Integrand size = 11, antiderivative size = 198 \[ \int \cot ^{-1}(c+d \cot (a+b x)) \, dx=x \cot ^{-1}(c+d \cot (a+b x))-\frac {1}{2} i x \log \left (1-\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )+\frac {1}{2} i x \log \left (1-\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )-\frac {\operatorname {PolyLog}\left (2,\frac {(1+i c-d) e^{2 i a+2 i b x}}{1+i c+d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {(c+i (1+d)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{4 b} \]
x*arccot(c+d*cot(b*x+a))-1/2*I*x*ln(1-(1+I*c-d)*exp(2*I*a+2*I*b*x)/(1+I*c+ d))+1/2*I*x*ln(1-(c+I*(1+d))*exp(2*I*a+2*I*b*x)/(c+I*(1-d)))-1/4*polylog(2 ,(1+I*c-d)*exp(2*I*a+2*I*b*x)/(1+I*c+d))/b+1/4*polylog(2,(c+I*(1+d))*exp(2 *I*a+2*I*b*x)/(c+I*(1-d)))/b
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1649\) vs. \(2(198)=396\).
Time = 12.13 (sec) , antiderivative size = 1649, normalized size of antiderivative = 8.33 \[ \int \cot ^{-1}(c+d \cot (a+b x)) \, dx =\text {Too large to display} \]
x*ArcCot[c + d*Cot[a + b*x]] - (d*(4*a*Sqrt[-d^2]*ArcTan[(c*d + Tan[a + b* x] + c^2*Tan[a + b*x])/d] + I*d*Log[1 + I*Tan[a + b*x]]*Log[(c*d - Sqrt[-d ^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d - Sqrt[-d^2])] + I *d*Log[1 - I*Tan[a + b*x]]*Log[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[ a + b*x])/(-I - I*c^2 + c*d + Sqrt[-d^2])] - I*d*Log[1 + I*Tan[a + b*x]]*L og[(c*d + Sqrt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(I + I*c^2 + c*d + Sqrt[-d^2])] - I*d*Log[1 - I*Tan[a + b*x]]*Log[(-(c*d) + Sqrt[-d^2] - (1 + c^2)*Tan[a + b*x])/(I + I*c^2 - c*d + Sqrt[-d^2])] - I*d*PolyLog[2, ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d - I*Sqrt[-d^2])] + I*d*PolyL og[2, ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c*d + I*Sqrt[-d^2])] - I*d*PolyLog[2, ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c*d - I*Sqrt [-d^2])] + I*d*PolyLog[2, ((1 + c^2)*(1 + I*Tan[a + b*x]))/(1 + c^2 - I*c* d + I*Sqrt[-d^2])])*((2*a)/(b*(-1 - c^2 - d^2 + Cos[2*(a + b*x)] + c^2*Cos [2*(a + b*x)] - d^2*Cos[2*(a + b*x)] - 2*c*d*Sin[2*(a + b*x)])) - (2*(a + b*x))/(b*(-1 - c^2 - d^2 + Cos[2*(a + b*x)] + c^2*Cos[2*(a + b*x)] - d^2*C os[2*(a + b*x)] - 2*c*d*Sin[2*(a + b*x)]))))/((d*Log[1 - ((1 + c^2)*(1 - I *Tan[a + b*x]))/(1 + c^2 + I*c*d - I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*T an[a + b*x]) - (d*Log[1 - ((1 + c^2)*(1 - I*Tan[a + b*x]))/(1 + c^2 + I*c* d + I*Sqrt[-d^2])]*Sec[a + b*x]^2)/(1 - I*Tan[a + b*x]) + (d*Log[(c*d + Sq rt[-d^2] + Tan[a + b*x] + c^2*Tan[a + b*x])/(-I - I*c^2 + c*d + Sqrt[-d...
Time = 0.66 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.40, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5693, 2620, 2715, 2838}
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.
