3.258 \(\int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx\)
Optimal. Leaf size=132 \[ -\frac{\text{PolyLog}\left (3,(1+i d) e^{2 i a+2 i b x}\right )}{8 b^2}+\frac{i x \text{PolyLog}\left (2,(1+i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac{1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac{1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)+\frac{1}{6} i b x^3 \]
[Out]
(I/6)*b*x^3 + (x^2*ArcCoth[1 + I*d + d*Cot[a + b*x]])/2 - (x^2*Log[1 - (1 + I*d)*E^((2*I)*a + (2*I)*b*x)])/4 +
((I/4)*x*PolyLog[2, (1 + I*d)*E^((2*I)*a + (2*I)*b*x)])/b - PolyLog[3, (1 + I*d)*E^((2*I)*a + (2*I)*b*x)]/(8*
b^2)
________________________________________________________________________________________
Rubi [A] time = 0.256505, antiderivative size = 132, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{6266, 2184, 2190, 2531, 2282, 6589} \[ -\frac{\text{PolyLog}\left (3,(1+i d) e^{2 i a+2 i b x}\right )}{8 b^2}+\frac{i x \text{PolyLog}\left (2,(1+i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac{1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac{1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)+\frac{1}{6} i b x^3 \]
Antiderivative was successfully verified.
[In]
Int[x*ArcCoth[1 + I*d + d*Cot[a + b*x]],x]
[Out]
(I/6)*b*x^3 + (x^2*ArcCoth[1 + I*d + d*Cot[a + b*x]])/2 - (x^2*Log[1 - (1 + I*d)*E^((2*I)*a + (2*I)*b*x)])/4 +
((I/4)*x*PolyLog[2, (1 + I*d)*E^((2*I)*a + (2*I)*b*x)])/b - PolyLog[3, (1 + I*d)*E^((2*I)*a + (2*I)*b*x)]/(8*
b^2)
Rule 6266
Int[ArcCoth[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^(m
+ 1)*ArcCoth[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 \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx &=\frac{1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))+\frac{1}{2} (i b) \int \frac{x^2}{1+(-1-i d) e^{2 i a+2 i b x}} \, dx\\ &=\frac{1}{6} i b x^3+\frac{1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))+\frac{1}{2} (b (i-d)) \int \frac{e^{2 i a+2 i b x} x^2}{1+(-1-i d) e^{2 i a+2 i b x}} \, dx\\ &=\frac{1}{6} i b x^3+\frac{1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac{1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac{1}{2} \int x \log \left (1+(-1-i d) e^{2 i a+2 i b x}\right ) \, dx\\ &=\frac{1}{6} i b x^3+\frac{1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac{1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac{i x \text{Li}_2\left ((1+i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac{i \int \text{Li}_2\left (-(-1-i d) e^{2 i a+2 i b x}\right ) \, dx}{4 b}\\ &=\frac{1}{6} i b x^3+\frac{1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac{1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac{i x \text{Li}_2\left ((1+i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2((1+i d) x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^2}\\ &=\frac{1}{6} i b x^3+\frac{1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac{1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac{i x \text{Li}_2\left ((1+i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac{\text{Li}_3\left ((1+i d) e^{2 i a+2 i b x}\right )}{8 b^2}\\ \end{align*}
Mathematica [A] time = 0.108773, size = 119, normalized size = 0.9 \[ \frac{1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)-\frac{2 i b x \text{PolyLog}\left (2,-\frac{i e^{-2 i (a+b x)}}{d-i}\right )+\text{PolyLog}\left (3,-\frac{i e^{-2 i (a+b x)}}{d-i}\right )+2 b^2 x^2 \log \left (1+\frac{i e^{-2 i (a+b x)}}{d-i}\right )}{8 b^2} \]
Antiderivative was successfully verified.
