2 + a*b^2)*f)*arctan(d*x + c)^2 - 
 12*(b^3*d^2*f*x^2 + b^3*c*d*e + (b^3*d^2*e + b^3*c*d*f)*x)*arctan(d*x + c 
)*log(d^2*x^2 + 2*c*d*x + c^2 + 1) - 3*(b^3*d*f*x + b^3*d*e - (b^3*d^2*f*x 
^2 + 2*b^3*c*d*f*x + (b^3*c^2 + b^3)*f)*arctan(d*x + c))*log(d^2*x^2 + 2*c 
*d*x + c^2 + 1)^2)/(d^2*f^3*x^4 + (c^2 + 1)*e^2*f + 2*(d^2*e*f^2 + c*d*f^3 
)*x^3 + (d^2*e^2*f + 4*c*d*e*f^2 + (c^2 + 1)*f^3)*x^2 + 2*(c*d*e^2*f + (c^ 
2 + 1)*e*f^2)*x), x))/(f^2*x + e*f)
 

3.1.40.8 Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{(e+f x)^2} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.1.40.9 Mupad [F(-1)]

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

input

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

output

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