Integrand size = 14, antiderivative size = 127 \[ \int (c+d x)^3 \cot (a+b x) \, dx=-\frac {i (c+d x)^4}{4 d}+\frac {(c+d x)^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i d^3 \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4} \]
-1/4*I*(d*x+c)^4/d+(d*x+c)^3*ln(1-exp(2*I*(b*x+a)))/b-3/2*I*d*(d*x+c)^2*po lylog(2,exp(2*I*(b*x+a)))/b^2+3/2*d^2*(d*x+c)*polylog(3,exp(2*I*(b*x+a)))/ b^3+3/4*I*d^3*polylog(4,exp(2*I*(b*x+a)))/b^4
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(560\) vs. \(2(127)=254\).
Time = 2.86 (sec) , antiderivative size = 560, normalized size of antiderivative = 4.41 \[ \int (c+d x)^3 \cot (a+b x) \, dx=\frac {6 i b^3 c^2 d \pi x+4 i b^4 c d^2 x^3+i b^4 d^3 x^4-12 i b^3 c^2 d x \arctan (\tan (a))+6 b^4 c^2 d x^2 \cot (a)+6 b^2 c^2 d \pi \log \left (1+e^{-2 i b x}\right )+12 b^3 c d^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+4 b^3 d^3 x^3 \log \left (1-e^{-i (a+b x)}\right )+12 b^3 c d^2 x^2 \log \left (1+e^{-i (a+b x)}\right )+4 b^3 d^3 x^3 \log \left (1+e^{-i (a+b x)}\right )+12 b^3 c^2 d x \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+12 b^2 c^2 d \arctan (\tan (a)) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )-6 b^2 c^2 d \pi \log (\cos (b x))+4 b^3 c^3 \log (\sin (a+b x))-12 b^2 c^2 d \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))+12 i b^2 d^2 x (2 c+d x) \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )+12 i b^2 d^2 x (2 c+d x) \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )-6 i b^2 c^2 d \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )+24 b c d^2 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+24 b d^3 x \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+24 b c d^2 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )+24 b d^3 x \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )-24 i d^3 \operatorname {PolyLog}\left (4,-e^{-i (a+b x)}\right )-24 i d^3 \operatorname {PolyLog}\left (4,e^{-i (a+b x)}\right )-6 b^4 c^2 d e^{i \arctan (\tan (a))} x^2 \cot (a) \sqrt {\sec ^2(a)}}{4 b^4} \]
((6*I)*b^3*c^2*d*Pi*x + (4*I)*b^4*c*d^2*x^3 + I*b^4*d^3*x^4 - (12*I)*b^3*c ^2*d*x*ArcTan[Tan[a]] + 6*b^4*c^2*d*x^2*Cot[a] + 6*b^2*c^2*d*Pi*Log[1 + E^ ((-2*I)*b*x)] + 12*b^3*c*d^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 4*b^3*d^3*x ^3*Log[1 - E^((-I)*(a + b*x))] + 12*b^3*c*d^2*x^2*Log[1 + E^((-I)*(a + b*x ))] + 4*b^3*d^3*x^3*Log[1 + E^((-I)*(a + b*x))] + 12*b^3*c^2*d*x*Log[1 - E ^((2*I)*(b*x + ArcTan[Tan[a]]))] + 12*b^2*c^2*d*ArcTan[Tan[a]]*Log[1 - E^( (2*I)*(b*x + ArcTan[Tan[a]]))] - 6*b^2*c^2*d*Pi*Log[Cos[b*x]] + 4*b^3*c^3* Log[Sin[a + b*x]] - 12*b^2*c^2*d*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a ]]]] + (12*I)*b^2*d^2*x*(2*c + d*x)*PolyLog[2, -E^((-I)*(a + b*x))] + (12* I)*b^2*d^2*x*(2*c + d*x)*PolyLog[2, E^((-I)*(a + b*x))] - (6*I)*b^2*c^2*d* PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))] + 24*b*c*d^2*PolyLog[3, -E^(( -I)*(a + b*x))] + 24*b*d^3*x*PolyLog[3, -E^((-I)*(a + b*x))] + 24*b*c*d^2* PolyLog[3, E^((-I)*(a + b*x))] + 24*b*d^3*x*PolyLog[3, E^((-I)*(a + b*x))] - (24*I)*d^3*PolyLog[4, -E^((-I)*(a + b*x))] - (24*I)*d^3*PolyLog[4, E^(( -I)*(a + b*x))] - 6*b^4*c^2*d*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^ 2])/(4*b^4)
Time = 0.68 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.35, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 25, 4202, 2620, 3011, 7163, 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.
