3.65 \(\int x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x)) \, dx\)

Optimal. Leaf size=154 \[ \frac{i x \text{PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac{\text{PolyLog}\left (4,i c e^{2 i a+2 i b x}\right )}{8 b^3}+\frac{x^2 \text{PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}+\frac{1}{6} i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{1}{3} x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{b x^4}{12} \]

[Out]

(b*x^4)/12 + (x^3*ArcTan[c + (1 - I*c)*Cot[a + b*x]])/3 + (I/6)*x^3*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)] + (x^
2*PolyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) + ((I/4)*x*PolyLog[3, I*c*E^((2*I)*a + (2*I)*b*x)])/b^2 - Pol
yLog[4, I*c*E^((2*I)*a + (2*I)*b*x)]/(8*b^3)

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Rubi [A]  time = 0.255871, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5173, 2184, 2190, 2531, 6609, 2282, 6589} \[ \frac{i x \text{PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac{\text{PolyLog}\left (4,i c e^{2 i a+2 i b x}\right )}{8 b^3}+\frac{x^2 \text{PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}+\frac{1}{6} i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{1}{3} x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{b x^4}{12} \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcTan[c + (1 - I*c)*Cot[a + b*x]],x]

[Out]

(b*x^4)/12 + (x^3*ArcTan[c + (1 - I*c)*Cot[a + b*x]])/3 + (I/6)*x^3*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)] + (x^
2*PolyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) + ((I/4)*x*PolyLog[3, I*c*E^((2*I)*a + (2*I)*b*x)])/b^2 - Pol
yLog[4, I*c*E^((2*I)*a + (2*I)*b*x)]/(8*b^3)

Rule 5173

Int[ArcTan[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^(m
+ 1)*ArcTan[c + d*Cot[a + b*x]])/(f*(m + 1)), x] - Dist[(I*b)/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - I*d - c*
E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, -1]

Rule 2184

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[((c + d*x)^m*(F^(g*(e + f*x)))^n)/(a + b*(F^(g*(e + f*x)))^n)
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x)) \, dx &=\frac{1}{3} x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))-\frac{1}{3} (i b) \int \frac{x^3}{-i (1-i c)+c-c e^{2 i a+2 i b x}} \, dx\\ &=\frac{b x^4}{12}+\frac{1}{3} x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{1}{3} (b c) \int \frac{e^{2 i a+2 i b x} x^3}{-i (1-i c)+c-c e^{2 i a+2 i b x}} \, dx\\ &=\frac{b x^4}{12}+\frac{1}{3} x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{1}{6} i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )-\frac{1}{2} i \int x^2 \log \left (1-\frac{c e^{2 i a+2 i b x}}{-i (1-i c)+c}\right ) \, dx\\ &=\frac{b x^4}{12}+\frac{1}{3} x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{1}{6} i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{x^2 \text{Li}_2\left (i c e^{2 i a+2 i b x}\right )}{4 b}-\frac{\int x \text{Li}_2\left (\frac{c e^{2 i a+2 i b x}}{-i (1-i c)+c}\right ) \, dx}{2 b}\\ &=\frac{b x^4}{12}+\frac{1}{3} x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{1}{6} i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{x^2 \text{Li}_2\left (i c e^{2 i a+2 i b x}\right )}{4 b}+\frac{i x \text{Li}_3\left (i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac{i \int \text{Li}_3\left (\frac{c e^{2 i a+2 i b x}}{-i (1-i c)+c}\right ) \, dx}{4 b^2}\\ &=\frac{b x^4}{12}+\frac{1}{3} x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{1}{6} i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{x^2 \text{Li}_2\left (i c e^{2 i a+2 i b x}\right )}{4 b}+\frac{i x \text{Li}_3\left (i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_3(i c x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^3}\\ &=\frac{b x^4}{12}+\frac{1}{3} x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{1}{6} i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{x^2 \text{Li}_2\left (i c e^{2 i a+2 i b x}\right )}{4 b}+\frac{i x \text{Li}_3\left (i c e^{2 i a+2 i b x}\right )}{4 b^2}-\frac{\text{Li}_4\left (i c e^{2 i a+2 i b x}\right )}{8 b^3}\\ \end{align*}

