\(\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx\) [12]
Optimal result
Integrand size = 18, antiderivative size = 223 \[
\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx=-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e}-\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}
\]
[Out]
-(a+b*arctan(c*x))^2*ln(2/(1-I*c*x))/e+(a+b*arctan(c*x))^2*ln(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/e+I*b*(a+b*arct
an(c*x))*polylog(2,1-2/(1-I*c*x))/e-I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/e-1/2*b
^2*polylog(3,1-2/(1-I*c*x))/e+1/2*b^2*polylog(3,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/e
Rubi [A] (verified)
Time = 0.03 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {4968}
\[
\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx=-\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{e}-\frac {\log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2}{e}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e}
\]
[In]
Int[(a + b*ArcTan[c*x])^2/(d + e*x),x]
[Out]
-(((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 -
I*c*x))])/e + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e - (I*b*(a + b*ArcTan[c*x])*PolyLog[2,
1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e - (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e) + (b^2*PolyLog[
3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e)
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]
Rubi steps \begin{align*}
\text {integral}& = -\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e}-\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e} \\
\end{align*}
Mathematica [F]
\[
\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx=\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx
\]
[In]
Integrate[(a + b*ArcTan[c*x])^2/(d + e*x),x]
[Out]
Integrate[(a + b*ArcTan[c*x])^2/(d + e*x), x]
Maple [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 68.26 (sec) , antiderivative size = 1199, normalized size of antiderivative =
5.38
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| method | result | size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(1199\) |
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\(\text {Expression too large to display}\) |
\(1199\) |
| parts |
\(\text {Expression too large to display}\) |
\(1202\) |
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[In]
int((a+b*arctan(c*x))^2/(e*x+d),x,method=_RETURNVERBOSE)
[Out]
1/c*(a^2*c*ln(c*e*x+c*d)/e+b^2*c*(ln(c*e*x+c*d)/e*arctan(c*x)^2-2/e*(1/2*arctan(c*x)^2*ln(-I*e*(1+I*c*x)^2/(c^
2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d)-1/4*I*Pi*csgn(I*(-I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c
^2*x^2+1)+I*e+c*d)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*(csgn(I*(-I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2
+1)+I*e+c*d))*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))-csgn(I*(-I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^
2+1)+I*e+c*d)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))-csgn(I*(-I*e*(1+I*c*x)^2/(c^2*x
^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d))*csgn(I*(-I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I
*e+c*d)/(1+(1+I*c*x)^2/(c^2*x^2+1)))+csgn(I*(-I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d)
/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2)*arctan(c*x)^2-1/2*I*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+1/4*polyl
og(3,-(1+I*c*x)^2/(c^2*x^2+1))+1/2*I*c*d/(c*d-I*e)*arctan(c*x)*polylog(2,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*
x^2+1))-1/2*c*d/(c*d-I*e)*arctan(c*x)^2*ln(1-(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))-1/4*c*d/(c*d-I*e)*po
lylog(3,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+1/2*I*e*arctan(c*x)*polylog(2,(I*e-c*d)/(c*d+I*e)*(1+I*c*
x)^2/(c^2*x^2+1))/(e+I*d*c)-1/2*e*arctan(c*x)^2*ln(1-(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(e+I*d*c)-1/
4*e*polylog(3,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(e+I*d*c)))+2*a*b*c*(ln(c*e*x+c*d)/e*arctan(c*x)-1/
2*I*ln(c*e*x+c*d)*(-ln((I*e-c*e*x)/(c*d+I*e))+ln((I*e+c*e*x)/(I*e-c*d)))/e+1/2*I*(dilog((I*e-c*e*x)/(c*d+I*e))
-dilog((I*e+c*e*x)/(I*e-c*d)))/e))
Fricas [F]
\[
\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{e x + d} \,d x }
\]
[In]
integrate((a+b*arctan(c*x))^2/(e*x+d),x, algorithm="fricas")
[Out]
integral((b^2*arctan(c*x)^2 + 2*a*b*arctan(c*x) + a^2)/(e*x + d), x)
Sympy [F]
\[
\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{d + e x}\, dx
\]
[In]
integrate((a+b*atan(c*x))**2/(e*x+d),x)
[Out]
Integral((a + b*atan(c*x))**2/(d + e*x), x)
Maxima [F]
\[
\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{e x + d} \,d x }
\]
[In]
integrate((a+b*arctan(c*x))^2/(e*x+d),x, algorithm="maxima")
[Out]
a^2*log(e*x + d)/e + integrate(1/16*(12*b^2*arctan(c*x)^2 + b^2*log(c^2*x^2 + 1)^2 + 32*a*b*arctan(c*x))/(e*x
+ d), x)
Giac [F]
\[
\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{e x + d} \,d x }
\]
[In]
integrate((a+b*arctan(c*x))^2/(e*x+d),x, algorithm="giac")
[Out]
sage0*x
Mupad [F(-1)]
Timed out. \[
\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{d+e\,x} \,d x
\]
[In]
int((a + b*atan(c*x))^2/(d + e*x),x)
[Out]
int((a + b*atan(c*x))^2/(d + e*x), x)