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

Optimal. Leaf size=139 \[ \frac{2 i d^2 \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac{2 i d^2 \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac{2 d (c+d x) \cos (a+b x)}{b^2}-\frac{4 d (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}+\frac{2 d^2 \sin (a+b x)}{b^3}-\frac{(c+d x)^2 \sin (a+b x)}{b}-\frac{(c+d x)^2 \csc (a+b x)}{b} \]

[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

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Rubi [A]  time = 0.14899, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {4408, 3296, 2637, 4410, 4183, 2279, 2391} \[ \frac{2 i d^2 \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac{2 i d^2 \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac{2 d (c+d x) \cos (a+b x)}{b^2}-\frac{4 d (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}+\frac{2 d^2 \sin (a+b x)}{b^3}-\frac{(c+d x)^2 \sin (a+b x)}{b}-\frac{(c+d x)^2 \csc (a+b x)}{b} \]

Antiderivative was successfully verified.

[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 4408

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 3296

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 2637

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

Rule 4410

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] /; Fr
eeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4183

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 2279

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 2391

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]

Rubi steps

\begin{align*} \int (c+d x)^2 \cos (a+b x) \cot ^2(a+b x) \, dx &=-\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) \tanh ^{-1}\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) \tanh ^{-1}\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 ) \operatorname{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 ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}\\ &=-\frac{4 d (c+d x) \tanh ^{-1}\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 \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^3}-\frac{2 i d^2 \text{Li}_2\left (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]  time = 3.95586, size = 310, normalized size = 2.23 \[ -\frac{-4 d^2 \left (2 \tan ^{-1}(\tan (a)) \tanh ^{-1}\left (\cos (a)-\sin (a) \tan \left (\frac{b x}{2}\right )\right )+\frac{\sec (a) \left (i \text{PolyLog}\left (2,-e^{i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )-i \text{PolyLog}\left (2,e^{i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )+\left (\tan ^{-1}(\tan (a))+b x\right ) \left (\log \left (1-e^{i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )-\log \left (1+e^{i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )\right )\right )}{\sqrt{\sec ^2(a)}}\right )+2 \cos (b x) \left (\sin (a) \left (b^2 (c+d x)^2-2 d^2\right )+2 b d \cos (a) (c+d x)\right )+2 \sin (b x) \left (\cos (a) \left (b^2 (c+d x)^2-2 d^2\right )-2 b d \sin (a) (c+d x)\right )+2 b^2 \csc (a) (c+d x)^2-b^2 \csc \left (\frac{a}{2}\right ) \sin \left (\frac{b x}{2}\right ) (c+d x)^2 \csc \left (\frac{1}{2} (a+b x)\right )+b^2 \sec \left (\frac{a}{2}\right ) \sin \left (\frac{b x}{2}\right ) (c+d x)^2 \sec \left (\frac{1}{2} (a+b x)\right )+8 b c d \tanh ^{-1}\left (\cos (a)-\sin (a) \tan \left (\frac{b x}{2}\right )\right )}{2 b^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(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]]*ArcTanh[C
os[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])/(2*b^3)

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Maple [B]  time = 0.16, size = 332, normalized size = 2.4 \begin{align*}{\frac{{\frac{i}{2}} \left ({d}^{2}{x}^{2}{b}^{2}+2\,{b}^{2}cdx+{b}^{2}{c}^{2}+2\,ib{d}^{2}x-2\,{d}^{2}+2\,ibcd \right ){{\rm e}^{i \left ( bx+a \right ) }}}{{b}^{3}}}-{\frac{{\frac{i}{2}} \left ({d}^{2}{x}^{2}{b}^{2}+2\,{b}^{2}cdx+{b}^{2}{c}^{2}-2\,ib{d}^{2}x-2\,{d}^{2}-2\,ibcd \right ){{\rm e}^{-i \left ( bx+a \right ) }}}{{b}^{3}}}-{\frac{2\,i \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2} \right ){{\rm e}^{i \left ( bx+a \right ) }}}{b \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) }}-4\,{\frac{cd{\it Artanh} \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}+2\,{\frac{{d}^{2}\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ) x}{{b}^{2}}}+2\,{\frac{{d}^{2}\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ) a}{{b}^{3}}}-{\frac{2\,i{d}^{2}{\it polylog} \left ( 2,{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}-2\,{\frac{{d}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) x}{{b}^{2}}}-2\,{\frac{{d}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) a}{{b}^{3}}}+{\frac{2\,i{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}+4\,{\frac{{d}^{2}a{\it Artanh} \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*(d^2*x^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*(d^2*x^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)))

