Integrand size = 22, antiderivative size = 137 \[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=-\frac {2 i d (c+d x)^3}{b^2}-\frac {2 d (c+d x)^3 \cot (a+b x)}{b^2}-\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}+\frac {6 d^2 (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^4 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^5} \]
-2*I*d*(d*x+c)^3/b^2-2*d*(d*x+c)^3*cot(b*x+a)/b^2-1/2*(d*x+c)^4*csc(b*x+a) ^2/b+6*d^2*(d*x+c)^2*ln(1-exp(2*I*(b*x+a)))/b^3-6*I*d^3*(d*x+c)*polylog(2, exp(2*I*(b*x+a)))/b^4+3*d^4*polylog(3,exp(2*I*(b*x+a)))/b^5
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(512\) vs. \(2(137)=274\).
Time = 6.64 (sec) , antiderivative size = 512, normalized size of antiderivative = 3.74 \[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=-\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}-\frac {d^4 e^{i a} \csc (a) \left (2 b^3 e^{-2 i a} x^3+3 i b^2 \left (1-e^{-2 i a}\right ) x^2 \log \left (1-e^{-i (a+b x)}\right )+3 i b^2 \left (1-e^{-2 i a}\right ) x^2 \log \left (1+e^{-i (a+b x)}\right )-6 b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )-6 b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+6 i \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+6 i \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )\right )}{b^5}+\frac {6 c^2 d^2 \csc (a) (-b x \cos (a)+\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a))}{b^3 \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {2 \csc (a) \csc (a+b x) \left (c^3 d \sin (b x)+3 c^2 d^2 x \sin (b x)+3 c d^3 x^2 \sin (b x)+d^4 x^3 \sin (b x)\right )}{b^2}-\frac {6 c d^3 \csc (a) \sec (a) \left (b^2 e^{i \arctan (\tan (a))} x^2+\frac {\left (i b x (-\pi +2 \arctan (\tan (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x+\arctan (\tan (a))) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+\pi \log (\cos (b x))+2 \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )\right ) \tan (a)}{\sqrt {1+\tan ^2(a)}}\right )}{b^4 \sqrt {\sec ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}} \]
-1/2*((c + d*x)^4*Csc[a + b*x]^2)/b - (d^4*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^( (2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - 6*b*(1 - E^ ((-2*I)*a))*x*PolyLog[2, -E^((-I)*(a + b*x))] - 6*b*(1 - E^((-2*I)*a))*x*P olyLog[2, E^((-I)*(a + b*x))] + (6*I)*(1 - E^((-2*I)*a))*PolyLog[3, -E^((- I)*(a + b*x))] + (6*I)*(1 - E^((-2*I)*a))*PolyLog[3, E^((-I)*(a + b*x))])) /b^5 + (6*c^2*d^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin [b*x]]*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (2*Csc[a]*Csc[a + b*x]*(c^3* d*Sin[b*x] + 3*c^2*d^2*x*Sin[b*x] + 3*c*d^3*x^2*Sin[b*x] + d^4*x^3*Sin[b*x ]))/b^2 - (6*c*d^3*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*( -Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan [a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*Arc Tan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^4*Sqrt[Sec[a]^2*(Cos[a] ^2 + Sin[a]^2)])
Time = 0.82 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.31, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {4910, 3042, 4672, 3042, 25, 4202, 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.
\(\displaystyle \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 4910 |
\(\displaystyle \frac {2 d \int (c+d x)^3 \csc ^2(a+b x)dx}{b}-\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \int (c+d x)^3 \csc (a+b x)^2dx}{b}-\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 4672 |
\(\displaystyle \frac {2 d \left (\frac {3 d \int (c+d x)^2 \cot (a+b x)dx}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}\right )}{b}-\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \left (\frac {3 d \int -(c+d x)^2 \tan \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}\right )}{b}-\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 d \left (-\frac {3 d \int (c+d x)^2 \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}\right )}{b}-\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle -\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}+\frac {2 d \left (-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \int \frac {e^{i (2 a+2 b x+\pi )} (c+d x)^2}{1+e^{i (2 a+2 b x+\pi )}}dx\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}+\frac {2 d \left (-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \int (c+d x) \log \left (1+e^{i (2 a+2 b x+\pi )}\right )dx}{b}-\frac {i (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}+\frac {2 d \left (-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \int \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}+\frac {2 d \left (-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {d \int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}\right )}{b}-\frac {i (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}+\frac {2 d \left (-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {d \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}\right )}{b}-\frac {i (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}\right )}{b}\) |
-1/2*((c + d*x)^4*Csc[a + b*x]^2)/b + (2*d*(-(((c + d*x)^3*Cot[a + b*x])/b ) - (3*d*(((I/3)*(c + d*x)^3)/d - (2*I)*(((-1/2*I)*(c + d*x)^2*Log[1 + E^( I*(2*a + Pi + 2*b*x))])/b + (I*d*(((I/2)*(c + d*x)*PolyLog[2, -E^(I*(2*a + Pi + 2*b*x))])/b - (d*PolyLog[3, -E^(I*(2*a + Pi + 2*b*x))])/(4*b^2)))/b) ))/b))/b
3.1.46.3.1 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[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d _.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Csc[a + b*x]^n/(b*n)), x ] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Free Q[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (127 ) = 254\).
