\(\int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx\) [113]

Optimal result

Integrand size = 19, antiderivative size = 367 \[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\frac {\cot ^{-1}(d+e x) \log \left (\frac {2 e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (i-d)+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1-i (d+e x))}\right )}{\sqrt {b^2-4 a c}}-\frac {\cot ^{-1}(d+e x) \log \left (\frac {2 e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (i-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1-i (d+e x))}\right )}{\sqrt {b^2-4 a c}}+\frac {i \operatorname {PolyLog}\left (2,1+\frac {2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 i c-2 c d+b e-\sqrt {b^2-4 a c} e\right ) (1-i (d+e x))}\right )}{2 \sqrt {b^2-4 a c}}-\frac {i \operatorname {PolyLog}\left (2,1+\frac {2 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c (i-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1-i (d+e x))}\right )}{2 \sqrt {b^2-4 a c}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {632, 212, 6860, 5156, 4967, 2449, 2352, 2497} \[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\frac {i \operatorname {PolyLog}\left (2,\frac {2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (-2 d c+2 i c+b e-\sqrt {b^2-4 a c} e\right ) (1-i (d+e x))}+1\right )}{2 \sqrt {b^2-4 a c}}-\frac {i \operatorname {PolyLog}\left (2,\frac {2 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c (i-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1-i (d+e x))}+1\right )}{2 \sqrt {b^2-4 a c}}+\frac {\cot ^{-1}(d+e x) \log \left (\frac {2 e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{(1-i (d+e x)) \left (e \left (b-\sqrt {b^2-4 a c}\right )+2 c (-d+i)\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {\cot ^{-1}(d+e x) \log \left (\frac {2 e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{(1-i (d+e x)) \left (e \left (\sqrt {b^2-4 a c}+b\right )+2 c (-d+i)\right )}\right )}{\sqrt {b^2-4 a c}} \]

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4967

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x]))*(Log[2/(1
- I*c*x)]/e), x] + (-Dist[b*(c/e), Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] + Dist[b*(c/e), Int[Log[2*c*(
(d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(1 + c^2*x^2), x], x] + Simp[(a + b*ArcCot[c*x])*(Log[2*c*((d + e*x)/((c
*d + I*e)*(1 - I*c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 5156

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x]
&& IGtQ[p, 0]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

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

Mathematica [F]

\[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx \]

[In]

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

[Out]

Integrate[ArcCot[d + e*x]/(a + b*x + c*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. 958 vs. \(2 (329 ) = 658\).

Time = 2.49 (sec) , antiderivative size = 959, normalized size of antiderivative = 2.61

method	result	size
risch	\(\frac {i e \pi \arctan \left (\frac {i b e -2 i c d -2 \left (-i e x -i d +1\right ) c +2 c}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \ln \left (-i e x -i d +1\right ) \ln \left (\frac {i b e -2 i c d -2 \left (-i e x -i d +1\right ) c +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}+2 c}{i b e -2 i c d +2 c +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}}\right )}{2 \sqrt {4 a c \,e^{2}-b^{2} e^{2}}}+\frac {e \ln \left (-i e x -i d +1\right ) \ln \left (\frac {i b e -2 i c d -2 \left (-i e x -i d +1\right ) c -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}+2 c}{i b e -2 i c d +2 c -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}}\right )}{2 \sqrt {4 a c \,e^{2}-b^{2} e^{2}}}-\frac {e \operatorname {dilog}\left (\frac {i b e -2 i c d -2 \left (-i e x -i d +1\right ) c +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}+2 c}{i b e -2 i c d +2 c +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}}\right )}{2 \sqrt {4 a c \,e^{2}-b^{2} e^{2}}}+\frac {e \operatorname {dilog}\left (\frac {i b e -2 i c d -2 \left (-i e x -i d +1\right ) c -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}+2 c}{i b e -2 i c d +2 c -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}}\right )}{2 \sqrt {4 a c \,e^{2}-b^{2} e^{2}}}-\frac {e \ln \left (i e x +i d +1\right ) \ln \left (\frac {i b e -2 i c d +2 \left (i e x +i d +1\right ) c -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}{i b e -2 i c d -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}\right )}{2 \sqrt {4 a c \,e^{2}-b^{2} e^{2}}}+\frac {e \ln \left (i e x +i d +1\right ) \ln \left (\frac {i b e -2 i c d +2 \left (i e x +i d +1\right ) c +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}{i b e -2 i c d +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}\right )}{2 \sqrt {4 a c \,e^{2}-b^{2} e^{2}}}-\frac {e \operatorname {dilog}\left (\frac {i b e -2 i c d +2 \left (i e x +i d +1\right ) c -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}{i b e -2 i c d -\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}\right )}{2 \sqrt {4 a c \,e^{2}-b^{2} e^{2}}}+\frac {e \operatorname {dilog}\left (\frac {i b e -2 i c d +2 \left (i e x +i d +1\right ) c +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}{i b e -2 i c d +\sqrt {4 a c \,e^{2}-b^{2} e^{2}}-2 c}\right )}{2 \sqrt {4 a c \,e^{2}-b^{2} e^{2}}}\)	\(959\)
derivativedivides	\(\text {Expression too large to display}\)	\(4611\)
default	\(\text {Expression too large to display}\)	\(4611\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\int { \frac {\operatorname {arccot}\left (e x + d\right )}{c x^{2} + b x + a} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

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(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\int \frac {\mathrm {acot}\left (d+e\,x\right )}{c\,x^2+b\,x+a} \,d x \]

[In]

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

[Out]

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