[In]
Integrate[x*ArcCoth[1 + I*d + d*Cot[a + b*x]],x]
[Out]
(x^2*ArcCoth[1 + I*d + d*Cot[a + b*x]])/2 - (2*b^2*x^2*Log[1 + I/((-I + d)*E^((2*I)*(a + b*x)))] + (2*I)*b*x*P
olyLog[2, (-I)/((-I + d)*E^((2*I)*(a + b*x)))] + PolyLog[3, (-I)/((-I + d)*E^((2*I)*(a + b*x)))])/(8*b^2)
________________________________________________________________________________________
Maple [C] time = 17.385, size = 2351, normalized size = 17.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*arccoth(1+I*d+d*cot(b*x+a)),x)
[Out]
1/2*I/b^2*a^2/(-d+I)*ln(1+I*exp(I*(b*x+a))*(I*(-d+I))^(1/2))+1/8*I*x^2*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*
(exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^2-1/4*I/b^2*a^2/(-d+I)*ln(I*exp(2*I*(b*x+a))-e
xp(2*I*(b*x+a))*d-I)-1/4*I/b^2/(-d+I)*ln(1+I*(-d+I)*exp(2*I*(b*x+a)))*a^2+1/2*I/b^2*a^2/(-d+I)*ln(1-I*exp(I*(b
*x+a))*(I*(-d+I))^(1/2))+1/4/b^2*a^2*d/(-d+I)*ln(I*exp(2*I*(b*x+a))-exp(2*I*(b*x+a))*d-I)-1/2/b^2*a^2*d/(-d+I)
*ln(1-I*exp(I*(b*x+a))*(I*(-d+I))^(1/2))-1/2/b^2*a^2*d/(-d+I)*ln(1+I*exp(I*(b*x+a))*(I*(-d+I))^(1/2))+1/4/b^2*
d/(-d+I)*ln(1+I*(-d+I)*exp(2*I*(b*x+a)))*a^2-1/8*I*x^2*Pi*csgn(I*(exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a))+I)/(ex
p(2*I*(b*x+a))-1))*csgn((exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))+1/8*I*x^2*Pi*csgn(I*ex
p(2*I*(b*x+a))/(exp(2*I*(b*x+a))-1))^3+1/8*I*x^2*Pi*csgn(I*d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))^3-1/8*I*x^
2*Pi*csgn((exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^2+1/4*x^2*ln(exp(2*I*(b*x+a))*d-I*ex
p(2*I*(b*x+a))+I)-1/4/b/(-d+I)*polylog(2,-I*(-d+I)*exp(2*I*(b*x+a)))*x-1/4/b^2/(-d+I)*polylog(2,-I*(-d+I)*exp(
2*I*(b*x+a)))*a+1/8/b^2*d/(-d+I)*polylog(3,-I*(-d+I)*exp(2*I*(b*x+a)))-1/8*I*x^2*Pi*csgn(I*(exp(2*I*(b*x+a))*d
-I*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^3-1/8*I/b^2/(-d+I)*polylog(3,-I*(-d+I)*exp(2*I*(b*x+a)))-1/4*I/(-
d+I)*ln(1+I*(-d+I)*exp(2*I*(b*x+a)))*x^2+1/6*I*b*x^3-1/4*I*x^2*Pi*csgn(I*exp(I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a
)))^2-1/8*I*x^2*Pi*csgn(I*d)*csgn(I*d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))^2-1/8*I*x^2*Pi*csgn(I*exp(2*I*(b*
x+a))/(exp(2*I*(b*x+a))-1))*csgn(I*d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))^2-1/8*I*x^2*Pi*csgn(d/(exp(2*I*(b*
x+a))-1)*exp(2*I*(b*x+a)))^3-1/4*x^2*ln(d)+1/8*I*x^2*Pi*csgn(I*exp(I*(b*x+a)))^2*csgn(I*exp(2*I*(b*x+a)))-1/8*
I*x^2*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))-1))^2-1/8*I*x^2*Pi*csgn(I/(exp(2*I
*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))-1))^2-1/8*I*x^2*Pi*csgn(I*d/(exp(2*I*(b*x+a))-1)*exp(2
*I*(b*x+a)))*csgn(d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))^2+1/8*I*x^2*Pi*csgn(I*d/(exp(2*I*(b*x+a))-1)*exp(2*
I*(b*x+a)))*csgn(d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))+1/8*I*x^2*Pi*csgn(I*exp(2*I*(b*x+a)))^3+1/8*I*x^2*Pi
*csgn(I*(exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))*csgn((exp(2*I*(b*x+a))*d-I*exp(2*I*(b*
x+a))+I)/(exp(2*I*(b*x+a))-1))^2-1/2*x^2*ln(exp(I*(b*x+a)))+1/8*I*x^2*Pi*csgn((exp(2*I*(b*x+a))*d-I*exp(2*I*(b
*x+a))+I)/(exp(2*I*(b*x+a))-1))^3+1/4*d/(-d+I)*ln(1+I*(-d+I)*exp(2*I*(b*x+a)))*x^2+1/2/b^2*a/(-d+I)*dilog(1-I*
exp(I*(b*x+a))*(I*(-d+I))^(1/2))+1/2/b^2*a/(-d+I)*dilog(1+I*exp(I*(b*x+a))*(I*(-d+I))^(1/2))+1/8*I*x^2*Pi*csgn
(d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))^2+1/8*I*x^2*Pi*csgn(I*(exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a))+I))*csg
n(I*(exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^2+1/2/b*d/(-d+I)*ln(1+I*(-d+I)*exp(2*I*(b*
x+a)))*x*a-1/2/b*a*d/(-d+I)*ln(1-I*exp(I*(b*x+a))*(I*(-d+I))^(1/2))*x-1/2/b*a*d/(-d+I)*ln(1+I*exp(I*(b*x+a))*(
I*(-d+I))^(1/2))*x+1/2*I/b*a/(-d+I)*ln(1-I*exp(I*(b*x+a))*(I*(-d+I))^(1/2))*x+1/2*I/b*a/(-d+I)*ln(1+I*exp(I*(b
