\(\int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx\) [39]

Optimal result

Integrand size = 20, antiderivative size = 208 \[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=-\frac {8 d (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^4 \csc (a+b x)}{b}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^4}-\frac {24 i d^4 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^5}+\frac {24 i d^4 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^5} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4495, 4268, 2611, 6744, 2320, 6724} \[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=-\frac {8 d (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {24 i d^4 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^5}+\frac {24 i d^4 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^5}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^4}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {(c+d x)^4 \csc (a+b x)}{b} \]

[In]

Int[(c + d*x)^4*Cot[a + b*x]*Csc[a + b*x],x]

[Out]

(-8*d*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b^2 - ((c + d*x)^4*Csc[a + b*x])/b + ((12*I)*d^2*(c + d*x)^2*PolyL
og[2, -E^(I*(a + b*x))])/b^3 - ((12*I)*d^2*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^3 - (24*d^3*(c + d*x)*Po
lyLog[3, -E^(I*(a + b*x))])/b^4 + (24*d^3*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^4 - ((24*I)*d^4*PolyLog[4,
-E^(I*(a + b*x))])/b^5 + ((24*I)*d^4*PolyLog[4, E^(I*(a + b*x))])/b^5

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 4268

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] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4495

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] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /;
FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[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 {(c+d x)^4 \csc (a+b x)}{b}+\frac {(4 d) \int (c+d x)^3 \csc (a+b x) \, dx}{b} \\ & = -\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 {\left (12 d^2\right ) \int (c+d x)^2 \log \left (1-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (12 d^2\right ) \int (c+d x)^2 \log \left (1+e^{i (a+b x)}\right ) \, dx}{b^2} \\ & = -\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 {\left (24 i d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right ) \, dx}{b^3}+\frac {\left (24 i d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right ) \, dx}{b^3} \\ & = -\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 {\left (24 d^4\right ) \int \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right ) \, dx}{b^4}-\frac {\left (24 d^4\right ) \int \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right ) \, dx}{b^4} \\ & = -\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 {\left (24 i d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}+\frac {\left (24 i d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.53 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.48 \[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\frac {-2 b (c+d x)^4 \csc (a)+8 i d \left (2 i (c+d x)^3 \text {arctanh}(\cos (a+b x)+i \sin (a+b x))+\frac {3 d \left (b^2 (c+d x)^2 \operatorname {PolyLog}(2,-\cos (a+b x)-i \sin (a+b x))+2 i b d (c+d x) \operatorname {PolyLog}(3,-\cos (a+b x)-i \sin (a+b x))-2 d^2 \operatorname {PolyLog}(4,-\cos (a+b x)-i \sin (a+b x))\right )}{b^3}-\frac {3 d \left (b^2 (c+d x)^2 \operatorname {PolyLog}(2,\cos (a+b x)+i \sin (a+b x))+2 i b d (c+d x) \operatorname {PolyLog}(3,\cos (a+b x)+i \sin (a+b x))-2 d^2 \operatorname {PolyLog}(4,\cos (a+b x)+i \sin (a+b x))\right )}{b^3}\right )+b (c+d x)^4 \csc \left (\frac {a}{2}\right ) \csc \left (\frac {1}{2} (a+b x)\right ) \sin \left (\frac {b x}{2}\right )-b (c+d x)^4 \sec \left (\frac {a}{2}\right ) \sec \left (\frac {1}{2} (a+b x)\right ) \sin \left (\frac {b x}{2}\right )}{2 b^2} \]

[In]

Integrate[(c + d*x)^4*Cot[a + b*x]*Csc[a + b*x],x]

[Out]

(-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*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 - (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]*S
ec[(a + b*x)/2]*Sin[(b*x)/2])/(2*b^2)

Maple [B] (verified)

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}}\)	\(716\)

[In]

int((d*x+c)^4*cos(b*x+a)*csc(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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+12*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*polylog(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

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. 1021 vs. \(2 (184) = 368\).

Time = 0.31 (sec) , antiderivative size = 1021, normalized size of antiderivative = 4.91 \[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^4*cos(b*x+a)*csc(b*x+a)^2,x, algorithm="fricas")

[Out]

-(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
*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) + 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)*log(-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*c*d^3 + a^3*d^4)*log(-cos(b*x + a) - I*sin(b*x + a) + 1)*sin(b
*x + a) - 12*(b*d^4*x + b*c*d^3)*polylog(3, cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) - 12*(b*d^4*x + b*c*d^
3)*polylog(3, cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) + 12*(b*d^4*x + b*c*d^3)*polylog(3, -cos(b*x + a) +
I*sin(b*x + a))*sin(b*x + a) + 12*(b*d^4*x + b*c*d^3)*polylog(3, -cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a))
/(b^5*sin(b*x + a))

Sympy [F]

\[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\int \left (c + d x\right )^{4} \cos {\left (a + b x \right )} \csc ^{2}{\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**4*cos(b*x+a)*csc(b*x+a)**2,x)

[Out]

Integral((c + d*x)**4*cos(a + b*x)*csc(a + b*x)**2, 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. 2948 vs. \(2 (184) = 368\).

