Integrand size = 20, antiderivative size = 146 \[ \int (c+d x)^3 \cot (a+b x) \csc (a+b x) \, dx=-\frac {6 d (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^3 \csc (a+b x)}{b}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {6 d^3 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^4}+\frac {6 d^3 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^4} \]
-6*d*(d*x+c)^2*arctanh(exp(I*(b*x+a)))/b^2-(d*x+c)^3*csc(b*x+a)/b+6*I*d^2* (d*x+c)*polylog(2,-exp(I*(b*x+a)))/b^3-6*I*d^2*(d*x+c)*polylog(2,exp(I*(b* x+a)))/b^3-6*d^3*polylog(3,-exp(I*(b*x+a)))/b^4+6*d^3*polylog(3,exp(I*(b*x +a)))/b^4
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(311\) vs. \(2(146)=292\).
Time = 1.34 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.13 \[ \int (c+d x)^3 \cot (a+b x) \csc (a+b x) \, dx=-\frac {b^3 c^3 \csc (a+b x)+3 b^3 c^2 d x \csc (a+b x)+3 b^3 c d^2 x^2 \csc (a+b x)+b^3 d^3 x^3 \csc (a+b x)-3 b^2 c^2 d \log \left (1-e^{i (a+b x)}\right )-6 b^2 c d^2 x \log \left (1-e^{i (a+b x)}\right )-3 b^2 d^3 x^2 \log \left (1-e^{i (a+b x)}\right )+3 b^2 c^2 d \log \left (1+e^{i (a+b x)}\right )+6 b^2 c d^2 x \log \left (1+e^{i (a+b x)}\right )+3 b^2 d^3 x^2 \log \left (1+e^{i (a+b x)}\right )-6 i b d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )+6 i b d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )+6 d^3 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )-6 d^3 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^4} \]
-((b^3*c^3*Csc[a + b*x] + 3*b^3*c^2*d*x*Csc[a + b*x] + 3*b^3*c*d^2*x^2*Csc [a + b*x] + b^3*d^3*x^3*Csc[a + b*x] - 3*b^2*c^2*d*Log[1 - E^(I*(a + b*x)) ] - 6*b^2*c*d^2*x*Log[1 - E^(I*(a + b*x))] - 3*b^2*d^3*x^2*Log[1 - E^(I*(a + b*x))] + 3*b^2*c^2*d*Log[1 + E^(I*(a + b*x))] + 6*b^2*c*d^2*x*Log[1 + E ^(I*(a + b*x))] + 3*b^2*d^3*x^2*Log[1 + E^(I*(a + b*x))] - (6*I)*b*d^2*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] + (6*I)*b*d^2*(c + d*x)*PolyLog[2, E^( I*(a + b*x))] + 6*d^3*PolyLog[3, -E^(I*(a + b*x))] - 6*d^3*PolyLog[3, E^(I *(a + b*x))])/b^4)
Time = 0.56 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4910, 3042, 4671, 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.
| \(\displaystyle \int (c+d x)^3 \cot (a+b x) \csc (a+b x) \, dx\) |
| \(\Big \downarrow \) 4910 |
| \(\displaystyle \frac {3 d \int (c+d x)^2 \csc (a+b x)dx}{b}-\frac {(c+d x)^3 \csc (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 d \int (c+d x)^2 \csc (a+b x)dx}{b}-\frac {(c+d x)^3 \csc (a+b x)}{b}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {(c+d x)^3 \csc (a+b x)}{b}+\frac {3 d \left (-\frac {2 d \int (c+d x) \log \left (1-e^{i (a+b x)}\right )dx}{b}+\frac {2 d \int (c+d x) \log \left (1+e^{i (a+b x)}\right )dx}{b}-\frac {2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {(c+d x)^3 \csc (a+b x)}{b}+\frac {3 d \left (\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b}-\frac {i d \int \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )dx}{b}\right )}{b}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b}-\frac {i d \int \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )dx}{b}\right )}{b}-\frac {2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {(c+d x)^3 \csc (a+b x)}{b}+\frac {3 d \left (\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b}-\frac {d \int e^{-i (a+b x)} \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}\right )}{b}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b}-\frac {d \int e^{-i (a+b x)} \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}\right )}{b}-\frac {2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {(c+d x)^3 \csc (a+b x)}{b}+\frac {3 d \left (-\frac {2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b}-\frac {d \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^2}\right )}{b}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b}-\frac {d \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^2}\right )}{b}\right )}{b}\) |
-(((c + d*x)^3*Csc[a + b*x])/b) + (3*d*((-2*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b + (2*d*((I*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b - (d*PolyLo g[3, -E^(I*(a + b*x))])/b^2))/b - (2*d*((I*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b - (d*PolyLog[3, E^(I*(a + b*x))])/b^2))/b))/b
3.1.40.3.1 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (134 ) = 268\).
