\(\int (c+d x)^3 \cot (a+b x) \, dx\) [33]

Optimal result

Integrand size = 14, antiderivative size = 127 \[ \int (c+d x)^3 \cot (a+b x) \, dx=-\frac {i (c+d x)^4}{4 d}+\frac {(c+d x)^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i d^3 \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4} \]

[Out]

-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*polylog(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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3798, 2221, 2611, 6744, 2320, 6724} \[ \int (c+d x)^3 \cot (a+b x) \, dx=\frac {3 i d^3 \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {(c+d x)^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i (c+d x)^4}{4 d} \]

[In]

Int[(c + d*x)^3*Cot[a + b*x],x]

[Out]

((-1/4*I)*(c + d*x)^4)/d + ((c + d*x)^3*Log[1 - E^((2*I)*(a + b*x))])/b - (((3*I)/2)*d*(c + d*x)^2*PolyLog[2,
E^((2*I)*(a + b*x))])/b^2 + (3*d^2*(c + d*x)*PolyLog[3, E^((2*I)*(a + b*x))])/(2*b^3) + (((3*I)/4)*d^3*PolyLog
[4, E^((2*I)*(a + b*x))])/b^4

Rule 2221

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]
- Dist[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]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

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] + Dist[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]

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rule 6744

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {i (c+d x)^4}{4 d}-2 i \int \frac {e^{2 i (a+b x)} (c+d x)^3}{1-e^{2 i (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 d) \int (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b} \\ & = -\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 {\left (3 i d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right ) \, dx}{b^2} \\ & = -\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 {\left (3 d^3\right ) \int \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right ) \, dx}{2 b^3} \\ & = -\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 {\left (3 i d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{4 b^4} \\ & = -\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} \\ \end{align*}

Mathematica [B] (verified)

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 \[ \int (c+d x)^3 \cot (a+b x) \, dx=\frac {6 i b^3 c^2 d \pi x+4 i b^4 c d^2 x^3+i b^4 d^3 x^4-12 i b^3 c^2 d x \arctan (\tan (a))+6 b^4 c^2 d x^2 \cot (a)+6 b^2 c^2 d \pi \log \left (1+e^{-2 i b x}\right )+12 b^3 c d^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+4 b^3 d^3 x^3 \log \left (1-e^{-i (a+b x)}\right )+12 b^3 c d^2 x^2 \log \left (1+e^{-i (a+b x)}\right )+4 b^3 d^3 x^3 \log \left (1+e^{-i (a+b x)}\right )+12 b^3 c^2 d x \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+12 b^2 c^2 d \arctan (\tan (a)) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )-6 b^2 c^2 d \pi \log (\cos (b x))+4 b^3 c^3 \log (\sin (a+b x))-12 b^2 c^2 d \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))+12 i b^2 d^2 x (2 c+d x) \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )+12 i b^2 d^2 x (2 c+d x) \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )-6 i b^2 c^2 d \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )+24 b c d^2 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+24 b d^3 x \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+24 b c d^2 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )+24 b d^3 x \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )-24 i d^3 \operatorname {PolyLog}\left (4,-e^{-i (a+b x)}\right )-24 i d^3 \operatorname {PolyLog}\left (4,e^{-i (a+b x)}\right )-6 b^4 c^2 d e^{i \arctan (\tan (a))} x^2 \cot (a) \sqrt {\sec ^2(a)}}{4 b^4} \]

[In]

Integrate[(c + d*x)^3*Cot[a + b*x],x]

[Out]

((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 + Arc
Tan[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]*Sq
rt[Sec[a]^2])/(4*b^4)

Maple [B] (verified)

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}}\)	\(792\)

[In]

int((d*x+c)^3*cos(b*x+a)*csc(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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*ln(exp(I*(b*x+a)))+6/b^3*d^3*polylog(3,-exp(I*(b*x+a)))*x+1/b^4*d^3*l
n(1-exp(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*polylog(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

Fricas [B] (verification not implemented)

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}} \]

[In]

integrate((d*x+c)^3*cos(b*x+a)*csc(b*x+a),x, algorithm="fricas")

[Out]

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*sin(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(-co
s(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*si
n(b*x + a)) + 6*(b*d^3*x + b*c*d^2)*polylog(3, -cos(b*x + a) - I*sin(b*x + a)))/b^4

Sympy [F]

\[ \int (c+d x)^3 \cot (a+b x) \, dx=\int \left (c + d x\right )^{3} \cos {\left (a + b x \right )} \csc {\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**3*cos(b*x+a)*csc(b*x+a),x)

[Out]

Integral((c + d*x)**3*cos(a + b*x)*csc(a + b*x), x)

Maxima [B] (verification not implemented)

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} \]

[In]

integrate((d*x+c)^3*cos(b*x+a)*csc(b*x+a),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int (c+d x)^3 \cot (a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \cos \left (b x + a\right ) \csc \left (b x + a\right ) \,d x } \]

[In]

integrate((d*x+c)^3*cos(b*x+a)*csc(b*x+a),x, algorithm="giac")

[Out]

integrate((d*x + c)^3*cos(b*x + a)*csc(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 \cot (a+b x) \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^3}{\sin \left (a+b\,x\right )} \,d x \]

[In]

int((cos(a + b*x)*(c + d*x)^3)/sin(a + b*x),x)

[Out]

int((cos(a + b*x)*(c + d*x)^3)/sin(a + b*x), x)