3.66 \(\int x \tan ^{-1}(c+(1-i c) \cot (a+b x)) \, dx\)
Optimal. Leaf size=123 \[ \frac{i \text{PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{8 b^2}+\frac{x \text{PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}+\frac{1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{1}{2} x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{b x^3}{6} \]
[Out]
(b*x^3)/6 + (x^2*ArcTan[c + (1 - I*c)*Cot[a + b*x]])/2 + (I/4)*x^2*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)] + (x*P
olyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) + ((I/8)*PolyLog[3, I*c*E^((2*I)*a + (2*I)*b*x)])/b^2
________________________________________________________________________________________
Rubi [A] time = 0.222363, antiderivative size = 123, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used =
{5173, 2184, 2190, 2531, 2282, 6589} \[ \frac{i \text{PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{8 b^2}+\frac{x \text{PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}+\frac{1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{1}{2} x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{b x^3}{6} \]
Antiderivative was successfully verified.
[In]
Int[x*ArcTan[c + (1 - I*c)*Cot[a + b*x]],x]
[Out]
(b*x^3)/6 + (x^2*ArcTan[c + (1 - I*c)*Cot[a + b*x]])/2 + (I/4)*x^2*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)] + (x*P
olyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) + ((I/8)*PolyLog[3, I*c*E^((2*I)*a + (2*I)*b*x)])/b^2
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 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 \tan ^{-1}(c+(1-i c) \cot (a+b x)) \, dx &=\frac{1}{2} x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x))-\frac{1}{2} (i b) \int \frac{x^2}{-i (1-i c)+c-c e^{2 i a+2 i b x}} \, dx\\ &=\frac{b x^3}{6}+\frac{1}{2} x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{1}{2} (b c) \int \frac{e^{2 i a+2 i b x} x^2}{-i (1-i c)+c-c e^{2 i a+2 i b x}} \, dx\\ &=\frac{b x^3}{6}+\frac{1}{2} x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )-\frac{1}{2} i \int x \log \left (1-\frac{c e^{2 i a+2 i b x}}{-i (1-i c)+c}\right ) \, dx\\ &=\frac{b x^3}{6}+\frac{1}{2} x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{x \text{Li}_2\left (i c e^{2 i a+2 i b x}\right )}{4 b}-\frac{\int \text{Li}_2\left (\frac{c e^{2 i a+2 i b x}}{-i (1-i c)+c}\right ) \, dx}{4 b}\\ &=\frac{b x^3}{6}+\frac{1}{2} x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{x \text{Li}_2\left (i c e^{2 i a+2 i b x}\right )}{4 b}+\frac{i \operatorname{Subst}\left (\int \frac{\text{Li}_2(i c x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^2}\\ &=\frac{b x^3}{6}+\frac{1}{2} x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x))+\frac{1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac{x \text{Li}_2\left (i c e^{2 i a+2 i b x}\right )}{4 b}+\frac{i \text{Li}_3\left (i c e^{2 i a+2 i b x}\right )}{8 b^2}\\ \end{align*}
Mathematica [A] time = 0.206048, size = 110, normalized size = 0.89 \[ \frac{i \left (2 i b x \text{PolyLog}\left (2,-\frac{i e^{-2 i (a+b x)}}{c}\right )+\text{PolyLog}\left (3,-\frac{i e^{-2 i (a+b x)}}{c}\right )+2 b^2 x^2 \log \left (1+\frac{i e^{-2 i (a+b x)}}{c}\right )\right )}{8 b^2}+\frac{1}{2} x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x)) \]
Antiderivative was successfully verified.
[In]
Integrate[x*ArcTan[c + (1 - I*c)*Cot[a + b*x]],x]
[Out]
(x^2*ArcTan[c + (1 - I*c)*Cot[a + b*x]])/2 + ((I/8)*(2*b^2*x^2*Log[1 + I/(c*E^((2*I)*(a + b*x)))] + (2*I)*b*x*
PolyLog[2, (-I)/(c*E^((2*I)*(a + b*x)))] + PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))]))/b^2
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Maple [C] time = 9.3, size = 1497, normalized size = 12.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-x*arctan(-c-(1-I*c)*cot(b*x+a)),x)
[Out]
-1/8*x^2*Pi*csgn(I*(I+c))*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))^2-1/8*x^2*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*
(I+c)/(exp(2*I*(b*x+a))-1))^2-1/8*x^2*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))^2-1/8*x^2*Pi*csgn((c*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^2+1/4*Pi*x^2-1/2*I/b*a*ln(1+I
*exp(I*(b*x+a))*(-I*c)^(1/2))*x+1/8*x^2*Pi*csgn(I*exp(2*I*(b*x+a)))^3-1/2*I/b*a*ln(1-I*exp(I*(b*x+a))*(-I*c)^(
1/2))*x+1/8*x^2*Pi*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^3-1/8*x^2*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+1/4*x*polylog(2,I*c*exp(2*I*(b*x
+a)))/b+1/2*I/b*ln(1-I*c*exp(2*I*(b*x+a)))*x*a-1/8*x^2*Pi*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^2+
1/8*x^2*Pi*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^3-1/8*x^2*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))+1/8*x^2*Pi*csgn(I*(I+c))*csgn(I/(e
