\(\int (c+d x)^2 \cot (a+b x) \, dx\) [34]

Optimal result

Integrand size = 14, antiderivative size = 93 \[ \int (c+d x)^2 \cot (a+b x) \, dx=-\frac {i (c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {d^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3} \]

[Out]

-1/3*I*(d*x+c)^3/d+(d*x+c)^2*ln(1-exp(2*I*(b*x+a)))/b-I*d*(d*x+c)*polylog(2,exp(2*I*(b*x+a)))/b^2+1/2*d^2*poly
log(3,exp(2*I*(b*x+a)))/b^3

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3798, 2221, 2611, 2320, 6724} \[ \int (c+d x)^2 \cot (a+b x) \, dx=\frac {d^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {(c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i (c+d x)^3}{3 d} \]

[In]

Int[(c + d*x)^2*Cot[a + b*x],x]

[Out]

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

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]

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(356\) vs. \(2(93)=186\).

Time = 1.61 (sec) , antiderivative size = 356, normalized size of antiderivative = 3.83 \[ \int (c+d x)^2 \cot (a+b x) \, dx=\frac {3 i b^2 c d \pi x+i b^3 d^2 x^3-6 i b^2 c d x \arctan (\tan (a))+3 b^3 c d x^2 \cot (a)+3 b c d \pi \log \left (1+e^{-2 i b x}\right )+3 b^2 d^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+3 b^2 d^2 x^2 \log \left (1+e^{-i (a+b x)}\right )+6 b^2 c d x \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+6 b c d \arctan (\tan (a)) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )-3 b c d \pi \log (\cos (b x))+3 b^2 c^2 \log (\sin (a+b x))-6 b c d \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))+6 i b d^2 x \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )+6 i b d^2 x \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )-3 i b c d \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )+6 d^2 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+6 d^2 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )-3 b^3 c d e^{i \arctan (\tan (a))} x^2 \cot (a) \sqrt {\sec ^2(a)}}{3 b^3} \]

[In]

Integrate[(c + d*x)^2*Cot[a + b*x],x]

[Out]

((3*I)*b^2*c*d*Pi*x + I*b^3*d^2*x^3 - (6*I)*b^2*c*d*x*ArcTan[Tan[a]] + 3*b^3*c*d*x^2*Cot[a] + 3*b*c*d*Pi*Log[1
+ E^((-2*I)*b*x)] + 3*b^2*d^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 3*b^2*d^2*x^2*Log[1 + E^((-I)*(a + b*x))] + 6
*b^2*c*d*x*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + 6*b*c*d*ArcTan[Tan[a]]*Log[1 - E^((2*I)*(b*x + ArcTan[T
an[a]]))] - 3*b*c*d*Pi*Log[Cos[b*x]] + 3*b^2*c^2*Log[Sin[a + b*x]] - 6*b*c*d*ArcTan[Tan[a]]*Log[Sin[b*x + ArcT
an[Tan[a]]]] + (6*I)*b*d^2*x*PolyLog[2, -E^((-I)*(a + b*x))] + (6*I)*b*d^2*x*PolyLog[2, E^((-I)*(a + b*x))] -
(3*I)*b*c*d*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))] + 6*d^2*PolyLog[3, -E^((-I)*(a + b*x))] + 6*d^2*PolyL
og[3, E^((-I)*(a + b*x))] - 3*b^3*c*d*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2])/(3*b^3)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (81 ) = 162\).

Time = 1.13 (sec) , antiderivative size = 477, normalized size of antiderivative = 5.13

method	result	size
risch	\(\frac {c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b}-\frac {2 c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b}+i c^{2} x +\frac {4 i d^{2} a^{3}}{3 b^{3}}-i d c \,x^{2}-\frac {d^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a^{2}}{b^{3}}+\frac {d^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x^{2}}{b}-\frac {2 i d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {2 i d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {2 d c \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b}+\frac {2 d c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}+\frac {4 c d a \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {2 c d a \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{2}}+\frac {2 d c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}-\frac {2 i d c \,a^{2}}{b^{2}}-\frac {2 i d c \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {2 i d c \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {2 i d^{2} a^{2} x}{b^{2}}+\frac {i c^{3}}{3 d}-\frac {i d^{2} x^{3}}{3}+\frac {2 d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {2 d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {2 d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{3}}-\frac {4 i d c x a}{b}\)	\(477\)

[In]

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

[Out]