Mathematica [A]  time = 0.32611, size = 136, normalized size = 0.88 \[ \frac{1}{24} \left (\frac{6 i x \text{PolyLog}\left (3,-\frac{i e^{-2 i (a+b x)}}{c}\right )}{b^2}+\frac{3 \text{PolyLog}\left (4,-\frac{i e^{-2 i (a+b x)}}{c}\right )}{b^3}-\frac{6 x^2 \text{PolyLog}\left (2,-\frac{i e^{-2 i (a+b x)}}{c}\right )}{b}+4 i x^3 \log \left (1+\frac{i e^{-2 i (a+b x)}}{c}\right )+8 x^3 \tan ^{-1}(c+(1-i c) \cot (a+b x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcTan[c + (1 - I*c)*Cot[a + b*x]],x]

[Out]

(8*x^3*ArcTan[c + (1 - I*c)*Cot[a + b*x]] + (4*I)*x^3*Log[1 + I/(c*E^((2*I)*(a + b*x)))] - (6*x^2*PolyLog[2, (
-I)/(c*E^((2*I)*(a + b*x)))])/b + ((6*I)*x*PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))])/b^2 + (3*PolyLog[4, (-I)/
(c*E^((2*I)*(a + b*x)))])/b^3)/24

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Maple [C]  time = 18.458, size = 1532, normalized size = 10. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^2*arctan(-c-(1-I*c)*cot(b*x+a)),x)

[Out]

1/2*I/b^3*a^3*ln(1-I*exp(I*(b*x+a))*(-I*c)^(1/2))+1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a)))^3+1/6*Pi*x^3-1/4/b^3*po
lylog(2,I*c*exp(2*I*(b*x+a)))*a^2+1/2/b^3*a^2*dilog(1-I*exp(I*(b*x+a))*(-I*c)^(1/2))+1/2/b^3*a^2*dilog(1+I*exp
(I*(b*x+a))*(-I*c)^(1/2))+1/4*x^2*polylog(2,I*c*exp(2*I*(b*x+a)))/b+1/6*I*x^3*ln(1-I*c*exp(2*I*(b*x+a)))-1/8*p
olylog(4,I*c*exp(2*I*(b*x+a)))/b^3-1/12*x^3*Pi*csgn((c*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^2-1/12*x^3*Pi
*csgn(I*(c*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^3+1/12*x^3*Pi*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))^3+1/12*b
*x^4-1/12*x^3*Pi*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^2+1/12*x^3*Pi*csgn((c*exp(2*I*(b*x+a))+I)/(
exp(2*I*(b*x+a))-1))^3+1/12*x^3*Pi*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^3+1/12*x^3*Pi*csgn(I*(c*e
xp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))*csgn((c*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^2+1/2*I/b^2*a^2*ln(
1+I*exp(I*(b*x+a))*(-I*c)^(1/2))*x-1/2*I/b^2*ln(1-I*c*exp(2*I*(b*x+a)))*x*a^2+1/2*I/b^2*a^2*ln(1-I*exp(I*(b*x+
a))*(-I*c)^(1/2))*x-1/6*I*x^3*ln(c*exp(2*I*(b*x+a))+I)+1/12*x^3*Pi*csgn(I*(I+c))*csgn(I/(exp(2*I*(b*x+a))-1))*
csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))+1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^3+1/3*I*x^3
*ln(exp(I*(b*x+a)))+1/6*I*x^3*ln(I+c)+1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))*
csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))-1/12*x^3*Pi*csgn(I*(c*exp(2*I*(b*x+a))+I))*csgn(I/(exp(2*I
*(b*x+a))-1))*csgn(I*(c*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))+1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a))*(I+c)/(e
xp(2*I*(b*x+a))-1))*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))-1/12*x^3*Pi*csgn(I*(I+c))*csgn(I*(I+c)/(
exp(2*I*(b*x+a))-1))^2-1/12*x^3*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))^2+1/12*x^3*
Pi*csgn(I*(c*exp(2*I*(b*x+a))+I))*csgn(I*(c*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^2+1/12*x^3*Pi*csgn(I/(ex
p(2*I*(b*x+a))-1))*csgn(I*(c*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^2+1/12*x^3*Pi*csgn(I*exp(I*(b*x+a)))^2*
csgn(I*exp(2*I*(b*x+a)))+1/2*I/b^3*a^3*ln(1+I*exp(I*(b*x+a))*(-I*c)^(1/2))-1/3*I/b^3*ln(1-I*c*exp(2*I*(b*x+a))
)*a^3-1/6*I/b^3*a^3*ln(c*exp(2*I*(b*x+a))+I)+1/4*I*x*polylog(3,I*c*exp(2*I*(b*x+a)))/b^2-1/12*x^3*Pi*csgn(I*(c
*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))*csgn((c*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))-1/12*x^3*Pi*csgn(
I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^2-1/12*x^3*Pi
*csgn(I*exp(2*I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^2-1/12*x^3*Pi*csgn(I*(I+c)/(exp(
2*I*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^2-1/6*x^3*Pi*csgn(I*exp(I*(b*x+a)))*csgn(
I*exp(2*I*(b*x+a)))^2