| \(\displaystyle \int (c+d x)^3 \cot (a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -(c+d x)^3 \tan \left (a+b x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int (c+d x)^3 \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle 2 i \int \frac {e^{i (2 a+2 b x+\pi )} (c+d x)^3}{1+e^{i (2 a+2 b x+\pi )}}dx-\frac {i (c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle 2 i \left (\frac {3 i d \int (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {i (c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {i (c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \left (\frac {i d \int \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {i (c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \left (\frac {d \int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {i (c+d x)^4}{4 d}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \left (\frac {d \operatorname {PolyLog}\left (4,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {i (c+d x)^4}{4 d}\) |
((-1/4*I)*(c + d*x)^4)/d + (2*I)*(((-1/2*I)*(c + d*x)^3*Log[1 + E^(I*(2*a + Pi + 2*b*x))])/b + (((3*I)/2)*d*(((I/2)*(c + d*x)^2*PolyLog[2, -E^(I*(2* a + Pi + 2*b*x))])/b - (I*d*(((-1/2*I)*(c + d*x)*PolyLog[3, -E^(I*(2*a + P i + 2*b*x))])/b + (d*PolyLog[4, -E^(I*(2*a + Pi + 2*b*x))])/(4*b^2)))/b))/ b)
3.1.33.3.1 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[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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[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]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 791 vs. \(2 (108 ) = 216\).
Time = 1.16 (sec) , antiderivative size = 792, normalized size of antiderivative = 6.24
| method | result | size |
| risch | \(-\frac {3 i d^{3} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b^{2}}-\frac {2 i d^{3} a^{3} x}{b^{3}}+\frac {3 c \,d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{3}}+\frac {6 c^{2} d a \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {3 c^{2} d a \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{2}}-\frac {d^{3} a^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{4}}+\frac {6 i d^{3} \operatorname {polylog}\left (4, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}-\frac {3 c \,d^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a^{2}}{b^{3}}+\frac {3 d \,c^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}-\frac {6 c \,d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {3 d \,c^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}-\frac {3 i d \,c^{2} a^{2}}{b^{2}}-\frac {3 i d \,c^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {3 i d \,c^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {4 i c \,d^{2} a^{3}}{b^{3}}+\frac {2 d^{3} a^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {6 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{3}}+\frac {d^{3} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a^{3}}{b^{4}}+\frac {6 d^{3} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{3}}-\frac {3 i d^{3} a^{4}}{2 b^{4}}+\frac {6 i d^{3} \operatorname {polylog}\left (4, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {d^{3} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x^{3}}{b}+\frac {d^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x^{3}}{b}+\frac {6 c \,d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {6 c \,d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {c^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b}-\frac {i d^{3} x^{4}}{4}+i c^{3} x +\frac {i c^{4}}{4 d}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b}+\frac {3 c \,d^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b}-\frac {3 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b^{2}}+\frac {c^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b}-\frac {2 c^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {3 c \,d^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x^{2}}{b}-i d^{2} c \,x^{3}-\frac {3 i d \,c^{2} x^{2}}{2}-\frac {6 i c \,d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {6 i d \,c^{2} x a}{b}+\frac {6 i c \,d^{2} a^{2} x}{b^{2}}-\frac {6 i c \,d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}\) | \(792\) |
1/b*c^3*ln(exp(I*(b*x+a))+1)-2/b*c^3*ln(exp(I*(b*x+a)))+1/b*c^3*ln(exp(I*( b*x+a))-1)+3/b*c*d^2*ln(exp(I*(b*x+a))+1)*x^2-3/b^3*c*d^2*ln(1-exp(I*(b*x+ a)))*a^2+3/b^2*d*c^2*ln(1-exp(I*(b*x+a)))*a-6/b^3*c*d^2*a^2*ln(exp(I*(b*x+ a)))-1/4*I*d^3*x^4-I*d^2*c*x^3-3/2*I*d*c^2*x^2-3*I/b^2*d*c^2*a^2-3*I/b^2*d *c^2*polylog(2,exp(I*(b*x+a)))-3*I/b^2*d*c^2*polylog(2,-exp(I*(b*x+a)))+4* I/b^3*c*d^2*a^3+3/b*d*c^2*ln(exp(I*(b*x+a))+1)*x+3/b*c*d^2*ln(1-exp(I*(b*x +a)))*x^2-3*I/b^2*d^3*polylog(2,-exp(I*(b*x+a)))*x^2-3*I/b^2*d^3*polylog(2 ,exp(I*(b*x+a)))*x^2-2*I/b^3*d^3*a^3*x+I*c^3*x+1/4*I/d*c^4+2/b^4*d^3*a^3*l n(exp(I*(b*x+a)))+6/b^3*d^3*polylog(3,-exp(I*(b*x+a)))*x+1/b^4*d^3*ln(1-ex p(I*(b*x+a)))*a^3+6/b^3*d^3*polylog(3,exp(I*(b*x+a)))*x-3/2*I/b^4*d^3*a^4+ 6*I/b^4*d^3*polylog(4,-exp(I*(b*x+a)))+1/b*d^3*ln(1-exp(I*(b*x+a)))*x^3+1/ b*d^3*ln(exp(I*(b*x+a))+1)*x^3+6/b^3*c*d^2*polylog(3,exp(I*(b*x+a)))+6/b^3 *c*d^2*polylog(3,-exp(I*(b*x+a)))-1/b^4*d^3*a^3*ln(exp(I*(b*x+a))-1)+3/b^3 *c*d^2*a^2*ln(exp(I*(b*x+a))-1)+6/b^2*c^2*d*a*ln(exp(I*(b*x+a)))-3/b^2*c^2 *d*a*ln(exp(I*(b*x+a))-1)+3/b*d*c^2*ln(1-exp(I*(b*x+a)))*x-6*I/b^2*c*d^2*p olylog(2,-exp(I*(b*x+a)))*x-6*I/b*d*c^2*x*a+6*I/b^2*c*d^2*a^2*x-6*I/b^2*c* d^2*polylog(2,exp(I*(b*x+a)))*x+6*I*d^3*polylog(4,exp(I*(b*x+a)))/b^4
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (104) = 208\).
