\(\int (c+d x)^2 \cos (a+b x) \cot ^2(a+b x) \, dx\) [173]

Optimal result

Integrand size = 22, antiderivative size = 139 \[ \int (c+d x)^2 \cos (a+b x) \cot ^2(a+b x) \, dx=-\frac {4 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {2 d (c+d x) \cos (a+b x)}{b^2}-\frac {(c+d x)^2 \csc (a+b x)}{b}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}+\frac {2 d^2 \sin (a+b x)}{b^3}-\frac {(c+d x)^2 \sin (a+b x)}{b} \]

[Out]

-4*d*(d*x+c)*arctanh(exp(I*(b*x+a)))/b^2-2*d*(d*x+c)*cos(b*x+a)/b^2-(d*x+c)^2*csc(b*x+a)/b+2*I*d^2*polylog(2,-
exp(I*(b*x+a)))/b^3-2*I*d^2*polylog(2,exp(I*(b*x+a)))/b^3+2*d^2*sin(b*x+a)/b^3-(d*x+c)^2*sin(b*x+a)/b

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4493, 3377, 2717, 4495, 4268, 2317, 2438} \[ \int (c+d x)^2 \cos (a+b x) \cot ^2(a+b x) \, dx=-\frac {4 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}+\frac {2 d^2 \sin (a+b x)}{b^3}-\frac {2 d (c+d x) \cos (a+b x)}{b^2}-\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {(c+d x)^2 \csc (a+b x)}{b} \]

[In]

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

[Out]

(-4*d*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b^2 - (2*d*(c + d*x)*Cos[a + b*x])/b^2 - ((c + d*x)^2*Csc[a + b*x])/
b + ((2*I)*d^2*PolyLog[2, -E^(I*(a + b*x))])/b^3 - ((2*I)*d^2*PolyLog[2, E^(I*(a + b*x))])/b^3 + (2*d^2*Sin[a
+ b*x])/b^3 - ((c + d*x)^2*Sin[a + b*x])/b

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, 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 4493

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Int[
(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 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]

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(310\) vs. \(2(139)=278\).

Time = 4.43 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.23 \[ \int (c+d x)^2 \cos (a+b x) \cot ^2(a+b x) \, dx=-\frac {8 b c d \text {arctanh}\left (\cos (a)-\sin (a) \tan \left (\frac {b x}{2}\right )\right )+2 b^2 (c+d x)^2 \csc (a)-4 d^2 \left (2 \arctan (\tan (a)) \text {arctanh}\left (\cos (a)-\sin (a) \tan \left (\frac {b x}{2}\right )\right )+\frac {\left ((b x+\arctan (\tan (a))) \left (\log \left (1-e^{i (b x+\arctan (\tan (a)))}\right )-\log \left (1+e^{i (b x+\arctan (\tan (a)))}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i (b x+\arctan (\tan (a)))}\right )-i \operatorname {PolyLog}\left (2,e^{i (b x+\arctan (\tan (a)))}\right )\right ) \sec (a)}{\sqrt {\sec ^2(a)}}\right )+2 \cos (b x) \left (2 b d (c+d x) \cos (a)+\left (-2 d^2+b^2 (c+d x)^2\right ) \sin (a)\right )-b^2 (c+d x)^2 \csc \left (\frac {a}{2}\right ) \csc \left (\frac {1}{2} (a+b x)\right ) \sin \left (\frac {b x}{2}\right )+b^2 (c+d x)^2 \sec \left (\frac {a}{2}\right ) \sec \left (\frac {1}{2} (a+b x)\right ) \sin \left (\frac {b x}{2}\right )+2 \left (\left (-2 d^2+b^2 (c+d x)^2\right ) \cos (a)-2 b d (c+d x) \sin (a)\right ) \sin (b x)}{2 b^3} \]

[In]

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

[Out]

