\(\int \frac {(a+b \cot ^{-1}(c+d x))^2}{e+f x} \, dx\) [139]
Optimal result
Integrand size = 20, antiderivative size = 261 \[
\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{e+f x} \, dx=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}-\frac {i b \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{f}+\frac {i b \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}
\]
[Out]
-(a+b*arccot(d*x+c))^2*ln(2/(1-I*(d*x+c)))/f+(a+b*arccot(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*arccot(d*x+c))*polylog(2,1-2/(1-I*(d*x+c)))/f+I*b*(a+b*arccot(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 = {5156, 4969}
\[
\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\frac {i b \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{f}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{f}-\frac {\log \left (\frac {2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{f}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 f}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 f}
\]
[In]
Int[(a + b*ArcCot[c + d*x])^2/(e + f*x),x]
[Out]
-(((a + b*ArcCot[c + d*x])^2*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcCot[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*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/f +
(I*b*(a + b*ArcCot[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 4969
Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^2)*(Log[
2/(1 - I*c*x)]/e), x] + (Simp[(a + b*ArcCot[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*ArcCot[c*x])*(PolyLog[2, 1 - 2/(1 - I*c*x)]/e), x] + Simp[I*b*(a + b*ArcCot[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 5156
Int[((a_.) + ArcCot[(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*ArcCot[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 {\left (a+b \cot ^{-1}(x)\right )^2}{\frac {d e-c f}{d}+\frac {f x}{d}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right )^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 \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{f}+\frac {i b \left (a+b \cot ^{-1}(c+d x)\right ) \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]
\[
\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{e+f x} \, dx
\]
[In]
Integrate[(a + b*ArcCot[c + d*x])^2/(e + f*x),x]
[Out]
Integrate[(a + b*ArcCot[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 = 14.06 (sec) , antiderivative size = 1911, normalized size of antiderivative =
7.32
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| method | result | size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(1911\) |
| default |
\(\text {Expression too large to display}\) |
\(1911\) |
| parts |
\(\text {Expression too large to display}\) |
\(2022\) |
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[In]
int((a+b*arccot(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*arccot(d*x+c)^2-2/f*(-1/2*arccot(d*x+c)^2*l
n(-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)+c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f-c*f+d*e)-1/4*
I*f/(-I*f+c*f-d*e)*polylog(3,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+1/2*arccot(d*x+c)^2*ln((d
*x+c+I)^2/(1+(d*x+c)^2)-1)-1/2*arccot(d*x+c)^2*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+I*d*e*arccot(d*x+c)*polylog
(2,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))/(-2*I*f+2*c*f-2*d*e)-polylog(3,-(d*x+c+I)/(1+(d*x+c
)^2)^(1/2))-1/2*arccot(d*x+c)^2*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+I*arccot(d*x+c)*polylog(2,-(d*x+c+I)/(1+(d
*x+c)^2)^(1/2))-polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-1/2*I*f/(-I*f+c*f-d*e)*arccot(d*x+c)^2*ln(1-(d*e+I*f-
c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+1/2*c*f/(-I*f+c*f-d*e)*arccot(d*x+c)^2*ln(1-(d*e+I*f-c*f)/(-c*f
+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+1/4*c*f/(-I*f+c*f-d*e)*polylog(3,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2
/(1+(d*x+c)^2))-1/2*f/(-I*f+c*f-d*e)*arccot(d*x+c)*polylog(2,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+
c)^2))-1/2*I*c*f/(-I*f+c*f-d*e)*arccot(d*x+c)*polylog(2,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2)
)+I*arccot(d*x+c)*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+1/4*I*Pi*csgn(I*(-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)+c*f
*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f-c*f+d*e)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))*(csgn(I*(
-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)+c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f-c*f+d*e))*csgn(
I/((d*x+c+I)^2/(1+(d*x+c)^2)-1))-csgn(I*(-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)+c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x
+c+I)^2/(1+(d*x+c)^2)-I*f-c*f+d*e)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))*csgn(I/((d*x+c+I)^2/(1+(d*x+c)^2)-1))-csgn(I
*(-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)+c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f-c*f+d*e))*csg
n(I*(-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)+c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f-c*f+d*e)/(
(d*x+c+I)^2/(1+(d*x+c)^2)-1))+csgn(I*(-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)+c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+
I)^2/(1+(d*x+c)^2)-I*f-c*f+d*e)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^2)*arccot(d*x+c)^2-1/2*d*e/(-I*f+c*f-d*e)*arcco
t(d*x+c)^2*ln(1-(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))-1/4*d*e/(-I*f+c*f-d*e)*polylog(3,(d*e+
I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))))-2*a*b*d*(-ln(c*f-d*e-f*(d*x+c))/f*arccot(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]
\[
\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x }
\]
[In]
integrate((a+b*arccot(d*x+c))^2/(f*x+e),x, algorithm="fricas")
[Out]
integral((b^2*arccot(d*x + c)^2 + 2*a*b*arccot(d*x + c) + a^2)/(f*x + e), x)
Sympy [F(-1)]
Timed out. \[
\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\text {Timed out}
\]
[In]
integrate((a+b*acot(d*x+c))**2/(f*x+e),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x }
\]
[In]
integrate((a+b*arccot(d*x+c))^2/(f*x+e),x, algorithm="maxima")
[Out]
a^2*log(f*x + e)/f + integrate(1/16*(12*b^2*arctan2(1, d*x + c)^2 + b^2*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 3
2*a*b*arctan2(1, d*x + c))/(f*x + e), x)
Giac [F]
\[
\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x }
\]
[In]
integrate((a+b*arccot(d*x+c))^2/(f*x+e),x, algorithm="giac")
[Out]
integrate((b*arccot(d*x + c) + a)^2/(f*x + e), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^2}{e+f\,x} \,d x
\]
[In]
int((a + b*acot(c + d*x))^2/(e + f*x),x)
[Out]
int((a + b*acot(c + d*x))^2/(e + f*x), x)