Integrand size = 13, antiderivative size = 159 \[ \int (e+f x) \cot ^{-1}(\coth (a+b x)) \, dx=\frac {(e+f x)^2 \cot ^{-1}(\coth (a+b x))}{2 f}-\frac {(e+f x)^2 \arctan \left (e^{2 a+2 b x}\right )}{2 f}+\frac {i (e+f x) \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}-\frac {i (e+f x) \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}-\frac {i f \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{8 b^2}+\frac {i f \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{8 b^2} \]
1/2*(f*x+e)^2*arccot(coth(b*x+a))/f-1/2*(f*x+e)^2*arctan(exp(2*b*x+2*a))/f +1/4*I*(f*x+e)*polylog(2,-I*exp(2*b*x+2*a))/b-1/4*I*(f*x+e)*polylog(2,I*ex p(2*b*x+2*a))/b-1/8*I*f*polylog(3,-I*exp(2*b*x+2*a))/b^2+1/8*I*f*polylog(3 ,I*exp(2*b*x+2*a))/b^2
Time = 0.10 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.49 \[ \int (e+f x) \cot ^{-1}(\coth (a+b x)) \, dx=e x \cot ^{-1}(\coth (a+b x))+\frac {1}{2} f x^2 \cot ^{-1}(\coth (a+b x))-\frac {i e \left (2 b x \left (\log \left (1-i e^{2 (a+b x)}\right )-\log \left (1+i e^{2 (a+b x)}\right )\right )-\operatorname {PolyLog}\left (2,-i e^{2 (a+b x)}\right )+\operatorname {PolyLog}\left (2,i e^{2 (a+b x)}\right )\right )}{4 b}-\frac {i f \left (2 b^2 x^2 \log \left (1-i e^{2 (a+b x)}\right )-2 b^2 x^2 \log \left (1+i e^{2 (a+b x)}\right )-2 b x \operatorname {PolyLog}\left (2,-i e^{2 (a+b x)}\right )+2 b x \operatorname {PolyLog}\left (2,i e^{2 (a+b x)}\right )+\operatorname {PolyLog}\left (3,-i e^{2 (a+b x)}\right )-\operatorname {PolyLog}\left (3,i e^{2 (a+b x)}\right )\right )}{8 b^2} \]
e*x*ArcCot[Coth[a + b*x]] + (f*x^2*ArcCot[Coth[a + b*x]])/2 - ((I/4)*e*(2* b*x*(Log[1 - I*E^(2*(a + b*x))] - Log[1 + I*E^(2*(a + b*x))]) - PolyLog[2, (-I)*E^(2*(a + b*x))] + PolyLog[2, I*E^(2*(a + b*x))]))/b - ((I/8)*f*(2*b ^2*x^2*Log[1 - I*E^(2*(a + b*x))] - 2*b^2*x^2*Log[1 + I*E^(2*(a + b*x))] - 2*b*x*PolyLog[2, (-I)*E^(2*(a + b*x))] + 2*b*x*PolyLog[2, I*E^(2*(a + b*x ))] + PolyLog[3, (-I)*E^(2*(a + b*x))] - PolyLog[3, I*E^(2*(a + b*x))]))/b ^2
Time = 0.58 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5709, 3042, 4668, 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.
| \(\displaystyle \int (e+f x) \cot ^{-1}(\coth (a+b x)) \, dx\) |
| \(\Big \downarrow \) 5709 |
| \(\displaystyle \frac {(e+f x)^2 \cot ^{-1}(\coth (a+b x))}{2 f}-\frac {b \int (e+f x)^2 \text {sech}(2 a+2 b x)dx}{2 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(e+f x)^2 \cot ^{-1}(\coth (a+b x))}{2 f}-\frac {b \int (e+f x)^2 \csc \left (2 i a+2 i b x+\frac {\pi }{2}\right )dx}{2 f}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {(e+f x)^2 \cot ^{-1}(\coth (a+b x))}{2 f}-\frac {b \left (-\frac {i f \int (e+f x) \log \left (1-i e^{2 a+2 b x}\right )dx}{b}+\frac {i f \int (e+f x) \log \left (1+i e^{2 a+2 b x}\right )dx}{b}+\frac {(e+f x)^2 \arctan \left (e^{2 a+2 b x}\right )}{b}\right )}{2 f}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {(e+f x)^2 \cot ^{-1}(\coth (a+b x))}{2 f}-\frac {b \left (\frac {i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )dx}{2 b}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{2 b}\right )}{b}-\frac {i f \left (\frac {f \int \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )dx}{2 b}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{2 b}\right )}{b}+\frac {(e+f x)^2 \arctan \left (e^{2 a+2 b x}\right )}{b}\right )}{2 f}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {(e+f x)^2 \cot ^{-1}(\coth (a+b x))}{2 f}-\frac {b \left (\frac {i f \left (\frac {f \int e^{-2 a-2 b