Integrand size = 18, antiderivative size = 132 \[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\frac {1}{6} i b x^3+\frac {1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac {1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac {i x \operatorname {PolyLog}\left (2,(1+i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (3,(1+i d) e^{2 i a+2 i b x}\right )}{8 b^2} \]
1/6*I*b*x^3+1/2*x^2*arccoth(1+I*d+d*cot(b*x+a))-1/4*x^2*ln(1-(1+I*d)*exp(2 *I*a+2*I*b*x))+1/4*I*x*polylog(2,(1+I*d)*exp(2*I*a+2*I*b*x))/b-1/8*polylog (3,(1+I*d)*exp(2*I*a+2*I*b*x))/b^2
Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.90 \[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\frac {1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac {2 b^2 x^2 \log \left (1+\frac {i e^{-2 i (a+b x)}}{-i+d}\right )+2 i b x \operatorname {PolyLog}\left (2,-\frac {i e^{-2 i (a+b x)}}{-i+d}\right )+\operatorname {PolyLog}\left (3,-\frac {i e^{-2 i (a+b x)}}{-i+d}\right )}{8 b^2} \]
(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*PolyLog[2, (-I)/((-I + d)*E^((2*I)*(a + b*x)))] + PolyLog[3, (-I)/((-I + d)*E^((2*I)*(a + b*x)))])/(8*b^2)
Time = 0.69 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.30, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6820, 2615, 2620, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \coth ^{-1}(d \cot (a+b x)+i d+1) \, dx\) |
\(\Big \downarrow \) 6820 |
\(\displaystyle \frac {1}{2} i b \int \frac {x^2}{1-(i d+1) e^{2 i a+2 i b x}}dx+\frac {1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)\) |
\(\Big \downarrow \) 2615 |
\(\displaystyle \frac {1}{2} i b \left (\frac {x^3}{3}+(1+i d) \int \frac {e^{2 i a+2 i b x} x^2}{1-(i d+1) e^{2 i a+2 i b x}}dx\right )+\frac {1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {1}{2} i b \left (\frac {x^3}{3}+(1+i d) \left (\frac {\int x \log \left (1-(i d+1) e^{2 i a+2 i b x}\right )dx}{b (-d+i)}-\frac {x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )}{2 b (-d+i)}\right )\right )+\frac {1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {1}{2} i b \left (\frac {x^3}{3}+(1+i d) \left (\frac {\frac {i x \operatorname {PolyLog}\left (2,(i d+1) e^{2 i a+2 i b x}\right )}{2 b}-\frac {i \int \operatorname {PolyLog}\left (2,(i d+1) e^{2 i a+2 i b x}\right )dx}{2 b}}{b (-d+i)}-\frac {x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )}{2 b (-d+i)}\right )\right )+\frac {1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {1}{2} i b \left (\frac {x^3}{3}+(1+i d) \left (\frac {\frac {i x \operatorname {PolyLog}\left (2,(i d+1) e^{2 i a+2 i b x}\right )}{2 b}-\frac {\int e^{-2 i a-2 i b x} \operatorname {PolyLog}\left (2,(i d+1) e^{2 i a+2 i b x}\right )de^{2 i a+2 i b x}}{4 b^2}}{b (-d+i)}-\frac {x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )}{2 b (-d+i)}\right )\right )+\frac {1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {1}{2} i b \left (\frac {x^3}{3}+(1+i d) \left (\frac {\frac {i x \operatorname {PolyLog}\left (2,(i d+1) e^{2 i a+2 i b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (3,(i d+1) e^{2 i a+2 i b x}\right )}{4 b^2}}{b (-d+i)}-\frac {x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )}{2 b (-d+i)}\right )\right )+\frac {1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)\) |
(x^2*ArcCoth[1 + I*d + d*Cot[a + b*x]])/2 + (I/2)*b*(x^3/3 + (1 + I*d)*(-1 /2*(x^2*Log[1 - (1 + I*d)*E^((2*I)*a + (2*I)*b*x)])/(b*(I - d)) + (((I/2)* 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)]/(4*b^2))/(b*(I - d))))
3.3.58.3.1 Defintions of rubi rules used
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] - Simp[ 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]
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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
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] + Simp[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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.09 (sec) , antiderivative size = 2285, normalized size of antiderivative = 17.31
-1/8*(I*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+I*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I/(exp(2*I*(b*x+a))-1)*exp(2* I*(b*x+a)))^2-I*Pi*csgn(I*exp(2*I*(b*x+a)))^3+2*I*Pi*csgn(I*exp(I*(b*x+a)) )*csgn(I*exp(2*I*(b*x+a)))^2-I*Pi*csgn(I/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x +a)))^3+I*Pi*csgn(I/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))*csgn(I*d/(exp(2 *I*(b*x+a))-1)*exp(2*I*(b*x+a)))^2-I*Pi*csgn(I*d/(exp(2*I*(b*x+a))-1)*exp( 2*I*(b*x+a)))^3+I*Pi*csgn(I*d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))*csgn( 1/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a))*d)^2+I*Pi*csgn(I*d)*csgn(I*d/(exp( 2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))^2-I*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+I*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn( I/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))^2+I*Pi*csgn(1/(exp(2*I*(b*x+a))-1 )*exp(2*I*(b*x+a))*d)^3-I*Pi*csgn(I*exp(2*I*(b*x+a)))*csgn(I/(exp(2*I*(b*x +a))-1))*csgn(I/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))-I*Pi*csgn(I*exp(I*( b*x+a)))^2*csgn(I*exp(2*I*(b*x+a)))-I*Pi*csgn(1/(exp(2*I*(b*x+a))-1)*exp(2 *I*(b*x+a))*d)^2-I*Pi*csgn(I*(exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a))+I))*cs gn(I*(exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^2+I*P 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))*c sgn((exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))+I*Pi*c sgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a))*d-I*exp(2*I*(b*x+a...
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.18 \[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\frac {4 i \, b^{3} x^{3} + 6 \, b^{2} x^{2} \log \left (\frac {{\left ({\left (d - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{d}\right ) + 4 i \, a^{3} + 6 i \, b x {\rm Li}_2\left (-{\left (-i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, a^{2} \log \left (\frac {{\left (d - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + i}{d - i}\right ) - 6 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left ({\left (-i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 3 \, {\rm polylog}\left (3, {\left (i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, b^{2}} \]
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
\[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\int x \operatorname {acoth}{\left (d \cot {\left (a + b x \right )} + i d + 1 \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (94) = 188\).
Time = 0.23 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.89 \[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\frac {\frac {12 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \operatorname {arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right )}{b} - \frac {-4 i \, {\left (b x + a\right )}^{3} + 12 i \, {\left (b x + a\right )}^{2} a - 6 i \, b x {\rm Li}_2\left ({\left (i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, {\left (i \, {\left (b x + a\right )}^{2} - 2 i \, {\left (b x + a\right )} a\right )} \arctan \left (d \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ), d \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left ({\left (d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (d^{2} + 1\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, d \sin \left (2 \, b x + 2 \, a\right ) - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\rm Li}_{3}({\left (i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )})}{b}}{24 \, b} \]
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 - 2*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*po lylog(3, (I*d + 1)*e^(2*I*b*x + 2*I*a)))/b)/b
\[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\int { x \operatorname {arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right ) \,d x } \]
Timed out. \[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\int x\,\mathrm {acoth}\left (d\,\mathrm {cot}\left (a+b\,x\right )+1+d\,1{}\mathrm {i}\right ) \,d x \]