3.12 \(\int \frac {(a+b \tan ^{-1}(c x))^2}{d+e x} \, dx\)

Optimal. Leaf size=223 \[ -\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e}+\frac {i b \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e}-\frac {\log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{e}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e} \]

[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]  time = 0.05, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {4858} \[ -\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e}+\frac {i b \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e}-\frac {\log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{e} \]

Antiderivative was successfully verified.

[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 4858

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] + Sim
p[(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] + Simp
[(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] && Ne
Q[c^2*d^2 + e^2, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{d+e x} \, dx &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 15.12, size = 741, normalized size = 3.32 \[ \frac {6 a^2 c d \log (d+e x)+12 a b c d \left (\tan ^{-1}(c x) \left (\frac {1}{2} \log \left (c^2 x^2+1\right )+\log \left (\sin \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )\right )\right )+\frac {1}{2} \left (-\log \left (\frac {2}{\sqrt {c^2 x^2+1}}\right ) \left (\pi -2 \tan ^{-1}(c x)\right )-i \text {Li}_2\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )^2+2 \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right ) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-2 \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right ) \log \left (2 \sin \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )\right )-i \text {Li}_2\left (-e^{-2 i \tan ^{-1}(c x)}\right )-\frac {1}{4} i \left (\pi -2 \tan ^{-1}(c x)\right )^2+\left (\pi -2 \tan ^{-1}(c x)\right ) \log \left (1+e^{-2 i \tan ^{-1}(c x)}\right )\right )\right )+b^2 \left (-2 \tan ^{-1}(c x) \left (\tan ^{-1}(c x)^2 \left (2 e \sqrt {\frac {c^2 d^2}{e^2}+1} e^{i \tan ^{-1}\left (\frac {c d}{e}\right )}-i c d-e\right )+3 c d \left (\pi \left (\log \left (-\frac {2 i}{c x-i}\right )-\log \left (1+e^{-2 i \tan ^{-1}(c x)}\right )\right )-2 \tan ^{-1}\left (\frac {c d}{e}\right ) \left (\log \left (\frac {e^{-i \tan ^{-1}\left (\frac {c d}{e}\right )} \left ((c x-i) e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )}+c x+i\right )}{2 \sqrt {c^2 x^2+1}}\right )+\log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-\log \left (\sin \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )\right )-\log \left (-i e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \sin \left (2 \tan ^{-1}(c x)\right )-e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \cos \left (2 \tan ^{-1}(c x)\right )+1\right )\right )\right )-3 c d \tan ^{-1}(c x) \left (2 \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-\log \left (-i e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \sin \left (2 \tan ^{-1}(c x)\right )-e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \cos \left (2 \tan ^{-1}(c x)\right )+1\right )\right )\right )-6 i c d \tan ^{-1}(c x) \text {Li}_2\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+3 c d \text {Li}_3\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+2 \tan ^{-1}(c x)^2 \left ((e+i c d) \tan ^{-1}(c x)-3 c d \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+6 i c d \tan ^{-1}(c x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )-3 c d \text {Li}_3\left (-e^{2 i \tan ^{-1}(c x)}\right )\right )}{6 c d e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*a^2*c*d*Log[d + e*x] + 12*a*b*c*d*(ArcTan[c*x]*(Log[1 + c^2*x^2]/2 + Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]
]) + ((-1/4*I)*(Pi - 2*ArcTan[c*x])^2 - I*(ArcTan[(c*d)/e] + ArcTan[c*x])^2 + (Pi - 2*ArcTan[c*x])*Log[1 + E^(
(-2*I)*ArcTan[c*x])] + 2*(ArcTan[(c*d)/e] + ArcTan[c*x])*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] -
(Pi - 2*ArcTan[c*x])*Log[2/Sqrt[1 + c^2*x^2]] - 2*(ArcTan[(c*d)/e] + ArcTan[c*x])*Log[2*Sin[ArcTan[(c*d)/e] +
ArcTan[c*x]]] - I*PolyLog[2, -E^((-2*I)*ArcTan[c*x])] - I*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))
])/2) + b^2*(2*ArcTan[c*x]^2*((I*c*d + e)*ArcTan[c*x] - 3*c*d*Log[1 + E^((2*I)*ArcTan[c*x])]) - 2*ArcTan[c*x]*
(((-I)*c*d - e + 2*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e]))*ArcTan[c*x]^2 - 3*c*d*ArcTan[c*x]*(2*Log[1
- E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2
*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]]) + 3*c*d*(Pi*(-Log[1 + E^((-2*I)*ArcTan[c*x])] + Log[(-2*I)/(-I + c*x
)]) - 2*ArcTan[(c*d)/e]*(Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + Log[(I + c*x + E^((2*I)*ArcTan[(
c*d)/e])*(-I + c*x))/(2*E^(I*ArcTan[(c*d)/e])*Sqrt[1 + c^2*x^2])] - Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*Ar
cTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] - Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]]))) + (6
*I)*c*d*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (6*I)*c*d*ArcTan[c*x]*PolyLog[2, E^((2*I)*(ArcTan[(c*
d)/e] + ArcTan[c*x]))] - 3*c*d*PolyLog[3, -E^((2*I)*ArcTan[c*x])] + 3*c*d*PolyLog[3, E^((2*I)*(ArcTan[(c*d)/e]
+ ArcTan[c*x]))]))/(6*c*d*e)