\(\displaystyle \int \cot ^{-1}(d \cot (a+b x)+c) \, dx\) |
\(\Big \downarrow \) 5693 |
\(\displaystyle -b (i c-d+1) \int \frac {e^{2 i a+2 i b x} x}{i c-(i c-d+1) e^{2 i a+2 i b x}+d+1}dx+b (-i c+d+1) \int \frac {e^{2 i a+2 i b x} x}{-i c-(-i c+d+1) e^{2 i a+2 i b x}-d+1}dx+x \cot ^{-1}(d \cot (a+b x)+c)\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -b (i c-d+1) \left (\frac {x \log \left (1-\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )}{2 b (c-i (1-d))}-\frac {\int \log \left (1-\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )dx}{2 b (c-i (1-d))}\right )+b (-i c+d+1) \left (\frac {\int \log \left (1-\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )dx}{2 b (c+i (d+1))}-\frac {x \log \left (1-\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{2 b (c+i (d+1))}\right )+x \cot ^{-1}(d \cot (a+b x)+c)\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -b (i c-d+1) \left (\frac {i \int e^{-2 i a-2 i b x} \log \left (1-\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )de^{2 i a+2 i b x}}{4 b^2 (c-i (1-d))}+\frac {x \log \left (1-\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )}{2 b (c-i (1-d))}\right )+b (-i c+d+1) \left (-\frac {i \int e^{-2 i a-2 i b x} \log \left (1-\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )de^{2 i a+2 i b x}}{4 b^2 (c+i (d+1))}-\frac {x \log \left (1-\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{2 b (c+i (d+1))}\right )+x \cot ^{-1}(d \cot (a+b x)+c)\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -b (i c-d+1) \left (\frac {x \log \left (1-\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )}{2 b (c-i (1-d))}-\frac {i \operatorname {PolyLog}\left (2,\frac {(i c-d+1) e^{2 i a+2 i b x}}{i c+d+1}\right )}{4 b^2 (c-i (1-d))}\right )+b (-i c+d+1) \left (\frac {i \operatorname {PolyLog}\left (2,\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{4 b^2 (c+i (d+1))}-\frac {x \log \left (1-\frac {(c+i (d+1)) e^{2 i a+2 i b x}}{c+i (1-d)}\right )}{2 b (c+i (d+1))}\right )+x \cot ^{-1}(d \cot (a+b x)+c)\) |
x*ArcCot[c + d*Cot[a + b*x]] - b*(1 + I*c - d)*((x*Log[1 - ((1 + I*c - d)* E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)])/(2*b*(c - I*(1 - d))) - ((I/4)*Po lyLog[2, ((1 + I*c - d)*E^((2*I)*a + (2*I)*b*x))/(1 + I*c + d)])/(b^2*(c - I*(1 - d)))) + b*(1 - I*c + d)*(-1/2*(x*Log[1 - ((c + I*(1 + d))*E^((2*I) *a + (2*I)*b*x))/(c + I*(1 - d))])/(b*(c + I*(1 + d))) + ((I/4)*PolyLog[2, ((c + I*(1 + d))*E^((2*I)*a + (2*I)*b*x))/(c + I*(1 - d))])/(b^2*(c + I*( 1 + d))))
3.2.73.3.1 Defintions of rubi rules used
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[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[ArcCot[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*ArcC ot[c + d*Cot[a + b*x]], x] + (-Simp[b*(1 + I*c - d) Int[x*(E^(2*I*a + 2*I *b*x)/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x))), x], x] + Simp[b*( 1 - I*c + d) Int[x*(E^(2*I*a + 2*I*b*x)/(1 - I*c - d - (1 - I*c + d)*E^(2 *I*a + 2*I*b*x))), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, -1 ]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1145 vs. \(2 (168 ) = 336\).
Time = 4.58 (sec) , antiderivative size = 1146, normalized size of antiderivative = 5.79
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1146\) |
default | \(\text {Expression too large to display}\) | \(1146\) |
risch | \(\text {Expression too large to display}\) | \(4982\) |
1/b/d*(-d*(1/2*Pi-arccot(cot(b*x+a)))*arccot(c+d*cot(b*x+a))-d^2*(-1/d*arc tan(d*((c+d*cot(b*x+a))/d-c/d)+c)*arctan(-(c+d*cot(b*x+a))/d+c/d)-1/d^2*(1 /2*I*d^2*ln(1-(c-I*d+I)*(1+I*(d*((c+d*cot(b*x+a))/d-c/d)+c))^2/((d*((c+d*c ot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))*arctan(d*((c+d*cot(b*x+a))/d-c/d)+c)/ (1+I*c+d)+1/2*I*d*ln(1-(c-I*d+I)*(1+I*(d*((c+d*cot(b*x+a))/d-c/d)+c))^2/(( d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))*arctan(d*((c+d*cot(b*x+a))/d -c/d)+c)/(1+I*c+d)+1/2*I*d/(c-I*d-I)*ln(1-(c-I*d+I)*(1+I*(d*((c+d*cot(b*x+ a))/d-c/d)+c))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))*c*arctan( d*((c+d*cot(b*x+a))/d-c/d)+c)+1/2*d^2*arctan(d*((c+d*cot(b*x+a))/d-c/d)+c) ^2/(1+I*c+d)+1/4*d^2*polylog(2,(c-I*d+I)*(1+I*(d*((c+d*cot(b*x+a))/d-c/d)+ c))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))/(1+I*c+d)+1/2*d*arct an(d*((c+d*cot(b*x+a))/d-c/d)+c)^2/(1+I*c+d)+1/2*d/(c-I*d-I)*c*arctan(d*(( c+d*cot(b*x+a))/d-c/d)+c)^2+1/4*d*polylog(2,(c-I*d+I)*(1+I*(d*((c+d*cot(b* x+a))/d-c/d)+c))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))/(1+I*c+ d)+1/4*d/(c-I*d-I)*polylog(2,(c-I*d+I)*(1+I*(d*((c+d*cot(b*x+a))/d-c/d)+c) )^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(I*d+I-c))*c-1/2*I*d*arctan(d*((c +d*cot(b*x+a))/d-c/d)+c)*ln(1-(I+c+I*d)*(1+I*(d*((c+d*cot(b*x+a))/d-c/d)+c ))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(-I*d+I-c))-1/2*d*arctan(d*((c+d *cot(b*x+a))/d-c/d)+c)^2-1/4*d*polylog(2,(I+c+I*d)*(1+I*(d*((c+d*cot(b*x+a ))/d-c/d)+c))^2/((d*((c+d*cot(b*x+a))/d-c/d)+c)^2+1)/(-I*d+I-c)))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 965 vs. \(2 (140) = 280\).
