Integrand size = 20, antiderivative size = 208 \[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=-\frac {8 d (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^4 \csc (a+b x)}{b}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^4}-\frac {24 i d^4 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^5}+\frac {24 i d^4 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^5} \]
-8*d*(d*x+c)^3*arctanh(exp(I*(b*x+a)))/b^2-(d*x+c)^4*csc(b*x+a)/b+12*I*d^2 *(d*x+c)^2*polylog(2,-exp(I*(b*x+a)))/b^3-12*I*d^2*(d*x+c)^2*polylog(2,exp (I*(b*x+a)))/b^3-24*d^3*(d*x+c)*polylog(3,-exp(I*(b*x+a)))/b^4+24*d^3*(d*x +c)*polylog(3,exp(I*(b*x+a)))/b^4-24*I*d^4*polylog(4,-exp(I*(b*x+a)))/b^5+ 24*I*d^4*polylog(4,exp(I*(b*x+a)))/b^5
Time = 1.53 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.48 \[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\frac {-2 b (c+d x)^4 \csc (a)+8 i d \left (2 i (c+d x)^3 \text {arctanh}(\cos (a+b x)+i \sin (a+b x))+\frac {3 d \left (b^2 (c+d x)^2 \operatorname {PolyLog}(2,-\cos (a+b x)-i \sin (a+b x))+2 i b d (c+d x) \operatorname {PolyLog}(3,-\cos (a+b x)-i \sin (a+b x))-2 d^2 \operatorname {PolyLog}(4,-\cos (a+b x)-i \sin (a+b x))\right )}{b^3}-\frac {3 d \left (b^2 (c+d x)^2 \operatorname {PolyLog}(2,\cos (a+b x)+i \sin (a+b x))+2 i b d (c+d x) \operatorname {PolyLog}(3,\cos (a+b x)+i \sin (a+b x))-2 d^2 \operatorname {PolyLog}(4,\cos (a+b x)+i \sin (a+b x))\right )}{b^3}\right )+b (c+d x)^4 \csc \left (\frac {a}{2}\right ) \csc \left (\frac {1}{2} (a+b x)\right ) \sin \left (\frac {b x}{2}\right )-b (c+d x)^4 \sec \left (\frac {a}{2}\right ) \sec \left (\frac {1}{2} (a+b x)\right ) \sin \left (\frac {b x}{2}\right )}{2 b^2} \]
(-2*b*(c + d*x)^4*Csc[a] + (8*I)*d*((2*I)*(c + d*x)^3*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]] + (3*d*(b^2*(c + d*x)^2*PolyLog[2, -Cos[a + b*x] - I*Si n[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x ]] - 2*d^2*PolyLog[4, -Cos[a + b*x] - I*Sin[a + b*x]]))/b^3 - (3*d*(b^2*(c + d*x)^2*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)* PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]] - 2*d^2*PolyLog[4, Cos[a + b*x] + I*Sin[a + b*x]]))/b^3) + b*(c + d*x)^4*Csc[a/2]*Csc[(a + b*x)/2]*Sin[(b* x)/2] - b*(c + d*x)^4*Sec[a/2]*Sec[(a + b*x)/2]*Sin[(b*x)/2])/(2*b^2)
Time = 0.80 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {4910, 3042, 4671, 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.
| \(\displaystyle \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx\) |
| \(\Big \downarrow \) 4910 |
| \(\displaystyle \frac {4 d \int (c+d x)^3 \csc (a+b x)dx}{b}-\frac {(c+d x)^4 \csc (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d \int (c+d x)^3 \csc (a+b x)dx}{b}-\frac {(c+d x)^4 \csc (a+b x)}{b}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {(c+d x)^4 \csc (a+b x)}{b}+\frac {4 d \left (-\frac {3 d \int (c+d x)^2 \log \left (1-e^{i (a+b x)}\right )dx}{b}+\frac {3 d \int (c+d x)^2 \log \left (1+e^{i (a+b x)}\right )dx}{b}-\frac {2 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {(c+d x)^4 \csc (a+b x)}{b}+\frac {4 d \left (\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b}-\frac {2 i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )dx}{b}\right )}{b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b}-\frac {2 i d \int (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )dx}{b}\right )}{b}-\frac {2 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle -\frac {(c+d x)^4 \csc (a+b x)}{b}+\frac {4 d \left (\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {i d \int \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )dx}{b}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {i d \int \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )dx}{b}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}-\frac {2 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {(c+d x)^4 \csc (a+b x)}{b}+\frac {4 d \left (\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {d \int e^{-i (a+b x)} \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {d \int e^{-i (a+b x)} \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}-\frac {2 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {(c+d x)^4 \csc (a+b x)}{b}+\frac {4 d \left (-\frac {2 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {d \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {d \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\) |
-(((c + d*x)^4*Csc[a + b*x])/b) + (4*d*((-2*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b + (3*d*((I*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b - ((2*I)* d*(((-I)*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b + (d*PolyLog[4, -E^(I*( a + b*x))])/b^2))/b))/b - (3*d*((I*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))] )/b - ((2*I)*d*(((-I)*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b + (d*PolyLo g[4, E^(I*(a + b*x))])/b^2))/b))/b))/b
3.1.39.3.1 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_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[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]
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. 715 vs. \(2 (190 ) = 380\).
