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

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

[Out]

-(b*x^4)/12 + (x^3*ArcCot[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 + Po
lyLog[4, I*c*E^((2*I)*a + (2*I)*b*x)]/(8*b^3)

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Rubi [A]  time = 0.265914, 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 = {5174, 2184, 2190, 2531, 6609, 2282, 6589} \[ -\frac{i x \text{PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{4 b^2}+\frac{\text{PolyLog}\left (4,i c e^{2 i a+2 i b x}\right )}{8 b^3}-\frac{x^2 \text{PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}-\frac{1}{6} i x^3 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{1}{3} x^3 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{b x^4}{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*x^4)/12 + (x^3*ArcCot[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 + Po
lyLog[4, I*c*E^((2*I)*a + (2*I)*b*x)]/(8*b^3)

Rule 5174

Int[ArcCot[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^(m
+ 1)*ArcCot[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 \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx &=\frac{1}{3} x^3 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 = 0.196723, size = 140, normalized size = 0.91 \[ \frac{1}{3} x^3 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac{-6 b^2 x^2 \text{PolyLog}\left (2,-\frac{i e^{-2 i (a+b x)}}{c}\right )+6 i b x \text{PolyLog}\left (3,-\frac{i e^{-2 i (a+b x)}}{c}\right )+3 \text{PolyLog}\left (4,-\frac{i e^{-2 i (a+b x)}}{c}\right )+4 i b^3 x^3 \log \left (1+\frac{i e^{-2 i (a+b x)}}{c}\right )}{24 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 22.095, size = 1526, normalized size = 9.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a)))^3+1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))*csgn(e
xp(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*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))-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))^2-1/4*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))*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*csg
n(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*(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+1/12*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*c
sgn(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/(exp(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*b*x^4+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)-1/2*I/b^3*a^3*ln(1-I*exp(I*(b*x+a))*(-I*c)^(1/2))-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*(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)))*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))-1/2/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(I+c)+
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-1/2*I/b^2*a^2*l
n(1+I*exp(I*(b*x+a))*(-I*c)^(1/2))*x-1/12*x^3*Pi*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^3-1/12*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(I*exp(I*(b*x+a)))^2*csgn(I*exp(2*I*(
b*x+a)))-1/3*I*x^3*ln(exp(I*(b*x+a)))-1/12*x^3*Pi*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^3+1/4/b^3*
polylog(2,I*c*exp(2*I*(b*x+a)))*a^2+1/12*x^3*Pi*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^2+1/8*polylo
g(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))^2-1/6*I*x^3*ln(1-I
*c*exp(2*I*(b*x+a)))+1/12*x^3*Pi*csgn(I*(c*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^3+1/6*I*x^3*ln(c*exp(2*I*
(b*x+a))+I)-1/12*x^3*Pi*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))^3+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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 2.50114, size = 482, normalized size = 3.13 \begin{align*} -\frac{2 \, b^{4} x^{4} - 8 \, \pi b^{3} x^{3} - 4 i \, b^{3} x^{3} \log \left (\frac{{\left (c e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{c + 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*(pi-arccot(-c-(1-I*c)*cot(b*x+a))),x, algorithm="fricas")

[Out]

-1/24*(2*b^4*x^4 - 8*pi*b^3*x^3 - 4*I*b^3*x^3*log((c*e^(2*I*b*x + 2*I*a) + I)*e^(-2*I*b*x - 2*I*a)/(c + 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*polylo
g(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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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