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Maxima [B]  time = 1.13643, size = 417, normalized size = 2.71 \begin{align*} \frac{\frac{{\left ({\left (b x + a\right )}^{3} - 3 \,{\left (b x + a\right )}^{2} a + 3 \,{\left (b x + a\right )} a^{2}\right )} \arctan \left ({\left (-i \, c + 1\right )} \cot \left (b x + a\right ) + c\right )}{b^{2}} + \frac{3 \,{\left (-3 i \,{\left (b x + a\right )}^{4} + 12 i \,{\left (b x + a\right )}^{3} a - 18 i \,{\left (b x + a\right )}^{2} a^{2} +{\left (-8 i \,{\left (b x + a\right )}^{3} + 18 i \,{\left (b x + a\right )}^{2} a - 18 i \,{\left (b x + a\right )} a^{2}\right )} \arctan \left (c \cos \left (2 \, b x + 2 \, a\right ), c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) +{\left (-12 i \,{\left (b x + a\right )}^{2} + 18 i \,{\left (b x + a\right )} a - 9 i \, a^{2}\right )}{\rm Li}_2\left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) +{\left (4 \,{\left (b x + a\right )}^{3} - 9 \,{\left (b x + a\right )}^{2} a + 9 \,{\left (b x + a\right )} a^{2}\right )} \log \left (c^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + c^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \,{\left (4 \, b x + a\right )}{\rm Li}_{3}(i \, c e^{\left (2 i \, b x + 2 i \, a\right )}) + 6 i \,{\rm Li}_{4}(i \, c e^{\left (2 i \, b x + 2 i \, a\right )})\right )}{\left (i \, c - 1\right )}}{b^{2}{\left (12 \, c + 12 i\right )}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x^2*arctan(-c-(1-I*c)*cot(b*x+a)),x, algorithm="maxima")

[Out]

1/3*(((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(b*x + a)*a^2)*arctan((-I*c + 1)*cot(b*x + a) + c)/b^2 + 3*(-3*I*(b*x
+ a)^4 + 12*I*(b*x + a)^3*a - 18*I*(b*x + a)^2*a^2 + (-8*I*(b*x + a)^3 + 18*I*(b*x + a)^2*a - 18*I*(b*x + a)*a
^2)*arctan2(c*cos(2*b*x + 2*a), c*sin(2*b*x + 2*a) + 1) + (-12*I*(b*x + a)^2 + 18*I*(b*x + a)*a - 9*I*a^2)*dil
og(I*c*e^(2*I*b*x + 2*I*a)) + (4*(b*x + a)^3 - 9*(b*x + a)^2*a + 9*(b*x + a)*a^2)*log(c^2*cos(2*b*x + 2*a)^2 +
c^2*sin(2*b*x + 2*a)^2 + 2*c*sin(2*b*x + 2*a) + 1) + 3*(4*b*x + a)*polylog(3, I*c*e^(2*I*b*x + 2*I*a)) + 6*I*
polylog(4, I*c*e^(2*I*b*x + 2*I*a)))*(I*c - 1)/(b^2*(12*c + 12*I)))/b

