Integrand size = 20, antiderivative size = 116 \[ \int x \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx=-\frac {1}{6} i b x^3+\frac {1}{2} x^2 \cot ^{-1}(c-(i-c) \coth (a+b x))+\frac {1}{4} i x^2 \log \left (1+i c e^{2 a+2 b x}\right )+\frac {i x \operatorname {PolyLog}\left (2,-i c e^{2 a+2 b x}\right )}{4 b}-\frac {i \operatorname {PolyLog}\left (3,-i c e^{2 a+2 b x}\right )}{8 b^2} \]
-1/6*I*b*x^3+1/2*x^2*arccot(c-(I-c)*coth(b*x+a))+1/4*I*x^2*ln(1+I*c*exp(2* b*x+2*a))+1/4*I*x*polylog(2,-I*c*exp(2*b*x+2*a))/b-1/8*I*polylog(3,-I*c*ex p(2*b*x+2*a))/b^2
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.89 \[ \int x \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx=\frac {2 b^2 x^2 \left (2 \cot ^{-1}(c+(-i+c) \coth (a+b x))+i \log \left (1-\frac {i e^{-2 (a+b x)}}{c}\right )\right )-2 i b x \operatorname {PolyLog}\left (2,\frac {i e^{-2 (a+b x)}}{c}\right )-i \operatorname {PolyLog}\left (3,\frac {i e^{-2 (a+b x)}}{c}\right )}{8 b^2} \]
(2*b^2*x^2*(2*ArcCot[c + (-I + c)*Coth[a + b*x]] + I*Log[1 - I/(c*E^(2*(a + b*x)))]) - (2*I)*b*x*PolyLog[2, I/(c*E^(2*(a + b*x)))] - I*PolyLog[3, I/ (c*E^(2*(a + b*x)))])/(8*b^2)
Time = 0.63 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5721, 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.
\(\displaystyle \int x \cot ^{-1}(c-(-c+i) \coth (a+b x)) \, dx\) |
\(\Big \downarrow \) 5721 |
\(\displaystyle \frac {1}{2} b \int \frac {x^2}{i-c e^{2 a+2 b x}}dx+\frac {1}{2} x^2 \cot ^{-1}(c-(-c+i) \coth (a+b x))\) |
\(\Big \downarrow \) 2615 |
\(\displaystyle \frac {1}{2} b \left (-i c \int \frac {e^{2 a+2 b x} x^2}{i-c e^{2 a+2 b x}}dx-\frac {i x^3}{3}\right )+\frac {1}{2} x^2 \cot ^{-1}(c-(-c+i) \coth (a+b x))\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {1}{2} b \left (-i c \left (\frac {\int x \log \left (i e^{2 a+2 b x} c+1\right )dx}{b c}-\frac {x^2 \log \left (1+i c e^{2 a+2 b x}\right )}{2 b c}\right )-\frac {i x^3}{3}\right )+\frac {1}{2} x^2 \cot ^{-1}(c-(-c+i) \coth (a+b x))\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {1}{2} b \left (-i c \left (\frac {\frac {\int \operatorname {PolyLog}\left (2,-i c e^{2 a+2 b x}\right )dx}{2 b}-\frac {x \operatorname {PolyLog}\left (2,-i c e^{2 a+2 b x}\right )}{2 b}}{b c}-\frac {x^2 \log \left (1+i c e^{2 a+2 b x}\right )}{2 b c}\right )-\frac {i x^3}{3}\right )+\frac {1}{2} x^2 \cot ^{-1}(c-(-c+i) \coth (a+b x))\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {1}{2} b \left (-i c \left (\frac {\frac {\int e^{-2 a-2 b x} \operatorname {PolyLog}\left (2,-i c e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,-i c e^{2 a+2 b x}\right )}{2 b}}{b c}-\frac {x^2 \log \left (1+i c e^{2 a+2 b x}\right )}{2 b c}\right )-\frac {i x^3}{3}\right )+\frac {1}{2} x^2 \cot ^{-1}(c-(-c+i) \coth (a+b x))\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {1}{2} b \left (-i c \left (\frac {\frac {\operatorname {PolyLog}\left (3,-i c e^{2 a+2 b x}\right )}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,-i c e^{2 a+2 b x}\right )}{2 b}}{b c}-\frac {x^2 \log \left (1+i c e^{2 a+2 b x}\right )}{2 b c}\right )-\frac {i x^3}{3}\right )+\frac {1}{2} x^2 \cot ^{-1}(c-(-c+i) \coth (a+b x))\) |
(x^2*ArcCot[c - (I - c)*Coth[a + b*x]])/2 + (b*((-1/3*I)*x^3 - I*c*(-1/2*( x^2*Log[1 + I*c*E^(2*a + 2*b*x)])/(b*c) + (-1/2*(x*PolyLog[2, (-I)*c*E^(2* a + 2*b*x)])/b + PolyLog[3, (-I)*c*E^(2*a + 2*b*x)]/(4*b^2))/(b*c))))/2
3.3.14.3.1 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_.) + Coth[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_ .), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(ArcCot[c + d*Coth[a + b*x]]/(f*(m + 1))), x] + Simp[b/(f*(m + 1)) Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - 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 = 1.91 (sec) , antiderivative size = 1374, normalized size of antiderivative = 11.84
1/2*I/b*ln(1+I*c*exp(2*b*x+2*a))*a*x-1/8*Pi*(csgn(I/(exp(2*b*x+2*a)-1))*cs gn(I*(2*exp(2*b*x+2*a)*c-2*I))*csgn(I*(2*exp(2*b*x+2*a)*c-2*I)/(exp(2*b*x+ 2*a)-1))-csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*(-2*I*exp(2*b*x+2*a)+2*exp(2*b* x+2*a)*c))*csgn(I*(-2*I*exp(2*b*x+2*a)+2*exp(2*b*x+2*a)*c)/(exp(2*b*x+2*a) -1))-csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*(2*exp(2*b*x+2*a)*c-2*I)/(exp(2*b*x +2*a)-1))^2+csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*(-2*I*exp(2*b*x+2*a)+2*exp(2 *b*x+2*a)*c)/(exp(2*b*x+2*a)-1))^2+csgn(I*(2*exp(2*b*x+2*a)*c-2*I))*csgn(I *(2*exp(2*b*x+2*a)*c-2*I)/(exp(2*b*x+2*a)-1))^2-csgn(I*(-2*I*exp(2*b*x+2*a )+2*exp(2*b*x+2*a)*c))*csgn(I*(-2*I*exp(2*b*x+2*a)+2*exp(2*b*x+2*a)*c)/(ex p(2*b*x+2*a)-1))^2-csgn(I*(2*exp(2*b*x+2*a)*c-2*I)/(exp(2*b*x+2*a)-1))^3+c sgn(I*(2*exp(2*b*x+2*a)*c-2*I)/(exp(2*b*x+2*a)-1))*csgn((2*exp(2*b*x+2*a)* c-2*I)/(exp(2*b*x+2*a)-1))^2+csgn(I*(2*exp(2*b*x+2*a)*c-2*I)/(exp(2*b*x+2* a)-1))*csgn((2*exp(2*b*x+2*a)*c-2*I)/(exp(2*b*x+2*a)-1))+csgn(I*(-2*I*exp( 2*b*x+2*a)+2*exp(2*b*x+2*a)*c)/(exp(2*b*x+2*a)-1))^3-csgn(I*(-2*I*exp(2*b* x+2*a)+2*exp(2*b*x+2*a)*c)/(exp(2*b*x+2*a)-1))*csgn((-2*I*exp(2*b*x+2*a)+2 *exp(2*b*x+2*a)*c)/(exp(2*b*x+2*a)-1))^2-csgn(I*(-2*I*exp(2*b*x+2*a)+2*exp (2*b*x+2*a)*c)/(exp(2*b*x+2*a)-1))*csgn((-2*I*exp(2*b*x+2*a)+2*exp(2*b*x+2 *a)*c)/(exp(2*b*x+2*a)-1))+csgn((-2*I*exp(2*b*x+2*a)+2*exp(2*b*x+2*a)*c)/( exp(2*b*x+2*a)-1))^3+csgn((-2*I*exp(2*b*x+2*a)+2*exp(2*b*x+2*a)*c)/(exp(2* b*x+2*a)-1))^2+csgn((2*exp(2*b*x+2*a)*c-2*I)/(exp(2*b*x+2*a)-1))^3+csgn...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (83) = 166\).
Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.12 \[ \int x \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx=\frac {-2 i \, b^{3} x^{3} + 3 i \, b^{2} x^{2} \log \left (\frac {{\left (c - i\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c e^{\left (2 \, b x + 2 \, a\right )} - i}\right ) - 2 i \, a^{3} + 6 i \, b x {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right ) + 6 i \, b x {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right ) + 3 i \, a^{2} \log \left (\frac {2 \, c e^{\left (b x + a\right )} + i \, \sqrt {-4 i \, c}}{2 \, c}\right ) + 3 i \, a^{2} \log \left (\frac {2 \, c e^{\left (b x + a\right )} - i \, \sqrt {-4 i \, c}}{2 \, c}\right ) - 3 \, {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \log \left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )} + 1\right ) - 3 \, {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )} + 1\right ) - 6 i \, {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right ) - 6 i \, {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (b x + a\right )}\right )}{12 \, b^{2}} \]
1/12*(-2*I*b^3*x^3 + 3*I*b^2*x^2*log((c - I)*e^(2*b*x + 2*a)/(c*e^(2*b*x + 2*a) - I)) - 2*I*a^3 + 6*I*b*x*dilog(1/2*sqrt(-4*I*c)*e^(b*x + a)) + 6*I* b*x*dilog(-1/2*sqrt(-4*I*c)*e^(b*x + a)) + 3*I*a^2*log(1/2*(2*c*e^(b*x + a ) + I*sqrt(-4*I*c))/c) + 3*I*a^2*log(1/2*(2*c*e^(b*x + a) - I*sqrt(-4*I*c) )/c) - 3*(-I*b^2*x^2 + I*a^2)*log(1/2*sqrt(-4*I*c)*e^(b*x + a) + 1) - 3*(- I*b^2*x^2 + I*a^2)*log(-1/2*sqrt(-4*I*c)*e^(b*x + a) + 1) - 6*I*polylog(3, 1/2*sqrt(-4*I*c)*e^(b*x + a)) - 6*I*polylog(3, -1/2*sqrt(-4*I*c)*e^(b*x + a)))/b^2
Exception generated. \[ \int x \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx=\text {Exception raised: CoercionFailed} \]
Exception raised: CoercionFailed >> Cannot convert 2*_t0**2*c*exp(2*a) - _ t0**2*I*exp(2*a) - I of type <class 'sympy.core.add.Add'> to QQ_I[x,b,c,_t 0,exp(a)]
Time = 1.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91 \[ \int x \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx={\left (\frac {2 \, x^{3}}{3 i \, c + 3} - \frac {2 \, b^{2} x^{2} \log \left (i \, c e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-i \, c e^{\left (2 \, b x + 2 \, a\right )}\right ) - {\rm Li}_{3}(-i \, c e^{\left (2 \, b x + 2 \, a\right )})}{-2 \, b^{3} {\left (-i \, c - 1\right )}}\right )} b {\left (c - i\right )} + \frac {1}{2} \, x^{2} \operatorname {arccot}\left ({\left (c - i\right )} \coth \left (b x + a\right ) + c\right ) \]
(2*x^3/(3*I*c + 3) - (2*b^2*x^2*log(I*c*e^(2*b*x + 2*a) + 1) + 2*b*x*dilog (-I*c*e^(2*b*x + 2*a)) - polylog(3, -I*c*e^(2*b*x + 2*a)))/(b^3*(2*I*c + 2 )))*b*(c - I) + 1/2*x^2*arccot((c - I)*coth(b*x + a) + c)
\[ \int x \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx=\int { x \operatorname {arccot}\left ({\left (c - i\right )} \coth \left (b x + a\right ) + c\right ) \,d x } \]
Timed out. \[ \int x \cot ^{-1}(c-(i-c) \coth (a+b x)) \, dx=\int x\,\mathrm {acot}\left (c+\mathrm {coth}\left (a+b\,x\right )\,\left (c-\mathrm {i}\right )\right ) \,d x \]