Integrand size = 19, antiderivative size = 123 \[ \int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=-\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )-\frac {x \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \operatorname {PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{8 b^2} \]
-1/6*b*x^3+1/2*x^2*(Pi-arccot(-c-(1-I*c)*cot(b*x+a)))-1/4*I*x^2*ln(1-I*c*e xp(2*I*a+2*I*b*x))-1/4*x*polylog(2,I*c*exp(2*I*a+2*I*b*x))/b-1/8*I*polylog (3,I*c*exp(2*I*a+2*I*b*x))/b^2
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {i \left (2 b^2 x^2 \log \left (1+\frac {i e^{-2 i (a+b x)}}{c}\right )+2 i b x \operatorname {PolyLog}\left (2,-\frac {i e^{-2 i (a+b x)}}{c}\right )+\operatorname {PolyLog}\left (3,-\frac {i e^{-2 i (a+b x)}}{c}\right )\right )}{8 b^2} \]
(x^2*ArcCot[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
Time = 0.62 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.26, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5697, 25, 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.
\(\displaystyle \int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx\) |
\(\Big \downarrow \) 5697 |
\(\displaystyle \frac {1}{2} i b \int -\frac {x^2}{e^{2 i a+2 i b x} c+i}dx+\frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{2} i b \int \frac {x^2}{e^{2 i a+2 i b x} c+i}dx\) |
\(\Big \downarrow \) 2615 |
\(\displaystyle \frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{2} i b \left (i c \int \frac {e^{2 i a+2 i b x} x^2}{e^{2 i a+2 i b x} c+i}dx-\frac {i x^3}{3}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{2} i b \left (i c \left (\frac {i \int x \log \left (1-i c e^{2 i a+2 i b x}\right )dx}{b c}-\frac {i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^3}{3}\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{2} i b \left (i c \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{2 b}-\frac {i \int \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )dx}{2 b}\right )}{b c}-\frac {i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^3}{3}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{2} i b \left (i c \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{2 b}-\frac {\int e^{-2 i a-2 i b x} \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )de^{2 i a+2 i b x}}{4 b^2}\right )}{b c}-\frac {i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^3}{3}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{2} i b \left (i c \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{4 b^2}\right )}{b c}-\frac {i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^3}{3}\right )\) |
(x^2*ArcCot[c + (1 - I*c)*Cot[a + b*x]])/2 - (I/2)*b*((-1/3*I)*x^3 + I*c*( ((-1/2*I)*x^2*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)])/(b*c) + (I*(((I/2)*x*P olyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)])/b - PolyLog[3, I*c*E^((2*I)*a + (2 *I)*b*x)]/(4*b^2)))/(b*c)))
3.2.76.3.1 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[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] + 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] && E qQ[(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 = 2.62 (sec) , antiderivative size = 1413, normalized size of antiderivative = 11.49
1/2*I/b*a*ln(1-I*exp(I*(b*x+a))*(-I*c)^(1/2))*x-1/8*(Pi*csgn(I*exp(I*(b*x+ a)))^2*csgn(I*exp(2*I*(b*x+a)))-2*Pi*csgn(I*exp(I*(b*x+a)))*csgn(I*exp(2*I *(b*x+a)))^2+Pi*csgn(I*exp(2*I*(b*x+a)))^3+Pi*csgn(I*exp(2*I*(b*x+a)))*csg n(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))-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-Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a)) *c+I))*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))+Pi*csgn(I/(exp( 2*I*(b*x+a))-1))*csgn(I*(I+c))*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))-Pi*csgn( I/(exp(2*I*(b*x+a))-1))*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))^2+Pi*csgn(I/(ex p(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))^2+P i*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))-Pi*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x +a))-1))^2+Pi*csgn(I*(exp(2*I*(b*x+a))*c+I))*csgn(I*(exp(2*I*(b*x+a))*c+I) /(exp(2*I*(b*x+a))-1))^2-Pi*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a) )-1))*csgn((exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))-Pi*csgn((exp(2*I*( b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))^2-Pi*csgn(I*(I+c))*csgn(I*(I+c)/(exp(2* I*(b*x+a))-1))^2+Pi*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))^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 -Pi*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))^3+Pi*csgn(I*(exp(2 *I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))*csgn((exp(2*I*(b*x+a))*c+I)/(exp...
Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.24 \[ \int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=-\frac {4 \, b^{3} x^{3} - 12 \, \pi b^{2} x^{2} - 6 i \, b^{2} x^{2} \log \left (\frac {{\left (c e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{c + i}\right ) + 4 \, a^{3} + 6 \, b x {\rm Li}_2\left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 6 i \, a^{2} \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} + i}{c}\right ) + 6 \, {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \log \left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) + 3 i \, {\rm polylog}\left (3, i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, b^{2}} \]
-1/24*(4*b^3*x^3 - 12*pi*b^2*x^2 - 6*I*b^2*x^2*log((c*e^(2*I*b*x + 2*I*a) + I)*e^(-2*I*b*x - 2*I*a)/(c + I)) + 4*a^3 + 6*b*x*dilog(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 - 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
Exception generated. \[ \int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\text {Exception raised: CoercionFailed} \]
Exception raised: CoercionFailed >> Cannot convert -_t0**4 + 3*_t0**2*I*c* exp(2*I*a) - _t0**2*exp(2*I*a) + 2*c**2*exp(4*I*a) + I*c*exp(4*I*a) of typ e <class 'sympy.core.add.Add'> to QQ_I[x,b,c,_t0,exp(I*a)]
Exception generated. \[ \int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c-1>0)', see `assume?` for more details)Is
\[ \int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\int { {\left (\pi - \operatorname {arccot}\left (-{\left (-i \, c + 1\right )} \cot \left (b x + a\right ) - c\right )\right )} x \,d x } \]
Timed out. \[ \int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\int x\,\left (\Pi +\mathrm {acot}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (-1+c\,1{}\mathrm {i}\right )\right )\right ) \,d x \]