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Fricas [C]  time = 1.94354, size = 459, normalized size = 2.98 \begin{align*} \frac{2 \, b^{4} x^{4} + 4 i \, b^{3} x^{3} \log \left (-\frac{{\left (c + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} + i}\right ) + 6 \, b^{2} x^{2}{\rm Li}_2\left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 2 \, a^{4} - 4 i \, a^{3} \log \left (\frac{c e^{\left (2 i \, b x + 2 i \, a\right )} + i}{c}\right ) + 6 i \, b x{\rm polylog}\left (3, i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) +{\left (4 i \, b^{3} x^{3} + 4 i \, a^{3}\right )} \log \left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 3 \,{\rm polylog}\left (4, i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x^2*arctan(-c-(1-I*c)*cot(b*x+a)),x, algorithm="fricas")

[Out]

1/24*(2*b^4*x^4 + 4*I*b^3*x^3*log(-(c + I)*e^(2*I*b*x + 2*I*a)/(c*e^(2*I*b*x + 2*I*a) + I)) + 6*b^2*x^2*dilog(
I*c*e^(2*I*b*x + 2*I*a)) - 2*a^4 - 4*I*a^3*log((c*e^(2*I*b*x + 2*I*a) + I)/c) + 6*I*b*x*polylog(3, I*c*e^(2*I*
b*x + 2*I*a)) + (4*I*b^3*x^3 + 4*I*a^3)*log(-I*c*e^(2*I*b*x + 2*I*a) + 1) - 3*polylog(4, I*c*e^(2*I*b*x + 2*I*
a)))/b^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{i b \int \frac{x^{3}}{c e^{2 i a} e^{2 i b x} + i}\, dx}{3} + \frac{i x^{3} \log{\left (- i c - \frac{i c}{e^{2 i a} e^{2 i b x} - 1} - \frac{i c e^{i a} e^{i b x}}{e^{i a} e^{i b x} - e^{- i a} e^{- i b x}} + 1 + \frac{1}{e^{2 i a} e^{2 i b x} - 1} + \frac{e^{i a} e^{i b x}}{e^{i a} e^{i b x} - e^{- i a} e^{- i b x}} \right )}}{6} - \frac{i x^{3} \log{\left (i c + \frac{i c}{e^{2 i a} e^{2 i b x} - 1} + \frac{i c e^{i a} e^{i b x}}{e^{i a} e^{i b x} - e^{- i a} e^{- i b x}} + 1 - \frac{1}{e^{2 i a} e^{2 i b x} - 1} - \frac{e^{i a} e^{i b x}}{e^{i a} e^{i b x} - e^{- i a} e^{- i b x}} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x**2*atan(-c-(1-I*c)*cot(b*x+a)),x)

[Out]

I*b*Integral(x**3/(c*exp(2*I*a)*exp(2*I*b*x) + I), x)/3 + I*x**3*log(-I*c - I*c/(exp(2*I*a)*exp(2*I*b*x) - 1)
- I*c*exp(I*a)*exp(I*b*x)/(exp(I*a)*exp(I*b*x) - exp(-I*a)*exp(-I*b*x)) + 1 + 1/(exp(2*I*a)*exp(2*I*b*x) - 1)
+ exp(I*a)*exp(I*b*x)/(exp(I*a)*exp(I*b*x) - exp(-I*a)*exp(-I*b*x)))/6 - I*x**3*log(I*c + I*c/(exp(2*I*a)*exp(
2*I*b*x) - 1) + I*c*exp(I*a)*exp(I*b*x)/(exp(I*a)*exp(I*b*x) - exp(-I*a)*exp(-I*b*x)) + 1 - 1/(exp(2*I*a)*exp(
2*I*b*x) - 1) - exp(I*a)*exp(I*b*x)/(exp(I*a)*exp(I*b*x) - exp(-I*a)*exp(-I*b*x)))/6

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -x^{2} \arctan \left (-{\left (-i \, c + 1\right )} \cot \left (b x + a\right ) - c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x^2*arctan(-c-(1-I*c)*cot(b*x+a)),x, algorithm="giac")

[Out]

integrate(-x^2*arctan(-(-I*c + 1)*cot(b*x + a) - c), x)