Time = 1.23 (sec) , antiderivative size = 716, normalized size of antiderivative = 3.44
| method | result | size |
| risch | \(-\frac {24 i d^{3} c \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{3}}+\frac {24 i d^{3} c \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{3}}+\frac {8 d^{4} a^{3} \operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{5}}-\frac {8 d \,c^{3} \operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {4 d^{4} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x^{3}}{b^{2}}+\frac {4 d^{4} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x^{3}}{b^{2}}-\frac {24 d^{3} c \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}-\frac {4 d^{4} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) a^{3}}{b^{5}}+\frac {4 d^{4} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a^{3}}{b^{5}}-\frac {24 d^{4} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{4}}+\frac {24 d^{4} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{4}}-\frac {12 d^{2} c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) a}{b^{3}}-\frac {2 i \left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}\right ) {\mathrm e}^{i \left (x b +a \right )}}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}-\frac {24 d^{3} c \,a^{2} \operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {24 d^{2} c^{2} a \,\operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {12 d^{3} c \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x^{2}}{b^{2}}+\frac {12 d^{3} c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b^{2}}+\frac {12 d^{2} c^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {12 d^{2} c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b^{2}}-\frac {12 d^{3} c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a^{2}}{b^{4}}+\frac {12 d^{2} c^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{3}}+\frac {12 d^{3} c \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) a^{2}}{b^{4}}+\frac {12 i d^{4} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b^{3}}+\frac {12 i d^{2} c^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {12 i d^{2} c^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {12 i d^{4} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b^{3}}+\frac {24 i d^{4} \operatorname {polylog}\left (4, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{5}}-\frac {24 i d^{4} \operatorname {polylog}\left (4, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{5}}+\frac {24 d^{3} c \operatorname {polylog}\left (3, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}\) | \(716\) |
24*I*d^4*polylog(4,exp(I*(b*x+a)))/b^5-24*I*d^4*polylog(4,-exp(I*(b*x+a))) /b^5-12*d^2/b^3*c^2*ln(exp(I*(b*x+a))+1)*a+24*d^3/b^4*c*polylog(3,exp(I*(b *x+a)))+8*d^4/b^5*a^3*arctanh(exp(I*(b*x+a)))-8*d/b^2*c^3*arctanh(exp(I*(b *x+a)))-4*d^4/b^2*ln(exp(I*(b*x+a))+1)*x^3+4*d^4/b^2*ln(1-exp(I*(b*x+a)))* x^3-24*d^3/b^4*c*polylog(3,-exp(I*(b*x+a)))-4*d^4/b^5*ln(exp(I*(b*x+a))+1) *a^3+4*d^4/b^5*ln(1-exp(I*(b*x+a)))*a^3-24*d^4/b^4*polylog(3,-exp(I*(b*x+a )))*x+24*d^4/b^4*polylog(3,exp(I*(b*x+a)))*x-2*I*(d^4*x^4+4*c*d^3*x^3+6*c^ 2*d^2*x^2+4*c^3*d*x+c^4)*exp(I*(b*x+a))/b/(exp(2*I*(b*x+a))-1)-24*d^3/b^4* c*a^2*arctanh(exp(I*(b*x+a)))+24*d^2/b^3*c^2*a*arctanh(exp(I*(b*x+a)))-12* d^3/b^2*c*ln(exp(I*(b*x+a))+1)*x^2+12*d^3/b^2*c*ln(1-exp(I*(b*x+a)))*x^2+1 2*d^2/b^2*c^2*ln(1-exp(I*(b*x+a)))*x-12*d^2/b^2*c^2*ln(exp(I*(b*x+a))+1)*x -12*d^3/b^4*c*ln(1-exp(I*(b*x+a)))*a^2+12*d^2/b^3*c^2*ln(1-exp(I*(b*x+a))) *a+12*d^3/b^4*c*ln(exp(I*(b*x+a))+1)*a^2+12*I*d^4/b^3*polylog(2,-exp(I*(b* x+a)))*x^2+12*I*d^2/b^3*c^2*polylog(2,-exp(I*(b*x+a)))-12*I*d^2/b^3*c^2*po lylog(2,exp(I*(b*x+a)))-12*I*d^4/b^3*polylog(2,exp(I*(b*x+a)))*x^2-24*I*d^ 3/b^3*c*polylog(2,exp(I*(b*x+a)))*x+24*I*d^3/b^3*c*polylog(2,-exp(I*(b*x+a )))*x
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1021 vs. \(2 (184) = 368\).