Time = 0.40 (sec) , antiderivative size = 965, normalized size of antiderivative = 4.87 \[ \int \cot ^{-1}(c+d \cot (a+b x)) \, dx=\text {Too large to display} \]
1/8*(8*b*x*arccot(d*cot(b*x + a) + c) + 2*I*a*log(1/2*c^2 + I*c*d - 1/2*d^ 2 - 1/2*(c^2 + d^2 + 2*d + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 + 2*I* d + I)*sin(2*b*x + 2*a) + 1/2) - 2*I*a*log(1/2*c^2 + I*c*d - 1/2*d^2 - 1/2 *(c^2 + d^2 - 2*d + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 - 2*I*d + I)* sin(2*b*x + 2*a) + 1/2) - 2*I*a*log(-1/2*c^2 + I*c*d + 1/2*d^2 + 1/2*(c^2 + d^2 + 2*d + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 + 2*I*d + I)*sin(2* b*x + 2*a) - 1/2) + 2*I*a*log(-1/2*c^2 + I*c*d + 1/2*d^2 + 1/2*(c^2 + d^2 - 2*d + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 - 2*I*d + I)*sin(2*b*x + 2*a) - 1/2) - 2*(I*b*x + I*a)*log((c^2 + d^2 - (c^2 + 2*I*c*d - d^2 + 1)*c os(2*b*x + 2*a) + (-I*c^2 + 2*c*d + I*d^2 - I)*sin(2*b*x + 2*a) + 2*d + 1) /(c^2 + d^2 + 2*d + 1)) - 2*(-I*b*x - I*a)*log((c^2 + d^2 - (c^2 - 2*I*c*d - d^2 + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*c*d - I*d^2 + I)*sin(2*b*x + 2*a ) + 2*d + 1)/(c^2 + d^2 + 2*d + 1)) - 2*(-I*b*x - I*a)*log((c^2 + d^2 - (c ^2 + 2*I*c*d - d^2 + 1)*cos(2*b*x + 2*a) + (-I*c^2 + 2*c*d + I*d^2 - I)*si n(2*b*x + 2*a) - 2*d + 1)/(c^2 + d^2 - 2*d + 1)) - 2*(I*b*x + I*a)*log((c^ 2 + d^2 - (c^2 - 2*I*c*d - d^2 + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*c*d - I* d^2 + I)*sin(2*b*x + 2*a) - 2*d + 1)/(c^2 + d^2 - 2*d + 1)) - dilog(-(c^2 + d^2 - (c^2 + 2*I*c*d - d^2 + 1)*cos(2*b*x + 2*a) + (-I*c^2 + 2*c*d + I*d ^2 - I)*sin(2*b*x + 2*a) + 2*d + 1)/(c^2 + d^2 + 2*d + 1) + 1) - dilog(-(c ^2 + d^2 - (c^2 - 2*I*c*d - d^2 + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*c*d ...
\[ \int \cot ^{-1}(c+d \cot (a+b x)) \, dx=\int \operatorname {acot}{\left (c + d \cot {\left (a + b x \right )} \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (140) = 280\).