-1/2*(8*b*c*d*ArcTanh[Cos[a] - Sin[a]*Tan[(b*x)/2]] + 2*b^2*(c + d*x)^2*Csc[a] - 4*d^2*(2*ArcTan[Tan[a]]*ArcTa
nh[Cos[a] - Sin[a]*Tan[(b*x)/2]] + (((b*x + ArcTan[Tan[a]])*(Log[1 - E^(I*(b*x + ArcTan[Tan[a]]))] - Log[1 + E
^(I*(b*x + ArcTan[Tan[a]]))]) + I*PolyLog[2, -E^(I*(b*x + ArcTan[Tan[a]]))] - I*PolyLog[2, E^(I*(b*x + ArcTan[
Tan[a]]))])*Sec[a])/Sqrt[Sec[a]^2]) + 2*Cos[b*x]*(2*b*d*(c + d*x)*Cos[a] + (-2*d^2 + b^2*(c + d*x)^2)*Sin[a])
- b^2*(c + d*x)^2*Csc[a/2]*Csc[(a + b*x)/2]*Sin[(b*x)/2] + b^2*(c + d*x)^2*Sec[a/2]*Sec[(a + b*x)/2]*Sin[(b*x)
/2] + 2*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] - 2*b*d*(c + d*x)*Sin[a])*Sin[b*x])/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. 331 vs. \(2 (131 ) = 262\).

Time = 1.78 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.39

method	result	size
risch	\(\frac {i \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +2 i b \,d^{2} x +b^{2} c^{2}+2 i b c d -2 d^{2}\right ) {\mathrm e}^{i \left (x b +a \right )}}{2 b^{3}}-\frac {i \left (x^{2} d^{2} b^{2}+2 b^{2} c d x -2 i b \,d^{2} x +b^{2} c^{2}-2 i b c d -2 d^{2}\right ) {\mathrm e}^{-i \left (x b +a \right )}}{2 b^{3}}-\frac {2 i \left (x^{2} d^{2}+2 c d x +c^{2}\right ) {\mathrm e}^{i \left (x b +a \right )}}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}-\frac {4 d c \,\operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{3}}-\frac {2 i d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {2 d^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b^{2}}-\frac {2 d^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) a}{b^{3}}+\frac {2 i d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {4 d^{2} a \,\operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}\)	\(332\)

[In]

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

[Out]

1/2*I*(x^2*d^2*b^2+2*b^2*c*d*x+b^2*c^2+2*I*b*d^2*x-2*d^2+2*I*b*c*d)/b^3*exp(I*(b*x+a))-1/2*I*(x^2*d^2*b^2+2*b^
2*c*d*x+b^2*c^2-2*I*b*d^2*x-2*d^2-2*I*b*c*d)/b^3*exp(-I*(b*x+a))-2*I*(d^2*x^2+2*c*d*x+c^2)*exp(I*(b*x+a))/b/(e
xp(2*I*(b*x+a))-1)-4*d/b^2*c*arctanh(exp(I*(b*x+a)))+2*d^2/b^2*ln(1-exp(I*(b*x+a)))*x+2*d^2/b^3*ln(1-exp(I*(b*
x+a)))*a-2*I*d^2*polylog(2,exp(I*(b*x+a)))/b^3-2*d^2/b^2*ln(exp(I*(b*x+a))+1)*x-2*d^2/b^3*ln(exp(I*(b*x+a))+1)
*a+2*I*d^2*polylog(2,-exp(I*(b*x+a)))/b^3+4*d^2/b^3*a*arctanh(exp(I*(b*x+a)))

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

Time = 0.28 (sec) , antiderivative size = 448, normalized size of antiderivative = 3.22 \[ \int (c+d x)^2 \cos (a+b x) \cot ^2(a+b x) \, dx=-\frac {2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} + i \, d^{2} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - i \, d^{2} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + i \, d^{2} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - i \, d^{2} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + {\left (b d^{2} x + b c d\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + {\left (b d^{2} x + b c d\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - {\left (b c d - a d^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) - {\left (b c d - a d^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) - \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) - {\left (b d^{2} x + a d^{2}\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - {\left (b d^{2} x + a d^{2}\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 2 \, d^{2}}{b^{3} \sin \left (b x + a\right )} \]

[In]

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

[Out]

