\(\int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx\) [46]

Optimal result

Integrand size = 22, antiderivative size = 137 \[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=-\frac {2 i d (c+d x)^3}{b^2}-\frac {2 d (c+d x)^3 \cot (a+b x)}{b^2}-\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}+\frac {6 d^2 (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^4 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^5} \] Output:

-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
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(512\) vs. \(2(137)=274\).

Time = 6.40 (sec) , antiderivative size = 512, normalized size of antiderivative = 3.74 \[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=-\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}-\frac {d^4 e^{i a} \csc (a) \left (2 b^3 e^{-2 i a} x^3+3 i b^2 \left (1-e^{-2 i a}\right ) x^2 \log \left (1-e^{-i (a+b x)}\right )+3 i b^2 \left (1-e^{-2 i a}\right ) x^2 \log \left (1+e^{-i (a+b x)}\right )-6 b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )-6 b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+6 i \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+6 i \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )\right )}{b^5}+\frac {6 c^2 d^2 \csc (a) (-b x \cos (a)+\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a))}{b^3 \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {2 \csc (a) \csc (a+b x) \left (c^3 d \sin (b x)+3 c^2 d^2 x \sin (b x)+3 c d^3 x^2 \sin (b x)+d^4 x^3 \sin (b x)\right )}{b^2}-\frac {6 c d^3 \csc (a) \sec (a) \left (b^2 e^{i \arctan (\tan (a))} x^2+\frac {\left (i b x (-\pi +2 \arctan (\tan (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x+\arctan (\tan (a))) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+\pi \log (\cos (b x))+2 \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )\right ) \tan (a)}{\sqrt {1+\tan ^2(a)}}\right )}{b^4 \sqrt {\sec ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}} \] Input:

Integrate[(c + d*x)^4*Cot[a + b*x]*Csc[a + b*x]^2,x]
 

Output:

-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)])
 

Rubi [A] (verified)

Time = 0.83 (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

\(\displaystyle -\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}+\frac {2 d \left (-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {d \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}\right )}{b}-\frac {i (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}\right )}{b}\)

Input:

Int[(c + d*x)^4*Cot[a + b*x]*Csc[a + b*x]^2,x]
 

Output:

-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
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2620

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]
 

rule 2720

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]]]
 

rule 3011

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4202

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]
 

rule 4672

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]
 

rule 4910

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]
 

rule 7143

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]
 

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. \(2 (127 ) = 254\).

Time = 0.63 (sec) , antiderivative size = 716, normalized size of antiderivative = 5.23

method	result	size
risch	\(\frac {24 d^{3} c a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {12 d^{3} c a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}-\frac {12 i d^{4} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{4}}+\frac {12 i d^{4} a^{2} x}{b^{4}}-\frac {12 i d^{3} c \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {12 i d^{3} c \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {12 d^{3} c \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {12 d^{3} c \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{3}}+\frac {12 d^{3} c \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{4}}+\frac {2 b \,d^{4} x^{4} {\mathrm e}^{2 i \left (b x +a \right )}+8 b c \,d^{3} x^{3} {\mathrm e}^{2 i \left (b x +a \right )}+12 b \,c^{2} d^{2} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}+8 b \,c^{3} d x \,{\mathrm e}^{2 i \left (b x +a \right )}-4 i d^{4} x^{3} {\mathrm e}^{2 i \left (b x +a \right )}+2 b \,c^{4} {\mathrm e}^{2 i \left (b x +a \right )}-12 i c \,d^{3} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}-12 i c^{2} d^{2} x \,{\mathrm e}^{2 i \left (b x +a \right )}+4 i d^{4} x^{3}-4 i c^{3} d \,{\mathrm e}^{2 i \left (b x +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 (b x +a \right )}-1\right )^{2}}+\frac {12 d^{4} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{5}}+\frac {12 d^{4} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{5}}-\frac {24 i d^{3} c a x}{b^{3}}+\frac {6 d^{4} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b^{3}}+\frac {6 d^{4} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{3}}-\frac {12 d^{2} c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {6 d^{2} c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}+\frac {6 d^{2} c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{3}}-\frac {6 d^{4} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{2}}{b^{5}}-\frac {12 d^{4} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{5}}+\frac {6 d^{4} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{5}}-\frac {4 i d^{4} x^{3}}{b^{2}}+\frac {8 i d^{4} a^{3}}{b^{5}}-\frac {12 i d^{3} c \,x^{2}}{b^{2}}-\frac {12 i d^{3} c \,a^{2}}{b^{4}}-\frac {12 i d^{4} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{4}}\)	\(716\)

Input:

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

Output:

12*d^3/b^3*c*ln(1-exp(I*(b*x+a)))*x-12*I*d^4/b^4*polylog(2,exp(I*(b*x+a))) 
*x-12*I*d^3/b^2*c*x^2-12*I*d^3/b^4*c*a^2-12*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^4*c*polylog(2,-exp(I*(b*x+a)))-12*I 
*d^3/b^4*c*polylog(2,exp(I*(b*x+a)))+12*d^3/b^3*c*ln(exp(I*(b*x+a))+1)*x+1 
2*d^3/b^4*c*ln(1-exp(I*(b*x+a)))*a+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)+6*d^4/b^3*ln(exp(I*(b*x+a))+1)*x^2+6*d^4/b^ 
3*ln(1-exp(I*(b*x+a)))*x^2-12*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)+6*d^2/b^3*c^2*ln(exp(I*(b*x+a))+1)-6*d^4/b^5*ln(1-ex 
p(I*(b*x+a)))*a^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)-4*I*d^4/b^2*x^3+8*I*d^4/b^5*a^3-24*I*d^3/b^3*c*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^4*polylog(3,-exp(I*(b*x+a)))/b^5+ 
12*d^4*polylog(3,exp(I*(b*x+a)))/b^5
 

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. 1075 vs. \(2 (124) = 248\).