Time = 0.31 (sec) , antiderivative size = 818, normalized size of antiderivative = 6.44 \[ \int (c+d x)^3 \cot (a+b x) \, dx=\frac {6 i \, d^{3} {\rm polylog}\left (4, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 6 i \, d^{3} {\rm polylog}\left (4, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 6 i \, d^{3} {\rm polylog}\left (4, -\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 6 i \, d^{3} {\rm polylog}\left (4, -\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (i \, b^{2} d^{3} x^{2} + 2 i \, b^{2} c d^{2} x + i \, b^{2} c^{2} d\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (-i \, b^{2} d^{3} x^{2} - 2 i \, b^{2} c d^{2} x - i \, b^{2} c^{2} d\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (-i \, b^{2} d^{3} x^{2} - 2 i \, b^{2} c d^{2} x - i \, b^{2} c^{2} d\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (i \, b^{2} d^{3} x^{2} + 2 i \, b^{2} c d^{2} x + i \, b^{2} c^{2} d\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) - \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) + 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, -\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, -\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{2 \, b^{4}} \]
1/2*(6*I*d^3*polylog(4, cos(b*x + a) + I*sin(b*x + a)) - 6*I*d^3*polylog(4 , cos(b*x + a) - I*sin(b*x + a)) - 6*I*d^3*polylog(4, -cos(b*x + a) + I*si n(b*x + a)) + 6*I*d^3*polylog(4, -cos(b*x + a) - I*sin(b*x + a)) - 3*(I*b^ 2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d)*dilog(cos(b*x + a) + I*sin(b*x + a)) - 3*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d)*dilog(cos(b*x + a) - I*sin(b*x + a)) - 3*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d) *dilog(-cos(b*x + a) + I*sin(b*x + a)) - 3*(I*b^2*d^3*x^2 + 2*I*b^2*c*d^2* x + I*b^2*c^2*d)*dilog(-cos(b*x + a) - I*sin(b*x + a)) + (b^3*d^3*x^3 + 3* b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(cos(b*x + a) + I*sin(b*x + a) + 1) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(cos( b*x + a) - I*sin(b*x + a) + 1) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(-1/2*cos(b*x + a) - 1/2*I*sin (b*x + a) + 1/2) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^ 2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*log(-cos(b*x + a) + I*sin(b*x + a) + 1) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2* b*c*d^2 + a^3*d^3)*log(-cos(b*x + a) - I*sin(b*x + a) + 1) + 6*(b*d^3*x + b*c*d^2)*polylog(3, cos(b*x + a) + I*sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2) *polylog(3, cos(b*x + a) - I*sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2)*polylog (3, -cos(b*x + a) + I*sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2)*polylog(3, ...
\[ \int (c+d x)^3 \cot (a+b x) \, dx=\int \left (c + d x\right )^{3} \cos {\left (a + b x \right )} \csc {\left (a + b x \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (104) = 208\).
