\(\int \frac {(a+b \arctan (c x))^3}{d+e x} \, dx\) [18]
Optimal result
Integrand size = 18, antiderivative size = 320 \[
\int \frac {(a+b \arctan (c x))^3}{d+e x} \, dx=-\frac {(a+b \arctan (c x))^3 \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {(a+b \arctan (c x))^3 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e}-\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}-\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e}+\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-i c x}\right )}{4 e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{4 e}
\]
[Out]
-(a+b*arctan(c*x))^3*ln(2/(1-I*c*x))/e+(a+b*arctan(c*x))^3*ln(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/e+3/2*I*b*(a+b*
arctan(c*x))^2*polylog(2,1-2/(1-I*c*x))/e-3/2*I*b*(a+b*arctan(c*x))^2*polylog(2,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c
*x))/e-3/2*b^2*(a+b*arctan(c*x))*polylog(3,1-2/(1-I*c*x))/e+3/2*b^2*(a+b*arctan(c*x))*polylog(3,1-2*c*(e*x+d)/
(c*d+I*e)/(1-I*c*x))/e-3/4*I*b^3*polylog(4,1-2/(1-I*c*x))/e+3/4*I*b^3*polylog(4,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c
*x))/e
Rubi [A] (verified)
Time = 0.04 (sec) , antiderivative size = 320, 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 = {4970}
\[
\int \frac {(a+b \arctan (c x))^3}{d+e x} \, dx=\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}-\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{2 e}-\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}+\frac {(a+b \arctan (c x))^3 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e}+\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2}{2 e}-\frac {\log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^3}{e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{4 e}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-i c x}\right )}{4 e}
\]
[In]
Int[(a + b*ArcTan[c*x])^3/(d + e*x),x]
[Out]
-(((a + b*ArcTan[c*x])^3*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])^3*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 -
I*c*x))])/e + (((3*I)/2)*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/e - (((3*I)/2)*b*(a + b*ArcTa
n[c*x])^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e - (3*b^2*(a + b*ArcTan[c*x])*PolyLog[3,
1 - 2/(1 - I*c*x)])/(2*e) + (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) - (((3*I)/4)*b^3*PolyLog[4, 1 - 2/(1 - I*c*x)])/e + (((3*I)/4)*b^3*PolyLog[4, 1 - (2*c*(d + e*x))/(
(c*d + I*e)*(1 - I*c*x))])/e
Rule 4970
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,
0]
Rubi steps \begin{align*}
\text {integral}& = -\frac {(a+b \arctan (c x))^3 \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {(a+b \arctan (c x))^3 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e}-\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}-\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e}+\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-i c x}\right )}{4 e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{4 e} \\
\end{align*}
Mathematica [F(-1)]
Timed out. \[
\int \frac {(a+b \arctan (c x))^3}{d+e x} \, dx=\text {\$Aborted}
\]
[In]
Integrate[(a + b*ArcTan[c*x])^3/(d + e*x),x]
[Out]
$Aborted
Maple [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 18.20 (sec) , antiderivative size = 2398, normalized size of antiderivative =
7.49
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| method | result | size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(2398\) |
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\(\text {Expression too large to display}\) |
\(2398\) |
| parts |
\(\text {Expression too large to display}\) |
\(2405\) |
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[In]
int((a+b*arctan(c*x))^3/(e*x+d),x,method=_RETURNVERBOSE)
[Out]
1/c*(a^3*c*ln(c*e*x+c*d)/e+b^3*c*(ln(c*e*x+c*d)/e*arctan(c*x)^3-3/e*(1/3*arctan(c*x)^3*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)-1/6*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)^3-1/2*I*arctan(c*x)^2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+1/2*arc
tan(c*x)*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+1/4*I*polylog(4,-(1+I*c*x)^2/(c^2*x^2+1))-1/3*c*d/(c*d-I*e)*arcta
n(c*x)^3*ln(1-(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)*polylog(3,(I*e-c*d)/(
c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+1/2*I*c*d/(c*d-I*e)*arctan(c*x)^2*polylog(2,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/
(c^2*x^2+1))-1/4*I*c*d/(c*d-I*e)*polylog(4,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))-1/3*e*arctan(c*x)^3*ln
(1-(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(e+I*d*c)-1/2*e*arctan(c*x)*polylog(3,(I*e-c*d)/(c*d+I*e)*(1+I
*c*x)^2/(c^2*x^2+1))/(e+I*d*c)+1/2*I*e*arctan(c*x)^2*polylog(2,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(e
+I*d*c)-1/4*I*e*polylog(4,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(e+I*d*c)))+3*a*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)))*(c
sgn(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*polylog(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)*polylog(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)))+3*a^2*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]
\[
\int \frac {(a+b \arctan (c x))^3}{d+e x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{e x + d} \,d x }
\]
[In]
integrate((a+b*arctan(c*x))^3/(e*x+d),x, algorithm="fricas")
[Out]
integral((b^3*arctan(c*x)^3 + 3*a*b^2*arctan(c*x)^2 + 3*a^2*b*arctan(c*x) + a^3)/(e*x + d), x)
Sympy [F]
\[
\int \frac {(a+b \arctan (c x))^3}{d+e x} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}}{d + e x}\, dx
\]
[In]
integrate((a+b*atan(c*x))**3/(e*x+d),x)
[Out]
Integral((a + b*atan(c*x))**3/(d + e*x), x)
Maxima [F]
\[
\int \frac {(a+b \arctan (c x))^3}{d+e x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{e x + d} \,d x }
\]
[In]
integrate((a+b*arctan(c*x))^3/(e*x+d),x, algorithm="maxima")
[Out]
a^3*log(e*x + d)/e + integrate(1/32*(28*b^3*arctan(c*x)^3 + 3*b^3*arctan(c*x)*log(c^2*x^2 + 1)^2 + 96*a*b^2*ar
ctan(c*x)^2 + 96*a^2*b*arctan(c*x))/(e*x + d), x)
Giac [F(-1)]
Timed out. \[
\int \frac {(a+b \arctan (c x))^3}{d+e x} \, dx=\text {Timed out}
\]
[In]
integrate((a+b*arctan(c*x))^3/(e*x+d),x, algorithm="giac")
[Out]
Timed out
Mupad [F(-1)]
Timed out. \[
\int \frac {(a+b \arctan (c x))^3}{d+e x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{d+e\,x} \,d x
\]
[In]
int((a + b*atan(c*x))^3/(d + e*x),x)
[Out]
int((a + b*atan(c*x))^3/(d + e*x), x)