3.1.12 \(\int \frac {(a+b \text {ArcTan}(c x))^2}{d+e x} \, dx\) [12]

Optimal. Leaf size=223 \[ -\frac {(a+b \text {ArcTan}(c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {(a+b \text {ArcTan}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {i b (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e}-\frac {i b (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e} \]

[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]
time = 0.03, 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 = {4968} \begin {gather*} -\frac {i b (a+b \text {ArcTan}(c x)) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {(a+b \text {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 \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) (a+b \text {ArcTan}(c x))}{e}-\frac {\log \left (\frac {2}{1-i c x}\right ) (a+b \text {ArcTan}(c x))^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} \end {gather*}

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 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*} \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 [F]
time = 78.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \text {ArcTan}(c x))^2}{d+e x} \, dx \end {gather*}

Verification is not applicable to the result.

[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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.90, size = 1324, normalized size = 5.94

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1324\)
default	\(\text {Expression too large to display}\)	\(1324\)

Verification of antiderivative is not currently implemented for this CAS.

[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-b^2*c/e*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)-1/2*I*b^2*c/e*arctan(c*x)^2*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+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*Pi-I*b^
2*c*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*c*d)+I*a*b*c*ln(c*e*x+c*d)/e*ln((I
*e-c*e*x)/(c*d+I*e))+1/2*I*b^2*c/e*arctan(c*x)^2*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+I*c*x)^2/(c^2*x^2+1)+1))^3*Pi-I*a*b*c*ln(c*e*x+c*d)/e*ln((I*e+c*e*x)/(I*e-c*d))-1/2*b^2*c/e*p
olylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+b^2*c^2/e*d/(-I*e+c*d)*arctan(c*x)^2*ln(1-(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(
c^2*x^2+1))+I*a*b*c/e*dilog((I*e-c*e*x)/(c*d+I*e))+1/2*b^2*c^2/e*d/(-I*e+c*d)*polylog(3,(I*e-c*d)/(c*d+I*e)*(1
+I*c*x)^2/(c^2*x^2+1))+b^2*c*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*c*d)+I*b^2*c
/e*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+1/2*b^2*c*polylog(3,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^
2+1))/(e+I*c*d)+2*a*b*c*ln(c*e*x+c*d)/e*arctan(c*x)-1/2*I*b^2*c/e*arctan(c*x)^2*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+I*c*x)^2/(c^2*x^2+1)+1))^2*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))*Pi-I*a*b*c/e*dilog((I*e+c*e*x)/(I*e-c*d))-I*b^2*c^2/e*d/(-I*e+c*d)*arc
tan(c*x)*polylog(2,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+1/2*I*b^2*c/e*arctan(c*x)^2*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+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I/((1+I*c*x)^2/(c^2*
x^2+1)+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))*Pi)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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*e^(-1)*log(x*e + d) + 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))/
(x*e + d), x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)/(x*e + d), x)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

sage0*x

________________________________________________________________________________________

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

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)

________________________________________________________________________________________