Time = 0.58 (sec) , antiderivative size = 2948, normalized size of antiderivative = 14.17 \[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^4*cos(b*x+a)*csc(b*x+a)^2,x, algorithm="maxima")

[Out]

-(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) -
(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))*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*c
os(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 + 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)*l
og(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^3*d^4/((cos(2*b*x + 2*a
)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*b^4) + c^4/sin(b*x + a) - 4*a*c^3*d/(b*sin(b*x + a)) + 6*a^
2*c^2*d^2/(b^2*sin(b*x + a)) - 4*a^3*c*d^3/(b^3*sin(b*x + a)) + a^4*d^4/(b^4*sin(b*x + a)) - 2*(2*((b*x + a)^3
*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a) - ((b*x + a)^3*d^4
+ 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (-I*
(b*x + a)^3*d^4 + 3*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^2 + 3*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x +
a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) + 2*((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x
+ a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a) - ((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)
^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (-I*(b*x + a)^3*d^4 + 3*(-I*b*c*d^3
+ I*a*d^4)*(b*x + a)^2 + 3*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(
sin(b*x + a), -cos(b*x + a) + 1) - ((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b
*c*d^3 + a^2*d^4)*(b*x + a)^2)*cos(b*x + a) - 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*
c*d^3 - a*d^4)*(b*x + a) - (b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x +
a))*cos(2*b*x + 2*a) - (I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*(b*x + a)^2*d^4 + I*a^2*d^4 + 2*(I*b*c*d^3 - I*a*d^
4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(-e^(I*b*x + I*a)) + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2
*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a) - (b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a
*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4 - I*a^2*d^4 + 2*(-I*b*
c*d^3 + I*a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(e^(I*b*x + I*a)) + (-I*(b*x + a)^3*d^4 + 3*(-I*b*c*d^3 + I
*a*d^4)*(b*x + a)^2 + 3*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a) + (I*(b*x + a)^3*d^4 + 3*(I*b*c
*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) - ((b*
x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*sin(2*b*x
+ 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + (I*(b*x + a)^3*d^4 + 3*(I*b*c*d^3 - I*a*d^
4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a) + (-I*(b*x + a)^3*d^4 + 3*(-I*b*c*d^3
+ I*a*d^4)*(b*x + a)^2 + 3*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) + ((b*x +
a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*sin(2*b*x + 2
*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - 12*(d^4*cos(2*b*x + 2*a) + I*d^4*sin(2*b*x +
2*a) - d^4)*polylog(4, -e^(I*b*x + I*a)) + 12*(d^4*cos(2*b*x + 2*a) + I*d^4*sin(2*b*x + 2*a) - d^4)*polylog(4,
e^(I*b*x + I*a)) + 12*(-I*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4 + (I*b*c*d^3 + I*(b*x + a)*d^4 - I*a*d^4)*cos(2
*b*x + 2*a) - (b*c*d^3 + (b*x + a)*d^4 - a*d^4)*sin(2*b*x + 2*a))*polylog(3, -e^(I*b*x + I*a)) + 12*(I*b*c*d^3
+ I*(b*x + a)*d^4 - I*a*d^4 + (-I*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4)*cos(2*b*x + 2*a) + (b*c*d^3 + (b*x + a
)*d^4 - a*d^4)*sin(2*b*x + 2*a))*polylog(3, e^(I*b*x + I*a)) + (-I*(b*x + a)^4*d^4 + 4*(-I*b*c*d^3 + I*a*d^4)*
(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2)*sin(b*x + a))/(-I*b^4*cos(2*b*x + 2*
a) + b^4*sin(2*b*x + 2*a) + I*b^4))/b

Giac [F]

\[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{2} \,d x } \]

[In]

integrate((d*x+c)^4*cos(b*x+a)*csc(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^4*cos(b*x + a)*csc(b*x + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^4}{{\sin \left (a+b\,x\right )}^2} \,d x \]

[In]

int((cos(a + b*x)*(c + d*x)^4)/sin(a + b*x)^2,x)

[Out]

int((cos(a + b*x)*(c + d*x)^4)/sin(a + b*x)^2, x)