x} \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{2 b}\right )}{b}-\frac {i f \left (\frac {f \int e^{-2 a-2 b x} \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{2 b}\right )}{b}+\frac {(e+f x)^2 \arctan \left (e^{2 a+2 b x}\right )}{b}\right )}{2 f}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {(e+f x)^2 \cot ^{-1}(\coth (a+b x))}{2 f}-\frac {b \left (\frac {(e+f x)^2 \arctan \left (e^{2 a+2 b x}\right )}{b}+\frac {i f \left (\frac {f \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{4 b^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{2 b}\right )}{b}-\frac {i f \left (\frac {f \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{4 b^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{2 b}\right )}{b}\right )}{2 f}\) |
((e + f*x)^2*ArcCot[Coth[a + b*x]])/(2*f) - (b*(((e + f*x)^2*ArcTan[E^(2*a + 2*b*x)])/b + (I*f*(-1/2*((e + f*x)*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/b + (f*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(4*b^2)))/b - (I*f*(-1/2*((e + f*x) *PolyLog[2, I*E^(2*a + 2*b*x)])/b + (f*PolyLog[3, I*E^(2*a + 2*b*x)])/(4*b ^2)))/b))/(2*f)
3.3.2.3.1 Defintions of rubi rules used
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[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[ArcCot[Coth[(a_.) + (b_.)*(x_)]]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(ArcCot[Coth[a + b*x]]/(f*(m + 1))), x] - Simp[b/ (f*(m + 1)) Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
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 = 2.32 (sec) , antiderivative size = 1776, normalized size of antiderivative = 11.17
1/4*I*f/b^2*ln(1+I*exp(2*b*x+2*a))*a^2+1/4*I*f/b*polylog(2,-I*exp(2*b*x+2* a))*x+1/4*I*f/b^2*polylog(2,-I*exp(2*b*x+2*a))*a+1/4*I*f/b^2*a^2*ln(-exp(2 *b*x+2*a)+I)+1/2*I*e/b*ln(1+exp(b*x+a)*(-1)^(3/4))*a+1/2*I*e/b*ln(1-exp(b* x+a)*(-1)^(3/4))*a-1/2*I*e/b*a*ln(-exp(2*b*x+2*a)+I)+1/2*I*(1/2*f*x^2+e*x) *ln(exp(2*b*x+2*a)+I)-1/4*Pi*(csgn(I*(exp(2*b*x+2*a)-I))*csgn(I/(exp(2*b*x +2*a)-1))*csgn(I*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2*a)-1))-csgn(I*(exp(2*b*x+ 2*a)-I))*csgn(I*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2*a)-1))^2-csgn(I*(exp(2*b*x +2*a)+I))*csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*(exp(2*b*x+2*a)+I)/(exp(2*b*x+ 2*a)-1))+csgn(I*(exp(2*b*x+2*a)+I))*csgn(I*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2 *a)-1))^2+csgn(I*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2*a)-1))*csgn((1+I)*(exp(2* b*x+2*a)-I)/(exp(2*b*x+2*a)-1))-csgn((1+I)*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2 *a)-1))^2-csgn(I*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a)-1))*csgn((1-I)*(exp(2* b*x+2*a)+I)/(exp(2*b*x+2*a)-1))-csgn((1-I)*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2 *a)-1))^2-csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*(exp(2*b*x+2*a)-I)/(exp(2*b*x+ 2*a)-1))^2+csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*(exp(2*b*x+2*a)+I)/(exp(2*b*x +2*a)-1))^2+csgn(I*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2*a)-1))^3-csgn(I*(exp(2* b*x+2*a)-I)/(exp(2*b*x+2*a)-1))*csgn((1+I)*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2 *a)-1))^2-csgn(I*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a)-1))^3+csgn(I*(exp(2*b* x+2*a)+I)/(exp(2*b*x+2*a)-1))*csgn((1-I)*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a )-1))^2+csgn((1+I)*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2*a)-1))^3+csgn((1-I)*...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (130) = 260\).