Time = 0.31 (sec) , antiderivative size = 1021, normalized size of antiderivative = 4.91 \[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\text {Too large to display} \]
-(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 - 12*I*d^4*polylog(4, cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) + 12 *I*d^4*polylog(4, cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) - 12*I*d^4*p olylog(4, -cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) + 12*I*d^4*polylog( 4, -cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) + 6*(I*b^2*d^4*x^2 + 2*I*b ^2*c*d^3*x + I*b^2*c^2*d^2)*dilog(cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) + 6*(-I*b^2*d^4*x^2 - 2*I*b^2*c*d^3*x - I*b^2*c^2*d^2)*dilog(cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) + 6*(I*b^2*d^4*x^2 + 2*I*b^2*c*d^3*x + I*b^2*c^2*d^2)*dilog(-cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) + 6*(-I* b^2*d^4*x^2 - 2*I*b^2*c*d^3*x - I*b^2*c^2*d^2)*dilog(-cos(b*x + a) - I*sin (b*x + a))*sin(b*x + a) + 2*(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)*log(cos(b*x + a) + I*sin(b*x + a) + 1)*sin(b*x + a) + 2*(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)*log(cos(b*x + a) - I*sin(b*x + a) + 1)*sin(b*x + a) - 2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3 *a^2*b*c*d^3 - a^3*d^4)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2)* sin(b*x + a) - 2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*l og(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2)*sin(b*x + a) - 2*(b^3*d^4 *x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3 + a^3*d^4)*log(-cos(b*x + a) + I*sin(b*x + a) + 1)*sin(b*x + a) - 2*(b^3* d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + 3*a*b^2*c^2*d^2 - 3*a^2*b...
\[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\int \left (c + d x\right )^{4} \cos {\left (a + b x \right )} \csc ^{2}{\left (a + b x \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2948 vs. \(2 (184) = 368\).
Time = 0.58 (sec) , antiderivative size = 2948, normalized size of antiderivative = 14.17 \[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\text {Too large to display} \]
-(2*(4*(b*x + a)*cos(b*x + a)*sin(2*b*x + 2*a) - 4*(b*x + a)*cos(2*b*x + 2 *a)*sin(b*x + a) + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - (c os(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b *x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 4*(b*x + a)*sin(b*x + a ))*c^3*d/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*b) - 6*(4*(b*x + a)*cos(b*x + a)*sin(2*b*x + 2*a) - 4*(b*x + a)*cos(2*b *x + 2*a)*sin(b*x + a) + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos( 2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1 ) - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log (cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 4*(b*x + a)*sin(b *x + a))*a*c^2*d^2/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*b^2) + 6*(4*(b*x + a)*cos(b*x + a)*sin(2*b*x + 2*a) - 4*(b*x + a)*cos(2*b*x + 2*a)*sin(b*x + a) + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a )^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos( b*x + a) + 1) - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2 *a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 4*(b* x + a)*sin(b*x + a))*a^2*c*d^3/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*b^3) - 2*(4*(b*x + a)*cos(b*x + a)*sin(2*b*x + 2* a) - 4*(b*x + a)*cos(2*b*x + 2*a)*sin(b*x + a) + (cos(2*b*x + 2*a)^2 + ...
\[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^4}{{\sin \left (a+b\,x\right )}^2} \,d x \]