\(\int \frac {(a+b \arctan (c+d x))^2}{e+f x} \, dx\) [34]

Optimal result

Integrand size = 20, antiderivative size = 261 \[ \int \frac {(a+b \arctan (c+d x))^2}{e+f x} \, dx=-\frac {(a+b \arctan (c+d x))^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {(a+b \arctan (c+d x))^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}+\frac {i b (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{f}-\frac {i b (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 261, 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, 4968} \[ \int \frac {(a+b \arctan (c+d x))^2}{e+f x} \, dx=-\frac {i b (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{f}+\frac {(a+b \arctan (c+d x))^2 \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{f}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right ) (a+b \arctan (c+d x))}{f}-\frac {\log \left (\frac {2}{1-i (c+d x)}\right ) (a+b \arctan (c+d x))^2}{f}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 f}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 f} \]

[In]

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

[Out]

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

Rule 4968

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^2)*(Log[
2/(1 - I*c*x)]/e), x] + (Simp[(a + b*ArcTan[c*x])^2*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] + S
imp[I*b*(a + b*ArcTan[c*x])*(PolyLog[2, 1 - 2/(1 - I*c*x)]/e), x] - Simp[I*b*(a + b*ArcTan[c*x])*(PolyLog[2, 1
- 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] - Simp[b^2*(PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e)), x] + Si
mp[b^2*(PolyLog[3, 1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e}, x] &&
NeQ[c^2*d^2 + e^2, 0]

Rule 5155

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]
&& IGtQ[p, 0]

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 4.59 (sec) , antiderivative size = 1877, normalized size of antiderivative = 7.19

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1877\)
default	\(\text {Expression too large to display}\)	\(1877\)
parts	\(\text {Expression too large to display}\)	\(1998\)

[In]

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

[Out]

1/d*(a^2*d*ln(c*f-d*e-f*(d*x+c))/f-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)))+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)^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)*arc
tan(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))))-2*a*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))-d
ilog((I*f-f*(d*x+c))/(d*e+I*f-c*f)))/f))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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