Time = 1.82 (sec) , antiderivative size = 716, normalized size of antiderivative = 5.23
method | result | size |
risch | \(-\frac {12 i d^{4} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{4}}+\frac {2 b \,d^{4} x^{4} {\mathrm e}^{2 i \left (x b +a \right )}+8 b c \,d^{3} x^{3} {\mathrm e}^{2 i \left (x b +a \right )}+12 b \,c^{2} d^{2} x^{2} {\mathrm e}^{2 i \left (x b +a \right )}+8 b \,c^{3} d x \,{\mathrm e}^{2 i \left (x b +a \right )}-4 i d^{4} x^{3} {\mathrm e}^{2 i \left (x b +a \right )}+2 b \,c^{4} {\mathrm e}^{2 i \left (x b +a \right )}-12 i c \,d^{3} x^{2} {\mathrm e}^{2 i \left (x b +a \right )}-12 i c^{2} d^{2} x \,{\mathrm e}^{2 i \left (x b +a \right )}+4 i d^{4} x^{3}-4 i c^{3} d \,{\mathrm e}^{2 i \left (x b +a \right )}+12 i c \,d^{3} x^{2}+12 i c^{2} d^{2} x +4 i c^{3} d}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{2}}+\frac {12 d^{4} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{5}}+\frac {12 d^{4} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{5}}-\frac {24 i d^{3} c x a}{b^{3}}+\frac {6 d^{4} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b^{3}}+\frac {6 d^{4} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x^{2}}{b^{3}}-\frac {12 d^{4} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{5}}+\frac {6 d^{4} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{5}}-\frac {6 d^{4} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a^{2}}{b^{5}}+\frac {6 d^{2} c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b^{3}}-\frac {12 d^{2} c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {6 d^{2} c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{3}}-\frac {4 i d^{4} x^{3}}{b^{2}}+\frac {8 i d^{4} a^{3}}{b^{5}}+\frac {12 d^{3} c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{4}}+\frac {12 d^{3} c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{3}}+\frac {12 d^{3} c \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b^{3}}+\frac {24 d^{3} c a \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}-\frac {12 d^{3} c a \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{4}}-\frac {12 i d^{3} c \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}-\frac {12 i d^{4} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{4}}+\frac {12 i d^{4} a^{2} x}{b^{4}}-\frac {12 i d^{3} c \,x^{2}}{b^{2}}-\frac {12 i d^{3} c \,a^{2}}{b^{4}}-\frac {12 i d^{3} c \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}\) | \(716\) |
-12*I*d^4/b^4*polylog(2,exp(I*(b*x+a)))*x+2*(b*d^4*x^4*exp(2*I*(b*x+a))+4* b*c*d^3*x^3*exp(2*I*(b*x+a))+6*b*c^2*d^2*x^2*exp(2*I*(b*x+a))+4*b*c^3*d*x* exp(2*I*(b*x+a))-2*I*d^4*x^3*exp(2*I*(b*x+a))+b*c^4*exp(2*I*(b*x+a))-6*I*c *d^3*x^2*exp(2*I*(b*x+a))-6*I*c^2*d^2*x*exp(2*I*(b*x+a))+2*I*d^4*x^3-2*I*c ^3*d*exp(2*I*(b*x+a))+6*I*c*d^3*x^2+6*I*c^2*d^2*x+2*I*c^3*d)/b^2/(exp(2*I* (b*x+a))-1)^2+12*d^3/b^4*c*ln(1-exp(I*(b*x+a)))*a+12*d^3/b^3*c*ln(1-exp(I* (b*x+a)))*x+12*d^3/b^3*c*ln(exp(I*(b*x+a))+1)*x+24*d^3/b^4*c*a*ln(exp(I*(b *x+a)))-12*d^3/b^4*c*a*ln(exp(I*(b*x+a))-1)-12*I*d^3/b^4*c*polylog(2,-exp( I*(b*x+a)))+6*d^4/b^3*ln(1-exp(I*(b*x+a)))*x^2+6*d^4/b^3*ln(exp(I*(b*x+a)) +1)*x^2-12*d^4/b^5*a^2*ln(exp(I*(b*x+a)))+6*d^4/b^5*a^2*ln(exp(I*(b*x+a))- 1)-6*d^4/b^5*ln(1-exp(I*(b*x+a)))*a^2+6*d^2/b^3*c^2*ln(exp(I*(b*x+a))+1)-1 2*d^2/b^3*c^2*ln(exp(I*(b*x+a)))+6*d^2/b^3*c^2*ln(exp(I*(b*x+a))-1)-4*I*d^ 4/b^2*x^3+8*I*d^4/b^5*a^3-12*I*d^4/b^4*polylog(2,-exp(I*(b*x+a)))*x+12*I*d ^4/b^4*a^2*x-12*I*d^3/b^2*c*x^2-12*I*d^3/b^4*c*a^2-12*I*d^3/b^4*c*polylog( 2,exp(I*(b*x+a)))-24*I*d^3/b^3*c*x*a+12*d^4*polylog(3,-exp(I*(b*x+a)))/b^5 +12*d^4*polylog(3,exp(I*(b*x+a)))/b^5
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1075 vs. \(2 (124) = 248\).