-2/b^3*d^2*a^2*ln(exp(I*(b*x+a)))+1/b^3*d^2*a^2*ln(exp(I*(b*x+a))-1)+4/3*I/b^3*d^2*a^3-I*d*c*x^2-2*I/b^2*d^2*p
olylog(2,exp(I*(b*x+a)))*x-2*I/b^2*d^2*polylog(2,-exp(I*(b*x+a)))*x+1/b*c^2*ln(exp(I*(b*x+a))+1)-2/b*c^2*ln(ex
p(I*(b*x+a)))+1/b*c^2*ln(exp(I*(b*x+a))-1)-1/b^3*d^2*ln(1-exp(I*(b*x+a)))*a^2+1/b*d^2*ln(1-exp(I*(b*x+a)))*x^2
+1/b*d^2*ln(exp(I*(b*x+a))+1)*x^2+I*c^2*x+1/3*I/d*c^3+2/b*d*c*ln(exp(I*(b*x+a))+1)*x+2/b^2*d*c*ln(1-exp(I*(b*x
+a)))*a+4/b^2*c*d*a*ln(exp(I*(b*x+a)))-2/b^2*c*d*a*ln(exp(I*(b*x+a))-1)+2/b*d*c*ln(1-exp(I*(b*x+a)))*x-2*I/b^2
*d*c*a^2-2*I/b^2*d*c*polylog(2,exp(I*(b*x+a)))-2*I/b^2*d*c*polylog(2,-exp(I*(b*x+a)))+2*I/b^2*d^2*a^2*x-1/3*I*
d^2*x^3-4*I/b*d*c*x*a+2*d^2*polylog(3,-exp(I*(b*x+a)))/b^3+2*d^2*polylog(3,exp(I*(b*x+a)))/b^3

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. 502 vs. \(2 (78) = 156\).

Time = 0.27 (sec) , antiderivative size = 502, normalized size of antiderivative = 5.40 \[ \int (c+d x)^2 \cot (a+b x) \, dx=\frac {2 \, d^{2} {\rm polylog}\left (3, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 2 \, d^{2} {\rm polylog}\left (3, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 2 \, d^{2} {\rm polylog}\left (3, -\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 2 \, d^{2} {\rm polylog}\left (3, -\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 2 \, {\left (i \, b d^{2} x + i \, b c d\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 2 \, {\left (-i \, b d^{2} x - i \, b c d\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 2 \, {\left (-i \, b d^{2} x - i \, b c d\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 2 \, {\left (i \, b d^{2} x + i \, b c d\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right )}{2 \, b^{3}} \]

[In]

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

[Out]

1/2*(2*d^2*polylog(3, cos(b*x + a) + I*sin(b*x + a)) + 2*d^2*polylog(3, cos(b*x + a) - I*sin(b*x + a)) + 2*d^2
*polylog(3, -cos(b*x + a) + I*sin(b*x + a)) + 2*d^2*polylog(3, -cos(b*x + a) - I*sin(b*x + a)) - 2*(I*b*d^2*x
+ I*b*c*d)*dilog(cos(b*x + a) + I*sin(b*x + a)) - 2*(-I*b*d^2*x - I*b*c*d)*dilog(cos(b*x + a) - I*sin(b*x + a)
) - 2*(-I*b*d^2*x - I*b*c*d)*dilog(-cos(b*x + a) + I*sin(b*x + a)) - 2*(I*b*d^2*x + I*b*c*d)*dilog(-cos(b*x +
a) - I*sin(b*x + a)) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(cos(b*x + a) + I*sin(b*x + a) + 1) + (b^2*d^2
*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(cos(b*x + a) - I*sin(b*x + a) + 1) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(-1/
2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(-1/2*cos(b*x + a) - 1/2*I*sin
(b*x + a) + 1/2) + (b^2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a^2*d^2)*log(-cos(b*x + a) + I*sin(b*x + a) + 1) +
(b^2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a^2*d^2)*log(-cos(b*x + a) - I*sin(b*x + a) + 1))/b^3

Sympy [F]

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

[In]

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

[Out]

Integral((c + d*x)**2*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. 411 vs. \(2 (78) = 156\).

Time = 0.30 (sec) , antiderivative size = 411, normalized size of antiderivative = 4.42 \[ \int (c+d x)^2 \cot (a+b x) \, dx=\frac {6 \, c^{2} \log \left (\sin \left (b x + a\right )\right ) - \frac {12 \, a c d \log \left (\sin \left (b x + a\right )\right )}{b} + \frac {6 \, a^{2} d^{2} \log \left (\sin \left (b x + a\right )\right )}{b^{2}} + \frac {-2 i \, {\left (b x + a\right )}^{3} d^{2} - 6 \, {\left (i \, b c d - i \, a d^{2}\right )} {\left (b x + a\right )}^{2} + 12 \, d^{2} {\rm Li}_{3}(-e^{\left (i \, b x + i \, a\right )}) + 12 \, d^{2} {\rm Li}_{3}(e^{\left (i \, b x + i \, a\right )}) - 6 \, {\left (-i \, {\left (b x + a\right )}^{2} d^{2} + 2 \, {\left (-i \, b c d + i \, a d^{2}\right )} {\left (b x + a\right )}\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - 6 \, {\left (i \, {\left (b x + a\right )}^{2} d^{2} + 2 \, {\left (i \, b c d - i \, a d^{2}\right )} {\left (b x + a\right )}\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 12 \, {\left (i \, b c d + i \, {\left (b x + a\right )} d^{2} - i \, a d^{2}\right )} {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) - 12 \, {\left (i \, b c d + i \, {\left (b x + a\right )} d^{2} - i \, a d^{2}\right )} {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) + 3 \, {\left ({\left (b x + a\right )}^{2} d^{2} + 2 \, {\left (b c d - a d^{2}\right )} {\left (b x + a\right )}\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) + 3 \, {\left ({\left (b x + a\right )}^{2} d^{2} + 2 \, {\left (b c d - a d^{2}\right )} {\left (b x + a\right )}\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )}{b^{2}}}{6 \, b} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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