*x+a))*(I*(-d+I))^(1/2))*x-1/4*I/b*d/(-d+I)*polylog(2,-I*(-d+I)*exp(2*I*(b*x+a)))*x-1/4*I/b^2*d/(-d+I)*polylog
(2,-I*(-d+I)*exp(2*I*(b*x+a)))*a+1/8*I*x^2*Pi*csgn(I*d)*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))-1))*csgn(I*d
/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))+1/8*I*x^2*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I/(exp(2*I*(b*x+a))-1))*csg
n(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))-1))-1/2*I/b/(-d+I)*ln(1+I*(-d+I)*exp(2*I*(b*x+a)))*x*a+1/2*I/b^2*a*d/(-
d+I)*dilog(1-I*exp(I*(b*x+a))*(I*(-d+I))^(1/2))+1/2*I/b^2*a*d/(-d+I)*dilog(1+I*exp(I*(b*x+a))*(I*(-d+I))^(1/2)
)-1/8*I*x^2*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a))+I))*csgn(I*(exp(2*I*
(b*x+a))*d-I*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))
________________________________________________________________________________________
Maxima [B] time = 1.1394, size = 335, normalized size = 2.54 \begin{align*} \frac{\frac{12 \,{\left ({\left (b x + a\right )}^{2} - 2 \,{\left (b x + a\right )} a\right )} \operatorname{arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right )}{b} - \frac{-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 ({\left (i \, d + 1\right )} 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 (d \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ), d \sin \left (2 \, b x + 2 \, a\right ) - \cos \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 ({\left (d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} +{\left (d^{2} + 1\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, d \sin \left (2 \, b x + 2 \, a\right ) - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \,{\rm Li}_{3}({\left (i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )})}{b}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*arccoth(1+I*d+d*cot(b*x+a)),x, algorithm="maxima")
[Out]
1/24*(12*((b*x + a)^2 - 2*(b*x + a)*a)*arccoth(d*cot(b*x + a) + I*d + 1)/b - (-4*I*(b*x + a)^3 + 12*I*(b*x + a
)^2*a - 6*I*b*x*dilog((I*d + 1)*e^(2*I*b*x + 2*I*a)) + (-6*I*(b*x + a)^2 + 12*I*(b*x + a)*a)*arctan2(d*cos(2*b
*x + 2*a) + sin(2*b*x + 2*a), d*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) + 1) + 3*((b*x + a)^2 - 2*(b*x + a)*a)*log
((d^2 + 1)*cos(2*b*x + 2*a)^2 + (d^2 + 1)*sin(2*b*x + 2*a)^2 + 2*d*sin(2*b*x + 2*a) - 2*cos(2*b*x + 2*a) + 1)
+ 3*polylog(3, (I*d + 1)*e^(2*I*b*x + 2*I*a)))/b)/b
________________________________________________________________________________________
Fricas [C] time = 1.69142, size = 429, normalized size = 3.25 \begin{align*} \frac{4 i \, b^{3} x^{3} + 6 \, b^{2} x^{2} \log \left (\frac{{\left ({\left (d - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{d}\right ) + 4 i \, a^{3} + 6 i \, b x{\rm Li}_2\left (-{\left (-i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, a^{2} \log \left (\frac{{\left (d - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + i}{d - i}\right ) - 6 \,{\left (b^{2} x^{2} - a^{2}\right )} \log \left ({\left (-i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 3 \,{\rm polylog}\left (3,{\left (i \, d + 1\right )} 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*arccoth(1+I*d+d*cot(b*x+a)),x, algorithm="fricas")
[Out]
1/24*(4*I*b^3*x^3 + 6*b^2*x^2*log(((d - I)*e^(2*I*b*x + 2*I*a) + I)*e^(-2*I*b*x - 2*I*a)/d) + 4*I*a^3 + 6*I*b*
x*dilog(-(-I*d - 1)*e^(2*I*b*x + 2*I*a)) - 6*a^2*log(((d - I)*e^(2*I*b*x + 2*I*a) + I)/(d - I)) - 6*(b^2*x^2 -
a^2)*log((-I*d - 1)*e^(2*I*b*x + 2*I*a) + 1) - 3*polylog(3, (I*d + 1)*e^(2*I*b*x + 2*I*a)))/b^2
________________________________________________________________________________________
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*acoth(1+I*d+d*cot(b*x+a)),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*arccoth(1+I*d+d*cot(b*x+a)),x, algorithm="giac")
[Out]
integrate(x*arccoth(d*cot(b*x + a) + I*d + 1), x)