-(2*b^2*d^2*x^2 + 4*b^2*c*d*x + 2*b^2*c^2 + I*d^2*dilog(cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) - I*d^2*di
log(cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) + I*d^2*dilog(-cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) - I
*d^2*dilog(-cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) - (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(b*
x + a)^2 + 2*(b*d^2*x + b*c*d)*cos(b*x + a)*sin(b*x + a) + (b*d^2*x + b*c*d)*log(cos(b*x + a) + I*sin(b*x + a)
+ 1)*sin(b*x + a) + (b*d^2*x + b*c*d)*log(cos(b*x + a) - I*sin(b*x + a) + 1)*sin(b*x + a) - (b*c*d - a*d^2)*l
og(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2)*sin(b*x + a) - (b*c*d - a*d^2)*log(-1/2*cos(b*x + a) - 1/2*I*
sin(b*x + a) + 1/2)*sin(b*x + a) - (b*d^2*x + a*d^2)*log(-cos(b*x + a) + I*sin(b*x + a) + 1)*sin(b*x + a) - (b
*d^2*x + a*d^2)*log(-cos(b*x + a) - I*sin(b*x + a) + 1)*sin(b*x + a) - 2*d^2)/(b^3*sin(b*x + a))

Sympy [F]

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

[In]

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

[Out]

Integral((c + d*x)**2*cos(a + b*x)*cot(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. 3199 vs. \(2 (127) = 254\).