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Maxima [B]  time = 5.39042, size = 4435, normalized size = 31.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*d^2*x^2*(-I*cos(a) + sin(a)) + b^2*c^2*(-I*cos(a) + sin(a)) - b*c*d*(2*cos(a) + 2*I*sin(a)) - 2*d^2*(-I*c
os(a) + sin(a)) - (2*b^2*c*d*(I*cos(a) - sin(a)) + b*d^2*(2*cos(a) + 2*I*sin(a)))*x - ((4*b*d^2*x*(-I*cos(a) +
sin(a)) + 4*b*c*d*(-I*cos(a) + sin(a)) - (-4*I*b*d^2*x - 4*I*b*c*d)*cos(2*b*x + 3*a) - 4*(b*d^2*x + b*c*d)*si
n(2*b*x + 3*a))*cos(3*b*x + 3*a) - ((4*I*b*d^2*x + 4*I*b*c*d)*cos(b*x + a) - 4*(b*d^2*x + b*c*d)*sin(b*x + a))
*cos(2*b*x + 3*a) + 4*(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*(4*cos
(a) + 4*I*sin(a)) + b*c*d*(4*cos(a) + 4*I*sin(a)) - 4*(b*d^2*x + b*c*d)*cos(2*b*x + 3*a) - (4*I*b*d^2*x + 4*I*
b*c*d)*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) + (4*(b*d^2*x + b*c*d)*cos(b*x + a) - (-4*I*b*d^2*x - 4*I*b*c*d)*sin
(b*x + a))*sin(2*b*x + 3*a) - (b*d^2*x*(4*cos(a) + 4*I*sin(a)) + b*c*d*(4*cos(a) + 4*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*(4*cos(a) + 4*I*s
in(a))*sin(b*x + a) + (4*b*c*d*(I*cos(a) - sin(a)) - 4*I*b*c*d*cos(2*b*x + 3*a) + 4*b*c*d*sin(2*b*x + 3*a))*co
s(3*b*x + 3*a) + 4*(I*b*c*d*cos(b*x + a) - b*c*d*sin(b*x + a))*cos(2*b*x + 3*a) - (b*c*d*(4*cos(a) + 4*I*sin(a
)) - 4*b*c*d*cos(2*b*x + 3*a) - 4*I*b*c*d*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) - (4*b*c*d*cos(b*x + a) + 4*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))*co
s(b*x + a) - b*d^2*x*(4*cos(a) + 4*I*sin(a))*sin(b*x + a) + 4*(b*d^2*x*(-I*cos(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) + 4*(-I*b*d^2*x*cos(b*x + a) + b*d^2*x*sin(b*x + a))*c
os(2*b*x + 3*a) + (b*d^2*x*(4*cos(a) + 4*I*sin(a)) - 4*b*d^2*x*cos(2*b*x + 3*a) - 4*I*b*d^2*x*sin(2*b*x + 3*a)
)*sin(3*b*x + 3*a) + (4*b*d^2*x*cos(b*x + a) + 4*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 - 2*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 + 2*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) + ((-6*I*b^2*d^2*x^2 - 12*I*b^2*c*d*x
- 6*I*b^2*c^2 + 4*I*d^2)*cos(b*x + 2*a) + 2*(3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*sin(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 + 2*b*d^2)*x)*cos(2*b*x + 3*
a) + ((6*I*b^2*d^2*x^2 + 12*I*b^2*c*d*x + 6*I*b^2*c^2 - 4*I*d^2)*cos(b*x + a) - 2*(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*(4*cos(a)
+ 4*I*sin(a))*sin(b*x + a) + (4*d^2*(I*cos(a) - sin(a)) - 4*I*d^2*cos(2*b*x + 3*a) + 4*d^2*sin(2*b*x + 3*a))*c
os(3*b*x + 3*a) - (-4*I*d^2*cos(b*x + a) + 4*d^2*sin(b*x + a))*cos(2*b*x + 3*a) - (d^2*(4*cos(a) + 4*I*sin(a))
- 4*d^2*cos(2*b*x + 3*a) - 4*I*d^2*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) - 4*(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*(4*cos(a) + 4*I