Time = 0.34 (sec) , antiderivative size = 759, normalized size of antiderivative = 5.98 \[ \int (c+d x)^3 \cot (a+b x) \, dx=\frac {4 \, c^{3} \log \left (\sin \left (b x + a\right )\right ) - \frac {12 \, a c^{2} d \log \left (\sin \left (b x + a\right )\right )}{b} + \frac {12 \, a^{2} c d^{2} \log \left (\sin \left (b x + a\right )\right )}{b^{2}} - \frac {4 \, a^{3} d^{3} \log \left (\sin \left (b x + a\right )\right )}{b^{3}} + \frac {-i \, {\left (b x + a\right )}^{4} d^{3} - 4 \, {\left (i \, b c d^{2} - i \, a d^{3}\right )} {\left (b x + a\right )}^{3} + 24 i \, d^{3} {\rm Li}_{4}(-e^{\left (i \, b x + i \, a\right )}) + 24 i \, d^{3} {\rm Li}_{4}(e^{\left (i \, b x + i \, a\right )}) - 6 \, {\left (i \, b^{2} c^{2} d - 2 i \, a b c d^{2} + i \, a^{2} d^{3}\right )} {\left (b x + a\right )}^{2} - 4 \, {\left (-i \, {\left (b x + a\right )}^{3} d^{3} + 3 \, {\left (-i \, b c d^{2} + i \, a d^{3}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (-i \, b^{2} c^{2} d + 2 i \, a b c d^{2} - i \, a^{2} d^{3}\right )} {\left (b x + a\right )}\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - 4 \, {\left (i \, {\left (b x + a\right )}^{3} d^{3} + 3 \, {\left (i \, b c d^{2} - i \, a d^{3}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (i \, b^{2} c^{2} d - 2 i \, a b c d^{2} + i \, a^{2} d^{3}\right )} {\left (b x + a\right )}\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 12 \, {\left (i \, b^{2} c^{2} d - 2 i \, a b c d^{2} + i \, {\left (b x + a\right )}^{2} d^{3} + i \, a^{2} d^{3} + 2 \, {\left (i \, b c d^{2} - i \, a d^{3}\right )} {\left (b x + a\right )}\right )} {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) - 12 \, {\left (i \, b^{2} c^{2} d - 2 i \, a b c d^{2} + i \, {\left (b x + a\right )}^{2} d^{3} + i \, a^{2} d^{3} + 2 \, {\left (i \, b c d^{2} - i \, a d^{3}\right )} {\left (b x + a\right )}\right )} {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) + 2 \, {\left ({\left (b x + a\right )}^{3} d^{3} + 3 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} {\left (b x + a\right )}\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) + 2 \, {\left ({\left (b x + a\right )}^{3} d^{3} + 3 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} {\left (b x + a\right )}\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) + 24 \, {\left (b c d^{2} + {\left (b x + a\right )} d^{3} - a d^{3}\right )} {\rm Li}_{3}(-e^{\left (i \, b x + i \, a\right )}) + 24 \, {\left (b c d^{2} + {\left (b x + a\right )} d^{3} - a d^{3}\right )} {\rm Li}_{3}(e^{\left (i \, b x + i \, a\right )})}{b^{3}}}{4 \, b} \]
1/4*(4*c^3*log(sin(b*x + a)) - 12*a*c^2*d*log(sin(b*x + a))/b + 12*a^2*c*d ^2*log(sin(b*x + a))/b^2 - 4*a^3*d^3*log(sin(b*x + a))/b^3 + (-I*(b*x + a) ^4*d^3 - 4*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^3 + 24*I*d^3*polylog(4, -e^(I*b *x + I*a)) + 24*I*d^3*polylog(4, e^(I*b*x + I*a)) - 6*(I*b^2*c^2*d - 2*I*a *b*c*d^2 + I*a^2*d^3)*(b*x + a)^2 - 4*(-I*(b*x + a)^3*d^3 + 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 + 3*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*a^2*d^3)*(b*x + a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 4*(I*(b*x + a)^3*d^3 + 3* (I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + 3*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*a^2 *d^3)*(b*x + a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 12*(I*b^2*c^2* d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3 + I*a^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3 )*(b*x + a))*dilog(-e^(I*b*x + I*a)) - 12*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + I *(b*x + a)^2*d^3 + I*a^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*dilog(e^ (I*b*x + I*a)) + 2*((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3* (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a))*log(cos(b*x + a)^2 + sin(b* x + a)^2 + 2*cos(b*x + a) + 1) + 2*((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)* (b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 24*(b*c*d^2 + (b*x + a)*d ^3 - a*d^3)*polylog(3, -e^(I*b*x + I*a)) + 24*(b*c*d^2 + (b*x + a)*d^3 - a *d^3)*polylog(3, e^(I*b*x + I*a)))/b^3)/b
\[ \int (c+d x)^3 \cot (a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \cos \left (b x + a\right ) \csc \left (b x + a\right ) \,d x } \]
Timed out. \[ \int (c+d x)^3 \cot (a+b x) \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^3}{\sin \left (a+b\,x\right )} \,d x \]