Time = 0.35 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.69 \[ \int \cot ^{-1}(c+d \cot (a+b x)) \, dx=\frac {d {\left (\frac {8 \, {\left (b x + a\right )} \arctan \left (\frac {c d + {\left (c^{2} + 1\right )} \tan \left (b x + a\right )}{d}\right )}{d} - \frac {8 \, {\left (b x + a\right )} \arctan \left (\frac {c d + {\left (c^{2} + 1\right )} \tan \left (b x + a\right )}{d}\right ) - 4 \, \arctan \left (\frac {c d + {\left (c^{2} + 1\right )} \tan \left (b x + a\right )}{d}\right ) \arctan \left (\frac {c d + {\left (c^{2} + d + 1\right )} \tan \left (b x + a\right )}{c^{2} + d^{2} + 2 \, d + 1}, -\frac {c d \tan \left (b x + a\right ) - c^{2} - d - 1}{c^{2} + d^{2} + 2 \, d + 1}\right ) + 4 \, \arctan \left (\frac {c d + {\left (c^{2} + 1\right )} \tan \left (b x + a\right )}{d}\right ) \arctan \left (-\frac {c d + {\left (c^{2} - d + 1\right )} \tan \left (b x + a\right )}{c^{2} + d^{2} - 2 \, d + 1}, -\frac {c d \tan \left (b x + a\right ) - c^{2} + d - 1}{c^{2} + d^{2} - 2 \, d + 1}\right ) - {\left (\log \left (\frac {{\left (c^{2} + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + 1}{c^{2} + d^{2} + 2 \, d + 1}\right ) - \log \left (\frac {{\left (c^{2} + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + 1}{c^{2} + d^{2} - 2 \, d + 1}\right )\right )} \log \left ({\left (c^{2} + 1\right )} d^{2} + 2 \, {\left (c^{3} + c\right )} d \tan \left (b x + a\right ) + {\left (c^{4} + 2 \, c^{2} + 1\right )} \tan \left (b x + a\right )^{2}\right ) - 2 \, {\rm Li}_2\left (\frac {{\left (i \, c - 1\right )} \tan \left (b x + a\right ) + i \, d}{c + i \, d + i}\right ) + 2 \, {\rm Li}_2\left (\frac {{\left (i \, c + 1\right )} \tan \left (b x + a\right ) + i \, d}{c + i \, d - i}\right ) + 2 \, {\rm Li}_2\left (-\frac {{\left (i \, c - 1\right )} \tan \left (b x + a\right ) + i \, d}{c - i \, d + i}\right ) - 2 \, {\rm Li}_2\left (-\frac {{\left (i \, c + 1\right )} \tan \left (b x + a\right ) + i \, d}{c - i \, d - i}\right )}{d}\right )} + 8 \, {\left (b x + a\right )} \operatorname {arccot}\left (c + \frac {d}{\tan \left (b x + a\right )}\right ) - 8 \, {\left (b x + a\right )} \arctan \left (\frac {c d + {\left (c^{2} + 1\right )} \tan \left (b x + a\right )}{d}\right )}{8 \, b} \]
1/8*(d*(8*(b*x + a)*arctan((c*d + (c^2 + 1)*tan(b*x + a))/d)/d - (8*(b*x + a)*arctan((c*d + (c^2 + 1)*tan(b*x + a))/d) - 4*arctan((c*d + (c^2 + 1)*t an(b*x + a))/d)*arctan2((c*d + (c^2 + d + 1)*tan(b*x + a))/(c^2 + d^2 + 2* d + 1), -(c*d*tan(b*x + a) - c^2 - d - 1)/(c^2 + d^2 + 2*d + 1)) + 4*arcta n((c*d + (c^2 + 1)*tan(b*x + a))/d)*arctan2(-(c*d + (c^2 - d + 1)*tan(b*x + a))/(c^2 + d^2 - 2*d + 1), -(c*d*tan(b*x + a) - c^2 + d - 1)/(c^2 + d^2 - 2*d + 1)) - (log(((c^2 + 1)*tan(b*x + a)^2 + c^2 + 1)/(c^2 + d^2 + 2*d + 1)) - log(((c^2 + 1)*tan(b*x + a)^2 + c^2 + 1)/(c^2 + d^2 - 2*d + 1)))*lo g((c^2 + 1)*d^2 + 2*(c^3 + c)*d*tan(b*x + a) + (c^4 + 2*c^2 + 1)*tan(b*x + a)^2) - 2*dilog(((I*c - 1)*tan(b*x + a) + I*d)/(c + I*d + I)) + 2*dilog(( (I*c + 1)*tan(b*x + a) + I*d)/(c + I*d - I)) + 2*dilog(-((I*c - 1)*tan(b*x + a) + I*d)/(c - I*d + I)) - 2*dilog(-((I*c + 1)*tan(b*x + a) + I*d)/(c - I*d - I)))/d) + 8*(b*x + a)*arccot(c + d/tan(b*x + a)) - 8*(b*x + a)*arct an((c*d + (c^2 + 1)*tan(b*x + a))/d))/b
\[ \int \cot ^{-1}(c+d \cot (a+b x)) \, dx=\int { \operatorname {arccot}\left (d \cot \left (b x + a\right ) + c\right ) \,d x } \]
Timed out. \[ \int \cot ^{-1}(c+d \cot (a+b x)) \, dx=\int \mathrm {acot}\left (c+d\,\mathrm {cot}\left (a+b\,x\right )\right ) \,d x \]