3.13.0 ∫ (a+b arctan(cx))2 _________________________

Integrand size = 20, antiderivative size = 460 \[ \int \frac {(a+b \arctan (c x))^2}{d+e x^2} \, dx=\frac {(a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \sqrt {e}} \]

1/2*(a+b*arctan(c*x))^2*ln(2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))/(-d)^(1/2)/e^(1/2)-1

/2*(a+b*arctan(c*x))^2*ln(2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^(1/2)))/(-d)^(1/2)/e^(1/2)-1/

2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))/(-d)^(1/2)/

e^(1/2)+1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^(1/2)))/(

-d)^(1/2)/e^(1/2)+1/4*b^2*polylog(3,1-2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))/(-d)^(1/2

)/e^(1/2)-1/4*b^2*polylog(3,1-2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^(1/2)))/(-d)^(1/2)/e^(1/2

Rubi [A] (veriﬁed)

Time = 0.18 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.100, Rules used = {5034, 4968} \[ \int \frac {(a+b \arctan (c x))^2}{d+e x^2} \, dx=-\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \sqrt {e}} \]

((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*Sqrt[-d]*S

qrt[e]) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*

Sqrt[-d]*Sqrt[e]) - ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*

Sqrt[e])*(1 - I*c*x))])/(Sqrt[-d]*Sqrt[e]) + ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt

[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqr

t[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] +

Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*Sqrt[e])

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] &&

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a

+ b*ArcTan[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-d} (a+b \arctan (c x))^2}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \arctan (c x))^2}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx \\ & = -\frac {\int \frac {(a+b \arctan (c x))^2}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d}}-\frac {\int \frac {(a+b \arctan (c x))^2}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d}} \\ & = \frac {(a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \sqrt {e}} \\ \end{align*}

Mathematica [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{d+e x^2} \, dx=\text {\$Aborted} \]

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

Maple [B] (veriﬁed)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2343 vs. $2 (362 ) = 724$.

Time = 225.41 (sec) , antiderivative size = 2344, normalized size of antiderivative = 5.10


method	result	size

parts	$\text {Expression too large to display}$	$2344$
derivativedivides	$\text {Expression too large to display}$	$5222$
default	$\text {Expression too large to display}$	$5222$

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

a^2/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2))+b^2/c*(-1/2*(-(c^2*d*e)^(1/2)*c^2*d+2*c^2*d*e-(c^2*d*e)^(1/2)*e)*c^2/e

/(c^4*d^2-2*c^2*d*e+e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))*arctan(c*x)

-1/4*I*(-(c^2*d*e)^(1/2)*c^2*d+2*c^2*d*e-(c^2*d*e)^(1/2)*e)*c^2*polylog(3,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-

c^2*d-2*(c^2*d*e)^(1/2)-e))/e/(c^4*d^2-2*c^2*d*e+e^2)+(c^2*d-2*(c^2*d*e)^(1/2)+e)/(c^4*d^2-2*c^2*d*e+e^2)*poly

log(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))*arctan(c*x)*c^2-1/4*I*(-(c^2*d*e)^(1/2)*

c^2*d+2*c^2*d*e-(c^2*d*e)^(1/2)*e)*polylog(3,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))/d

/(c^4*d^2-2*c^2*d*e+e^2)-1/3*(-(c^2*d*e)^(1/2)*c^2*d+2*c^2*d*e-(c^2*d*e)^(1/2)*e)*c^2/e/(c^4*d^2-2*c^2*d*e+e^2

)*arctan(c*x)^3-1/2*(c^2*d*e)^(1/2)/d/e*arctan(c*x)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2

*d*e)^(1/2)-e))-1/3*(-(c^2*d*e)^(1/2)*c^2*d+2*c^2*d*e-(c^2*d*e)^(1/2)*e)/d/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)

^3-1/2*I*(-(c^2*d*e)^(1/2)*c^2*d+2*c^2*d*e-(c^2*d*e)^(1/2)*e)*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2

*(c^2*d*e)^(1/2)-e))*arctan(c*x)^2/d/(c^4*d^2-2*c^2*d*e+e^2)-1/4*I*(c^2*d*e)^(1/2)/d/e*polylog(3,(c^2*d-e)*(1+

I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))+I*(c^2*d-2*(c^2*d*e)^(1/2)+e)*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c

^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))*arctan(c*x)^2*c^2/(c^4*d^2-2*c^2*d*e+e^2)-1/3*(c^2*d*e)^(1/2)/d/e*arct

an(c*x)^3+2/3*(c^2*d-2*(c^2*d*e)^(1/2)+e)/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)^3*c^2-1/2*(-(c^2*d*e)^(1/2)*c^2*

d+2*c^2*d*e-(c^2*d*e)^(1/2)*e)/d/(c^4*d^2-2*c^2*d*e+e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2

*(c^2*d*e)^(1/2)-e))*arctan(c*x)+I*(c^2*d-2*(c^2*d*e)^(1/2)+e)*polylog(3,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c

^2*d-2*(c^2*d*e)^(1/2)-e))*c^2/(2*c^4*d^2-4*c^2*d*e+2*e^2)-1/2*I*(c^2*d*e)^(1/2)/d/e*arctan(c*x)^2*ln(1-(c^2*d

-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))-1/2*I*(-(c^2*d*e)^(1/2)*c^2*d+2*c^2*d*e-(c^2*d*e)^(1

/2)*e)*c^2*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))*arctan(c*x)^2/e/(c^4*d^2-2*c^2

*d*e+e^2))+I*a*b*c^3*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))*arctan(c*x)/e/(c^4*d

^2-2*c^2*d*e+e^2)*(c^2*d*e)^(1/2)*d-2*I*a*b*c*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)

-e))*arctan(c*x)/(c^4*d^2-2*c^2*d*e+e^2)*(c^2*d*e)^(1/2)+a*b*c^3/e/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)^2*(c^2*

d*e)^(1/2)*d-2*a*b*c/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)^2*(c^2*d*e)^(1/2)-I*a*b/c*(c^2*d*e)^(1/2)/d/e*arctan(

c*x)*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))+I*a*b/c*ln(1-(c^2*d-e)*(1+I*c*x)^2/(

c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))*arctan(c*x)/d/(c^4*d^2-2*c^2*d*e+e^2)*(c^2*d*e)^(1/2)*e+1/2*a*b*c^3/e

/(c^4*d^2-2*c^2*d*e+e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))*(c^2*d*e)^(

1/2)*d-a*b*c/(c^4*d^2-2*c^2*d*e+e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))

*(c^2*d*e)^(1/2)+a*b/c/d/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)^2*(c^2*d*e)^(1/2)*e+1/2*a*b/c/d/(c^4*d^2-2*c^2*d*

e+e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))*(c^2*d*e)^(1/2)*e-a*b/c*(c^2*

d*e)^(1/2)/d/e*arctan(c*x)^2-1/2*a*b/c*(c^2*d*e)^(1/2)/d/e*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d

Fricas [F]

\[ \int \frac {(a+b \arctan (c x))^2}{d+e x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{e x^{2} + d} \,d x } \]

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

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

Sympy [F]

\[ \int \frac {(a+b \arctan (c x))^2}{d+e x^2} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{d + e x^{2}}\, dx \]

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

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \arctan (c x))^2}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]

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

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more

Giac [F]

\[ \int \frac {(a+b \arctan (c x))^2}{d+e x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{e x^{2} + d} \,d x } \]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{d+e x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{e\,x^2+d} \,d x \]

\(\int \frac {(a+b \arctan (c x))^2}{d+e x^2} \, dx\) [1264]

Optimal result