Time = 1.14 (sec) , antiderivative size = 3199, normalized size of antiderivative = 23.01 \[ \int (c+d x)^2 \cos (a+b x) \cot ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(b^2*d^2*x^2*(-I*cos(a) + sin(a)) + b^2*c^2*(-I*cos(a) + sin(a)) - 2*b*c*d*(cos(a) + I*sin(a)) - 2*d^2*(-I
*cos(a) + sin(a)) - 2*(b^2*c*d*(I*cos(a) - sin(a)) + b*d^2*(cos(a) + I*sin(a)))*x - 4*((b*d^2*x*(-I*cos(a) + s
in(a)) + b*c*d*(-I*cos(a) + sin(a)) + (I*b*d^2*x + I*b*c*d)*cos(2*b*x + 3*a) - (b*d^2*x + b*c*d)*sin(2*b*x + 3
*a))*cos(3*b*x + 3*a) + ((-I*b*d^2*x - I*b*c*d)*cos(b*x + a) + (b*d^2*x + b*c*d)*sin(b*x + a))*cos(2*b*x + 3*a
) + (b*d^2*x*(I*cos(a) - sin(a)) + b*c*d*(I*cos(a) - sin(a)))*cos(b*x + a) + (b*d^2*x*(cos(a) + I*sin(a)) + b*
c*d*(cos(a) + I*sin(a)) - (b*d^2*x + b*c*d)*cos(2*b*x + 3*a) + (-I*b*d^2*x - I*b*c*d)*sin(2*b*x + 3*a))*sin(3*
b*x + 3*a) + ((b*d^2*x + b*c*d)*cos(b*x + a) + (I*b*d^2*x + I*b*c*d)*sin(b*x + a))*sin(2*b*x + 3*a) - (b*d^2*x
*(cos(a) + I*sin(a)) + b*c*d*(cos(a) + I*sin(a)))*sin(b*x + a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 4*(b
*c*d*(-I*cos(a) + sin(a))*cos(b*x + a) + b*c*d*(cos(a) + I*sin(a))*sin(b*x + a) + (b*c*d*(I*cos(a) - sin(a)) -
I*b*c*d*cos(2*b*x + 3*a) + b*c*d*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) + (I*b*c*d*cos(b*x + a) - b*c*d*sin(b*x +
a))*cos(2*b*x + 3*a) - (b*c*d*(cos(a) + I*sin(a)) - b*c*d*cos(2*b*x + 3*a) - I*b*c*d*sin(2*b*x + 3*a))*sin(3*
b*x + 3*a) - (b*c*d*cos(b*x + a) + I*b*c*d*sin(b*x + a))*sin(2*b*x + 3*a))*arctan2(sin(b*x + a), cos(b*x + a)
- 1) - 4*(b*d^2*x*(I*cos(a) - sin(a))*cos(b*x + a) - b*d^2*x*(cos(a) + I*sin(a))*sin(b*x + a) + (b*d^2*x*(-I*c
os(a) + sin(a)) + I*b*d^2*x*cos(2*b*x + 3*a) - b*d^2*x*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) + (-I*b*d^2*x*cos(b*
x + a) + b*d^2*x*sin(b*x + a))*cos(2*b*x + 3*a) + (b*d^2*x*(cos(a) + I*sin(a)) - b*d^2*x*cos(2*b*x + 3*a) - I*
b*d^2*x*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) + (b*d^2*x*cos(b*x + a) + I*b*d^2*x*sin(b*x + a))*sin(2*b*x + 3*a))
*arctan2(sin(b*x + a), -cos(b*x + a) + 1) + ((I*b^2*d^2*x^2 + I*b^2*c^2 - 2*b*c*d - 2*I*d^2 - 2*(-I*b^2*c*d +
b*d^2)*x)*cos(3*b*x + 3*a) + (-I*b^2*d^2*x^2 - I*b^2*c^2 + 2*b*c*d + 2*I*d^2 - 2*(I*b^2*c*d - b*d^2)*x)*cos(b*
x + a) - (b^2*d^2*x^2 + b^2*c^2 + 2*I*b*c*d - 2*d^2 + 2*(b^2*c*d + I*b*d^2)*x)*sin(3*b*x + 3*a) + (b^2*d^2*x^2
+ b^2*c^2 + 2*I*b*c*d - 2*d^2 + 2*(b^2*c*d + I*b*d^2)*x)*sin(b*x + a))*cos(3*b*x + 4*a) - 2*((3*I*b^2*d^2*x^2
+ 6*I*b^2*c*d*x + 3*I*b^2*c^2 - 2*I*d^2)*cos(b*x + 2*a) - (3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*s
in(b*x + 2*a))*cos(3*b*x + 3*a) + (I*b^2*d^2*x^2 + I*b^2*c^2 + 2*b*c*d - 2*I*d^2 - 2*(-I*b^2*c*d - b*d^2)*x)*c
os(2*b*x + 3*a) - 2*((-3*I*b^2*d^2*x^2 - 6*I*b^2*c*d*x - 3*I*b^2*c^2 + 2*I*d^2)*cos(b*x + a) + (3*b^2*d^2*x^2
+ 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*sin(b*x + a))*cos(b*x + 2*a) - 4*(d^2*(-I*cos(a) + sin(a))*cos(b*x + a) + d
^2*(cos(a) + I*sin(a))*sin(b*x + a) + (d^2*(I*cos(a) - sin(a)) - I*d^2*cos(2*b*x + 3*a) + d^2*sin(2*b*x + 3*a)
)*cos(3*b*x + 3*a) + (I*d^2*cos(b*x + a) - d^2*sin(b*x + a))*cos(2*b*x + 3*a) - (d^2*(cos(a) + I*sin(a)) - d^2
*cos(2*b*x + 3*a) - I*d^2*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) - (d^2*cos(b*x + a) + I*d^2*sin(b*x + a))*sin(2*b