Time = 0.32 (sec) , antiderivative size = 600, normalized size of antiderivative = 3.77 \[ \int (e+f x) \cot ^{-1}(\coth (a+b x)) \, dx=\frac {2 \, {\left (b^{2} f x^{2} + 2 \, b^{2} e x\right )} \arctan \left (\frac {\sinh \left (b x + a\right )}{\cosh \left (b x + a\right )}\right ) - 2 \, {\left (i \, b f x + i \, b e\right )} {\rm Li}_2\left (\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 \, {\left (i \, b f x + i \, b e\right )} {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 \, {\left (-i \, b f x - i \, b e\right )} {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 \, {\left (-i \, b f x - i \, b e\right )} {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + {\left (-i \, b^{2} f x^{2} - 2 i \, b^{2} e x - 2 i \, a b e + i \, a^{2} f\right )} \log \left (\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (-i \, b^{2} f x^{2} - 2 i \, b^{2} e x - 2 i \, a b e + i \, a^{2} f\right )} \log \left (-\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (i \, b^{2} f x^{2} + 2 i \, b^{2} e x + 2 i \, a b e - i \, a^{2} f\right )} \log \left (\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (i \, b^{2} f x^{2} + 2 i \, b^{2} e x + 2 i \, a b e - i \, a^{2} f\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (2 i \, a b e - i \, a^{2} f\right )} \log \left (i \, \sqrt {4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + {\left (2 i \, a b e - i \, a^{2} f\right )} \log \left (-i \, \sqrt {4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + {\left (-2 i \, a b e + i \, a^{2} f\right )} \log \left (i \, \sqrt {-4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + {\left (-2 i \, a b e + i \, a^{2} f\right )} \log \left (-i \, \sqrt {-4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + 2 i \, f {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 2 i \, f {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 i \, f {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 i \, f {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{4 \, b^{2}} \]
1/4*(2*(b^2*f*x^2 + 2*b^2*e*x)*arctan(sinh(b*x + a)/cosh(b*x + a)) - 2*(I* b*f*x + I*b*e)*dilog(1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 2*(I *b*f*x + I*b*e)*dilog(-1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 2* (-I*b*f*x - I*b*e)*dilog(1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 2*(-I*b*f*x - I*b*e)*dilog(-1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a) )) + (-I*b^2*f*x^2 - 2*I*b^2*e*x - 2*I*a*b*e + I*a^2*f)*log(1/2*sqrt(4*I)* (cosh(b*x + a) + sinh(b*x + a)) + 1) + (-I*b^2*f*x^2 - 2*I*b^2*e*x - 2*I*a *b*e + I*a^2*f)*log(-1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (I*b^2*f*x^2 + 2*I*b^2*e*x + 2*I*a*b*e - I*a^2*f)*log(1/2*sqrt(-4*I)*(cosh (b*x + a) + sinh(b*x + a)) + 1) + (I*b^2*f*x^2 + 2*I*b^2*e*x + 2*I*a*b*e - I*a^2*f)*log(-1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (2*I* a*b*e - I*a^2*f)*log(I*sqrt(4*I) + 2*cosh(b*x + a) + 2*sinh(b*x + a)) + (2 *I*a*b*e - I*a^2*f)*log(-I*sqrt(4*I) + 2*cosh(b*x + a) + 2*sinh(b*x + a)) + (-2*I*a*b*e + I*a^2*f)*log(I*sqrt(-4*I) + 2*cosh(b*x + a) + 2*sinh(b*x + a)) + (-2*I*a*b*e + I*a^2*f)*log(-I*sqrt(-4*I) + 2*cosh(b*x + a) + 2*sinh (b*x + a)) + 2*I*f*polylog(3, 1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a) )) + 2*I*f*polylog(3, -1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 2* I*f*polylog(3, 1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 2*I*f*pol ylog(3, -1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))))/b^2
\[ \int (e+f x) \cot ^{-1}(\coth (a+b x)) \, dx=\int \left (e + f x\right ) \operatorname {acot}{\left (\coth {\left (a + b x \right )} \right )}\, dx \]
\[ \int (e+f x) \cot ^{-1}(\coth (a+b x)) \, dx=\int { {\left (f x + e\right )} \operatorname {arccot}\left (\coth \left (b x + a\right )\right ) \,d x } \]
1/2*(f*x^2 + 2*e*x)*arctan((e^(2*b*x + 2*a) - 1)/(e^(2*b*x + 2*a) + 1)) - integrate((b*f*x^2*e^(2*a) + 2*b*e*x*e^(2*a))*e^(2*b*x)/(e^(4*b*x + 4*a) + 1), x)
\[ \int (e+f x) \cot ^{-1}(\coth (a+b x)) \, dx=\int { {\left (f x + e\right )} \operatorname {arccot}\left (\coth \left (b x + a\right )\right ) \,d x } \]
Timed out. \[ \int (e+f x) \cot ^{-1}(\coth (a+b x)) \, dx=\int \mathrm {acot}\left (\mathrm {coth}\left (a+b\,x\right )\right )\,\left (e+f\,x\right ) \,d x \]