Time = 0.30 (sec) , antiderivative size = 1075, normalized size of antiderivative = 7.85 \[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=\text {Too large to display} \]
1/2*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b ^4*c^4 + 4*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3*d)*c os(b*x + a)*sin(b*x + a) - 12*(-I*b*d^4*x - I*b*c*d^3 + (I*b*d^4*x + I*b*c *d^3)*cos(b*x + a)^2)*dilog(cos(b*x + a) + I*sin(b*x + a)) - 12*(I*b*d^4*x + I*b*c*d^3 + (-I*b*d^4*x - I*b*c*d^3)*cos(b*x + a)^2)*dilog(cos(b*x + a) - I*sin(b*x + a)) - 12*(I*b*d^4*x + I*b*c*d^3 + (-I*b*d^4*x - I*b*c*d^3)* cos(b*x + a)^2)*dilog(-cos(b*x + a) + I*sin(b*x + a)) - 12*(-I*b*d^4*x - I *b*c*d^3 + (I*b*d^4*x + I*b*c*d^3)*cos(b*x + a)^2)*dilog(-cos(b*x + a) - I *sin(b*x + a)) - 6*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - (b^2*d^4*x ^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(b*x + a)^2)*log(cos(b*x + a) + I*sin (b*x + a) + 1) - 6*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - (b^2*d^4*x ^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(b*x + a)^2)*log(cos(b*x + a) - I*sin (b*x + a) + 1) - 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4 - (b^2*c^2*d^2 - 2 *a*b*c*d^3 + a^2*d^4)*cos(b*x + a)^2)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b* x + a) + 1/2) - 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4 - (b^2*c^2*d^2 - 2* a*b*c*d^3 + a^2*d^4)*cos(b*x + a)^2)*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2) - 6*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + 2*a*b*c*d^3 - a^2*d^4 - (b ^2*d^4*x^2 + 2*b^2*c*d^3*x + 2*a*b*c*d^3 - a^2*d^4)*cos(b*x + a)^2)*log(-c os(b*x + a) + I*sin(b*x + a) + 1) - 6*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + 2*a*b *c*d^3 - a^2*d^4 - (b^2*d^4*x^2 + 2*b^2*c*d^3*x + 2*a*b*c*d^3 - a^2*d^4...
Timed out. \[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=\text {Timed out} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4551 vs. \(2 (124) = 248\).
Time = 0.55 (sec) , antiderivative size = 4551, normalized size of antiderivative = 33.22 \[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=\text {Too large to display} \]
1/2*(8*(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*(b*x + a)*sin(2*b*x + 2*a)^2 - (2*(b*x + a)*cos(2*b*x + 2*a) + sin(2*b*x + 2*a))*cos(4*b*x + 4*a) - 2*(b* x + a)*cos(2*b*x + 2*a) - (2*(b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) + 1)*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*c^3*d/((2*(2*cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - cos(4*b*x + 4*a)^2 - 4*cos(2*b*x + 2*a)^2 - sin(4*b *x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) - 1)*b) - 24*(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*(b*x + a)*sin(2*b*x + 2*a)^2 - (2*(b*x + a)*cos(2*b*x + 2*a) + sin(2*b*x + 2*a) )*cos(4*b*x + 4*a) - 2*(b*x + a)*cos(2*b*x + 2*a) - (2*(b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) + 1)*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*a*c^2 *d^2/((2*(2*cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - cos(4*b*x + 4*a)^2 - 4*cos(2*b*x + 2*a)^2 - sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) - 1)*b^2) + 24*(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*(b*x + a)*sin(2*b*x + 2*a)^2 - (2*(b*x + a)*cos (2*b*x + 2*a) + sin(2*b*x + 2*a))*cos(4*b*x + 4*a) - 2*(b*x + a)*cos(2*b*x + 2*a) - (2*(b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) + 1)*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*a^2*c*d^3/((2*(2*cos(2*b*x + 2*a) - 1)*cos(4*b* x + 4*a) - cos(4*b*x + 4*a)^2 - 4*cos(2*b*x + 2*a)^2 - sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) - 1)*b^3) - 8*(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*(b*x + a)*sin...
\[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^4}{{\sin \left (a+b\,x\right )}^3} \,d x \]