*sin(a))*sin(b*x + a) + (4*d^2*(-I*cos(a) + sin(a)) + 4*I*d^2*cos(2*b*x + 3*a) - 4*d^2*sin(2*b*x + 3*a))*cos(3
*b*x + 3*a) - (4*I*d^2*cos(b*x + a) - 4*d^2*sin(b*x + a))*cos(2*b*x + 3*a) + (d^2*(4*cos(a) + 4*I*sin(a)) - 4*
d^2*cos(2*b*x + 3*a) - 4*I*d^2*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) + 4*(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)) + ((b*d^2*x*(2*cos(a) + 2*I*sin(a)) + b*c*d*(2*cos(a) + 2*I*sin(a)) -
2*(b*d^2*x + b*c*d)*cos(2*b*x + 3*a) + (-2*I*b*d^2*x - 2*I*b*c*d)*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) + (2*(b*
d^2*x + b*c*d)*cos(b*x + a) + (2*I*b*d^2*x + 2*I*b*c*d)*sin(b*x + a))*cos(2*b*x + 3*a) - (b*d^2*x*(2*cos(a) +
2*I*sin(a)) + b*c*d*(2*cos(a) + 2*I*sin(a)))*cos(b*x + a) - (2*b*d^2*x*(-I*cos(a) + sin(a)) + 2*b*c*d*(-I*cos(
a) + sin(a)) - (-2*I*b*d^2*x - 2*I*b*c*d)*cos(2*b*x + 3*a) - 2*(b*d^2*x + b*c*d)*sin(2*b*x + 3*a))*sin(3*b*x +
3*a) + ((2*I*b*d^2*x + 2*I*b*c*d)*cos(b*x + a) - 2*(b*d^2*x + b*c*d)*sin(b*x + a))*sin(2*b*x + 3*a) - 2*(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*d^2*x*(2*cos(a) + 2*I*sin(a)) + b*c*d*(2*cos(a) + 2*I*sin(a)) - 2*(b*d^2*x + b*c*d)*cos(2
*b*x + 3*a) - (2*I*b*d^2*x + 2*I*b*c*d)*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) + (2*(b*d^2*x + b*c*d)*cos(b*x + a)
- (-2*I*b*d^2*x - 2*I*b*c*d)*sin(b*x + a))*cos(2*b*x + 3*a) - (b*d^2*x*(2*cos(a) + 2*I*sin(a)) + b*c*d*(2*cos
(a) + 2*I*sin(a)))*cos(b*x + a) + (2*b*d^2*x*(I*cos(a) - sin(a)) + 2*b*c*d*(I*cos(a) - sin(a)) - (2*I*b*d^2*x
+ 2*I*b*c*d)*cos(2*b*x + 3*a) + 2*(b*d^2*x + b*c*d)*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) - ((-2*I*b*d^2*x - 2*I*
b*c*d)*cos(b*x + a) + 2*(b*d^2*x + b*c*d)*sin(b*x + a))*sin(2*b*x + 3*a) + 2*(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 + 2*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 - 2*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)
+ (6*I*b^2*d^2*x^2 + 12*I*b^2*c*d*x + 6*I*b^2*c^2 - 4*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) - (-6*I*b^2*d^2*x^2 - 12*I*b^2*c*d*x - 6*I*b^2*c^2 + 4*I*d^2)*sin(b*x + a))*sin
(b*x + 2*a))/(b^3*(2*cos(a) + 2*I*sin(a))*cos(b*x + a) + 2*b^3*(I*cos(a) - sin(a))*sin(b*x + a) - (b^3*(2*cos(
a) + 2*I*sin(a)) - 2*b^3*cos(2*b*x + 3*a) - 2*I*b^3*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) - (2*b^3*cos(b*x + a) +
2*I*b^3*sin(b*x + a))*cos(2*b*x + 3*a) + (2*b^3*(-I*cos(a) + sin(a)) + 2*I*b^3*cos(2*b*x + 3*a) - 2*b^3*sin(2
*b*x + 3*a))*sin(3*b*x + 3*a) + 2*(-I*b^3*cos(b*x + a) + b^3*sin(b*x + a))*sin(2*b*x + 3*a))

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Fricas [B]  time = 0.61068, size = 1172, normalized size = 8.43 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right )^{2} \cos{\left (a + b x \right )} \cot ^{2}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2} \cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)