Time = 0.12 (sec) , antiderivative size = 1075, normalized size of antiderivative = 7.85 \[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

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...
 

Sympy [F]

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

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

Output:

Integral((c + d*x)**4*cot(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. 4549 vs. \(2 (124) = 248\).

Time = 0.29 (sec) , antiderivative size = 4549, normalized size of antiderivative = 33.20 \[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(c^4*cot(b*x + a)^2 - 4*a*c^3*d*cot(b*x + a)^2/b + 6*a^2*c^2*d^2*cot( 
b*x + a)^2/b^2 - 4*a^3*c*d^3*cot(b*x + a)^2/b^3 + a^4*d^4*cot(b*x + a)^2/b 
^4 - 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)...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx=\frac {-48 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1\right ) \sin \left (b x +a \right )^{2} d^{4}+96 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{2} d^{4}-48 \cos \left (b x +a \right ) b^{2} c \,d^{3} x -24 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1\right ) \sin \left (b x +a \right )^{2} b^{2} c^{2} d^{2}+24 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{2} b^{2} c^{2} d^{2}-8 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{3} c^{3} d -8 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{3} d^{4} x^{3}-48 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b c \,d^{3}-48 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b \,d^{4} x -8 b^{4} c^{3} d x -12 b^{4} c^{2} d^{2} x^{2}-8 b^{4} c \,d^{3} x^{3}-48 b^{2} c \,d^{3} x -2 b^{4} c^{4}-12 \left (\int \frac {x^{2}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}d x \right ) \sin \left (b x +a \right )^{2} b^{3} d^{4}-12 \left (\int \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) x^{2}d x \right ) \sin \left (b x +a \right )^{2} b^{3} d^{4}-24 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{3} c^{2} d^{2} x -24 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{3} c \,d^{3} x^{2}+\sin \left (b x +a \right )^{2} b^{4} c^{4}-24 \cos \left (b x +a \right ) b^{2} d^{4} x^{2}-48 \sin \left (b x +a \right ) b c \,d^{3}-48 \sin \left (b x +a \right ) b \,d^{4} x -2 b^{4} d^{4} x^{4}-24 b^{2} d^{4} x^{2}-24 \left (\int \frac {x}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}d x \right ) \sin \left (b x +a \right )^{2} b^{3} c \,d^{3}-24 \left (\int \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) x d x \right ) \sin \left (b x +a \right )^{2} b^{3} c \,d^{3}}{4 \sin \left (b x +a \right )^{2} b^{5}} \] Input:

int((d*x+c)^4*cot(b*x+a)*csc(b*x+a)^2,x)
 

Output:

( - 8*cos(a + b*x)*sin(a + b*x)*b**3*c**3*d - 24*cos(a + b*x)*sin(a + b*x) 
*b**3*c**2*d**2*x - 24*cos(a + b*x)*sin(a + b*x)*b**3*c*d**3*x**2 - 8*cos( 
a + b*x)*sin(a + b*x)*b**3*d**4*x**3 - 48*cos(a + b*x)*sin(a + b*x)*b*c*d* 
*3 - 48*cos(a + b*x)*sin(a + b*x)*b*d**4*x - 48*cos(a + b*x)*b**2*c*d**3*x 
 - 24*cos(a + b*x)*b**2*d**4*x**2 - 12*int(x**2/tan((a + b*x)/2)**3,x)*sin 
(a + b*x)**2*b**3*d**4 - 24*int(x/tan((a + b*x)/2)**3,x)*sin(a + b*x)**2*b 
**3*c*d**3 - 12*int(tan((a + b*x)/2)*x**2,x)*sin(a + b*x)**2*b**3*d**4 - 2 
4*int(tan((a + b*x)/2)*x,x)*sin(a + b*x)**2*b**3*c*d**3 - 24*log(tan((a + 
b*x)/2)**2 + 1)*sin(a + b*x)**2*b**2*c**2*d**2 - 48*log(tan((a + b*x)/2)** 
2 + 1)*sin(a + b*x)**2*d**4 + 24*log(tan((a + b*x)/2))*sin(a + b*x)**2*b** 
2*c**2*d**2 + 96*log(tan((a + b*x)/2))*sin(a + b*x)**2*d**4 + sin(a + b*x) 
**2*b**4*c**4 - 48*sin(a + b*x)*b*c*d**3 - 48*sin(a + b*x)*b*d**4*x - 2*b* 
*4*c**4 - 8*b**4*c**3*d*x - 12*b**4*c**2*d**2*x**2 - 8*b**4*c*d**3*x**3 - 
2*b**4*d**4*x**4 - 48*b**2*c*d**3*x - 24*b**2*d**4*x**2)/(4*sin(a + b*x)** 
2*b**5)