*x + 3*a))*dilog(-e^(I*b*x + I*a)) - 4*(d^2*(I*cos(a) - sin(a))*cos(b*x + a) - d^2*(cos(a) + I*sin(a))*sin(b*x
+ a) + (d^2*(-I*cos(a) + sin(a)) + I*d^2*cos(2*b*x + 3*a) - d^2*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) + (-I*d^2*
cos(b*x + a) + d^2*sin(b*x + a))*cos(2*b*x + 3*a) + (d^2*(cos(a) + I*sin(a)) - d^2*cos(2*b*x + 3*a) - I*d^2*si
n(2*b*x + 3*a))*sin(3*b*x + 3*a) + (d^2*cos(b*x + a) + I*d^2*sin(b*x + a))*sin(2*b*x + 3*a))*dilog(e^(I*b*x +
I*a)) + 2*((b*d^2*x*(cos(a) + I*sin(a)) + b*c*d*(cos(a) + I*sin(a)) - (b*d^2*x + b*c*d)*cos(2*b*x + 3*a) - (I*
b*d^2*x + I*b*c*d)*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) + ((b*d^2*x + b*c*d)*cos(b*x + a) - (-I*b*d^2*x - I*b*c*
d)*sin(b*x + a))*cos(2*b*x + 3*a) - (b*d^2*x*(cos(a) + I*sin(a)) + b*c*d*(cos(a) + I*sin(a)))*cos(b*x + a) - (
b*d^2*x*(-I*cos(a) + sin(a)) + b*c*d*(-I*cos(a) + sin(a)) + (I*b*d^2*x + I*b*c*d)*cos(2*b*x + 3*a) - (b*d^2*x
+ b*c*d)*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) - ((-I*b*d^2*x - I*b*c*d)*cos(b*x + a) + (b*d^2*x + b*c*d)*sin(b*x
+ a))*sin(2*b*x + 3*a) - (b*d^2*x*(I*cos(a) - sin(a)) + b*c*d*(I*cos(a) - sin(a)))*sin(b*x + a))*log(cos(b*x
+ a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - 2*((b*d^2*x*(cos(a) + I*sin(a)) + b*c*d*(cos(a) + I*sin(a)) -
(b*d^2*x + b*c*d)*cos(2*b*x + 3*a) + (-I*b*d^2*x - I*b*c*d)*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) + ((b*d^2*x + b
*c*d)*cos(b*x + a) + (I*b*d^2*x + I*b*c*d)*sin(b*x + a))*cos(2*b*x + 3*a) - (b*d^2*x*(cos(a) + I*sin(a)) + b*c
*d*(cos(a) + I*sin(a)))*cos(b*x + a) + (b*d^2*x*(I*cos(a) - sin(a)) + b*c*d*(I*cos(a) - sin(a)) + (-I*b*d^2*x
- I*b*c*d)*cos(2*b*x + 3*a) + (b*d^2*x + b*c*d)*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) + ((I*b*d^2*x + I*b*c*d)*co
s(b*x + a) - (b*d^2*x + b*c*d)*sin(b*x + a))*sin(2*b*x + 3*a) + (b*d^2*x*(-I*cos(a) + sin(a)) + b*c*d*(-I*cos(
a) + sin(a)))*sin(b*x + a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - ((b^2*d^2*x^2 + b^2*c^
2 + 2*I*b*c*d - 2*d^2 + 2*(b^2*c*d + I*b*d^2)*x)*cos(3*b*x + 3*a) - (b^2*d^2*x^2 + b^2*c^2 + 2*I*b*c*d - 2*d^2
+ 2*(b^2*c*d + I*b*d^2)*x)*cos(b*x + a) - (-I*b^2*d^2*x^2 - I*b^2*c^2 + 2*b*c*d + 2*I*d^2 - 2*(I*b^2*c*d - b*
d^2)*x)*sin(3*b*x + 3*a) - (I*b^2*d^2*x^2 + I*b^2*c^2 - 2*b*c*d - 2*I*d^2 - 2*(-I*b^2*c*d + b*d^2)*x)*sin(b*x
+ a))*sin(3*b*x + 4*a) + 2*((3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*cos(b*x + 2*a) - (-3*I*b^2*d^2*x
^2 - 6*I*b^2*c*d*x - 3*I*b^2*c^2 + 2*I*d^2)*sin(b*x + 2*a))*sin(3*b*x + 3*a) - (b^2*d^2*x^2 + b^2*c^2 - 2*I*b*
c*d - 2*d^2 + 2*(b^2*c*d - I*b*d^2)*x)*sin(2*b*x + 3*a) - 2*((3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)
*cos(b*x + a) + (3*I*b^2*d^2*x^2 + 6*I*b^2*c*d*x + 3*I*b^2*c^2 - 2*I*d^2)*sin(b*x + a))*sin(b*x + 2*a))/(b^3*(
cos(a) + I*sin(a))*cos(b*x + a) + b^3*(I*cos(a) - sin(a))*sin(b*x + a) - (b^3*(cos(a) + I*sin(a)) - b^3*cos(2*
b*x + 3*a) - I*b^3*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) - (b^3*cos(b*x + a) + I*b^3*sin(b*x + a))*cos(2*b*x + 3*
a) + (b^3*(-I*cos(a) + sin(a)) + I*b^3*cos(2*b*x + 3*a) - b^3*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) + (-I*b^3*cos
(b*x + a) + b^3*sin(b*x + a))*sin(2*b*x + 3*a))

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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