xp(2*I*(b*x+a))-1))*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))+1/8*x^2*Pi*csgn((c*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a)
)-1))^3+1/8*x^2*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))+1/6*b*x^3-1/4*I*x^2*ln(c*exp(2*I*(b*x+a))+I)+1/8*x^2*Pi*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1)
)^3+1/4/b^2*polylog(2,I*c*exp(2*I*(b*x+a)))*a+1/4*I*x^2*ln(1-I*c*exp(2*I*(b*x+a)))-1/8*x^2*Pi*csgn(I*(c*exp(2*
I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^3-1/2/b^2*a*dilog(1-I*exp(I*(b*x+a))*(-I*c)^(1/2))-1/2/b^2*a*dilog(1+I*exp
(I*(b*x+a))*(-I*c)^(1/2))+1/8*I*polylog(3,I*c*exp(2*I*(b*x+a)))/b^2+1/8*x^2*Pi*csgn(I*exp(I*(b*x+a)))^2*csgn(I
*exp(2*I*(b*x+a)))-1/4*x^2*Pi*csgn(I*exp(I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a)))^2-1/8*x^2*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))+1/4*I/b^2*a^2*ln(c*exp(2*I*
(b*x+a))+I)-1/2*I/b^2*a^2*ln(1-I*exp(I*(b*x+a))*(-I*c)^(1/2))-1/2*I/b^2*a^2*ln(1+I*exp(I*(b*x+a))*(-I*c)^(1/2)
)+1/8*x^2*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))+1/8*x^2*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/b^2*ln(1-I*c*exp(2*I*(b*x+a)))*a^2+1/8*x^2*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+1/8*x^2*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(c*exp(2*I*(b*x+a))+I)/(ex
p(2*I*(b*x+a))-1))^2-1/8*x^2*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
+1/2*I*x^2*ln(exp(I*(b*x+a)))+1/4*I*x^2*ln(I+c)
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Maxima [B] time = 1.08603, size = 294, normalized size = 2.39 \begin{align*} \frac{\frac{{\left ({\left (b x + a\right )}^{2} - 2 \,{\left (b x + a\right )} a\right )} \arctan \left ({\left (-i \, c + 1\right )} \cot \left (b x + a\right ) + c\right )}{b} + \frac{2 \,{\left (-4 i \,{\left (b x + a\right )}^{3} + 12 i \,{\left (b x + a\right )}^{2} a - 6 i \, b x{\rm Li}_2\left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) +{\left (-6 i \,{\left (b x + a\right )}^{2} + 12 i \,{\left (b x + a\right )} a\right )} \arctan \left (c \cos \left (2 \, b x + 2 \, a\right ), c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \,{\left ({\left (b x + a\right )}^{2} - 2 \,{\left (b x + a\right )} a\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 \,{\rm Li}_{3}(i \, c e^{\left (2 i \, b x + 2 i \, a\right )})\right )}{\left (i \, c - 1\right )}}{b{\left (12 \, c + 12 i\right )}}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-x*arctan(-c-(1-I*c)*cot(b*x+a)),x, algorithm="maxima")
[Out]
1/2*(((b*x + a)^2 - 2*(b*x + a)*a)*arctan((-I*c + 1)*cot(b*x + a) + c)/b + 2*(-4*I*(b*x + a)^3 + 12*I*(b*x + a
)^2*a - 6*I*b*x*dilog(I*c*e^(2*I*b*x + 2*I*a)) + (-6*I*(b*x + a)^2 + 12*I*(b*x + a)*a)*arctan2(c*cos(2*b*x + 2
*a), c*sin(2*b*x + 2*a) + 1) + 3*((b*x + a)^2 - 2*(b*x + a)*a)*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*polylog(3, I*c*e^(2*I*b*x + 2*I*a)))*(I*c - 1)/(b*(12*c + 12*I)))/b
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Fricas [C] time = 1.90519, size = 394, normalized size = 3.2 \begin{align*} \frac{4 \, b^{3} x^{3} + 6 i \, b^{2} x^{2} \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 ) + 4 \, a^{3} + 6 \, b x{\rm Li}_2\left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 6 i \, a^{2} \log \left (\frac{c e^{\left (2 i \, b x + 2 i \, a\right )} + i}{c}\right ) +{\left (6 i \, b^{2} x^{2} - 6 i \, a^{2}\right )} \log \left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) + 3 i \,{\rm polylog}\left (3, i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-x*arctan(-c-(1-I*c)*cot(b*x+a)),x, algorithm="fricas")
[Out]
1/24*(4*b^3*x^3 + 6*I*b^2*x^2*log(-(c + I)*e^(2*I*b*x + 2*I*a)/(c*e^(2*I*b*x + 2*I*a) + I)) + 4*a^3 + 6*b*x*di
log(I*c*e^(2*I*b*x + 2*I*a)) + 6*I*a^2*log((c*e^(2*I*b*x + 2*I*a) + I)/c) + (6*I*b^2*x^2 - 6*I*a^2)*log(-I*c*e
^(2*I*b*x + 2*I*a) + 1) + 3*I*polylog(3, I*c*e^(2*I*b*x + 2*I*a)))/b^2
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{i b \int \frac{x^{2}}{c e^{2 i a} e^{2 i b x} + i}\, dx}{2} + \frac{i x^{2} \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 )}}{4} - \frac{i x^{2} \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 )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-x*atan(-c-(1-I*c)*cot(b*x+a)),x)
[Out]
I*b*Integral(x**2/(c*exp(2*I*a)*exp(2*I*b*x) + I), x)/2 + I*x**2*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)))/4 - I*x**2*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)))/4
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -x \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*arctan(-c-(1-I*c)*cot(b*x+a)),x, algorithm="giac")
[Out]
integrate(-x*arctan(-(-I*c + 1)*cot(b*x + a) - c), x)