Time = 1.13 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.97
| method | result | size |
| risch | \(-\frac {2 i \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right ) {\mathrm e}^{i \left (x b +a \right )}}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}+\frac {6 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{3}}+\frac {6 i d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {12 d^{2} c a \,\operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{3}}-\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) a}{b^{3}}-\frac {6 i d^{2} c \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {6 i d^{3} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{3}}-\frac {6 d^{3} a^{2} \operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b^{2}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b^{2}}+\frac {6 d^{3} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}-\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x^{2}}{b^{2}}-\frac {6 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}-\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a^{2}}{b^{4}}+\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) a^{2}}{b^{4}}-\frac {6 d \,c^{2} \operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}\) | \(433\) |
-2*I*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)*exp(I*(b*x+a))/b/(exp(2*I*(b*x+a) )-1)+6*I*d^3/b^3*polylog(2,-exp(I*(b*x+a)))*x+6*I*d^2/b^3*c*polylog(2,-exp (I*(b*x+a)))+12*d^2/b^3*c*a*arctanh(exp(I*(b*x+a)))+6*d^2/b^3*c*ln(1-exp(I *(b*x+a)))*a-6*d^2/b^3*c*ln(exp(I*(b*x+a))+1)*a-6*I*d^2/b^3*c*polylog(2,ex p(I*(b*x+a)))-6*I*d^3/b^3*polylog(2,exp(I*(b*x+a)))*x-6*d^3/b^4*a^2*arctan h(exp(I*(b*x+a)))+6*d^2/b^2*c*ln(1-exp(I*(b*x+a)))*x-6*d^2/b^2*c*ln(exp(I* (b*x+a))+1)*x+3*d^3/b^2*ln(1-exp(I*(b*x+a)))*x^2+6*d^3*polylog(3,exp(I*(b* x+a)))/b^4-3*d^3/b^2*ln(exp(I*(b*x+a))+1)*x^2-6*d^3*polylog(3,-exp(I*(b*x+ a)))/b^4-3*d^3/b^4*ln(1-exp(I*(b*x+a)))*a^2+3*d^3/b^4*ln(exp(I*(b*x+a))+1) *a^2-6*d/b^2*c^2*arctanh(exp(I*(b*x+a)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 669 vs. \(2 (130) = 260\).
Time = 0.29 (sec) , antiderivative size = 669, normalized size of antiderivative = 4.58 \[ \int (c+d x)^3 \cot (a+b x) \csc (a+b x) \, dx=-\frac {2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{2} d x + 2 \, b^{3} c^{3} - 6 \, d^{3} {\rm polylog}\left (3, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - 6 \, d^{3} {\rm polylog}\left (3, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 6 \, d^{3} {\rm polylog}\left (3, -\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 6 \, d^{3} {\rm polylog}\left (3, -\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 6 \, {\left (i \, b d^{3} x + i \, b c d^{2}\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 6 \, {\left (-i \, b d^{3} x - i \, b c d^{2}\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 6 \, {\left (i \, b d^{3} x + i \, b c d^{2}\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 6 \, {\left (-i \, b d^{3} x - i \, b c d^{2}\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} 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 ) \sin \left (b x + a\right ) - 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} 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 ) \sin \left (b x + a\right ) - 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + 2 \, a b c d^{2} - a^{2} d^{3}\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + 2 \, a b c d^{2} - a^{2} d^{3}\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right )}{2 \, b^{4} \sin \left (b x + a\right )} \]
-1/2*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 6*b^3*c^2*d*x + 2*b^3*c^3 - 6*d^3* polylog(3, cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) - 6*d^3*polylog(3, cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) + 6*d^3*polylog(3, -cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) + 6*d^3*polylog(3, -cos(b*x + a) - I*sin (b*x + a))*sin(b*x + a) + 6*(I*b*d^3*x + I*b*c*d^2)*dilog(cos(b*x + a) + I *sin(b*x + a))*sin(b*x + a) + 6*(-I*b*d^3*x - I*b*c*d^2)*dilog(cos(b*x + a ) - I*sin(b*x + a))*sin(b*x + a) + 6*(I*b*d^3*x + I*b*c*d^2)*dilog(-cos(b* x + a) + I*sin(b*x + a))*sin(b*x + a) + 6*(-I*b*d^3*x - I*b*c*d^2)*dilog(- cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2 *x + b^2*c^2*d)*log(cos(b*x + a) + I*sin(b*x + a) + 1)*sin(b*x + a) + 3*(b ^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*log(cos(b*x + a) - I*sin(b*x + a) + 1)*sin(b*x + a) - 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2)*sin(b*x + a) - 3*(b^2*c^2*d - 2*a*b*c*d^ 2 + a^2*d^3)*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2)*sin(b*x + a ) - 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*log(-cos(b*x + a) + I*sin(b*x + a) + 1)*sin(b*x + a) - 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*log(-cos(b*x + a) - I*sin(b*x + a) + 1)*sin(b*x + a ))/(b^4*sin(b*x + a))
\[ \int (c+d x)^3 \cot (a+b x) \csc (a+b x) \, dx=\int \left (c + d x\right )^{3} \cos {\left (a + b x \right )} \csc ^{2}{\left (a + b x \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1770 vs. \(2 (130) = 260\).
Time = 0.38 (sec) , antiderivative size = 1770, normalized size of antiderivative = 12.12 \[ \int (c+d x)^3 \cot (a+b x) \csc (a+b x) \, dx=\text {Too large to display} \]
-1/2*(3*(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(c os(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 4*(b*x + a)*sin(b*x + a))*c^2*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)*s in(b*x + a))*a*c*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) + 3*(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*co s(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*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*c^3/sin(b*x + a) - 6*a*c^2*d/(b*sin(b*x + a)) + 6*a^2*c*d^2/(b^2*sin(b*x + a)) - 2*a^3*d^3/(b^3*sin(b*x + a)) -...
\[ \int (c+d x)^3 \cot (a+b x) \csc (a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int (c+d x)^3 \cot (a+b x) \csc (a+b x) \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^3}{{\sin \left (a+b\,x\right )}^2} \,d x \]