________________________________________________________________________________________

fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{2}}{e x + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [C]  time = 1.23, size = 1297, normalized size = 5.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*ln(c*e*x+c*d)/e+b^2*ln(c*e*x+c*d)/e*arctan(c*x)^2-b^2/e*arctan(c*x)^2*ln(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*
(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)+I*a*b*ln(c*e*x+c*d)/e*ln((I*e-c*e*x)/(d*c+I*e))-I*a*b/e*dilog((I*e+c*e*x)/(I*
e-d*c))+1/2*I*b^2/e*Pi*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/
(c^2*x^2+1)+1))^3*arctan(c*x)^2-I*a*b*ln(c*e*x+c*d)/e*ln((I*e+c*e*x)/(I*e-d*c))-1/2*I*b^2/e*Pi*csgn(I/((1+I*c*
x)^2/(c^2*x^2+1)+1))*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c
^2*x^2+1)+1))^2*arctan(c*x)^2-1/2*b^2/e*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+c*b^2/e*d/(d*c-I*e)*arctan(c*x)^2*
ln(1-(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+1/2*I*b^2/e*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*(-
I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*(-I*(1+I*
c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c))*arctan(c*x)^2+1/2*c*b^2/e*d/(d*c-I*e)*polylog(3,(I*
e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+b^2*arctan(c*x)^2*ln(1-(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/
(e+I*d*c)-I*b^2*arctan(c*x)*polylog(2,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(e+I*d*c)+1/2*b^2*polylog(3
,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(e+I*d*c)+2*a*b*ln(c*e*x+c*d)/e*arctan(c*x)+I*a*b/e*dilog((I*e-c
*e*x)/(d*c+I*e))+I*b^2/e*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-I*c*b^2/e*d/(d*c-I*e)*arctan(c*x)*pol
ylog(2,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))-1/2*I*b^2/e*Pi*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1
+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*
x)^2/(c^2*x^2+1)+I*e+d*c))*arctan(c*x)^2

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{2} \log \left (e x + d\right )}{e} + \int \frac {12 \, b^{2} \arctan \left (c x\right )^{2} + b^{2} \log \left (c^{2} x^{2} + 1\right )^{2} + 32 \, a b \arctan \left (c x\right )}{16 \, {\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{d+e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x))^2/(d + e*x),x)

[Out]

int((a + b*atan(c*x))^2/(d + e*x), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________