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

Optimal. Leaf size=139 \[ \frac {2 i d^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^3}-\frac {2 i d^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^3}+\frac {2 d^2 \sin (a+b x)}{b^3}-\frac {2 d (c+d x) \cos (a+b x)}{b^2}-\frac {4 d (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {(c+d x)^2 \csc (a+b x)}{b} \]

[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

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Rubi [A]  time = 0.15, antiderivative size = 139, normalized size of antiderivative = 1.00, 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 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]

Rule 2637

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

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 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 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 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]

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*}

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Mathematica [B]  time = 3.92, size = 310, normalized size = 2.23 \[ -\frac {2 \cos (b x) \left (\sin (a) \left (b^2 (c+d x)^2-2 d^2\right )+2 b d \cos (a) (c+d x)\right )+2 \sin (b x) \left (\cos (a) \left (b^2 (c+d x)^2-2 d^2\right )-2 b d \sin (a) (c+d x)\right )+2 b^2 \csc (a) (c+d x)^2-b^2 \csc \left (\frac {a}{2}\right ) \sin \left (\frac {b x}{2}\right ) (c+d x)^2 \csc \left (\frac {1}{2} (a+b x)\right )+b^2 \sec \left (\frac {a}{2}\right ) \sin \left (\frac {b x}{2}\right ) (c+d x)^2 \sec \left (\frac {1}{2} (a+b x)\right )+8 b c d \tanh ^{-1}\left (\cos (a)-\sin (a) \tan \left (\frac {b x}{2}\right )\right )-4 d^2 \left (2 \tan ^{-1}(\tan (a)) \tanh ^{-1}\left (\cos (a)-\sin (a) \tan \left (\frac {b x}{2}\right )\right )+\frac {\sec (a) \left (i \text {Li}_2\left (-e^{i \left (b x+\tan ^{-1}(\tan (a))\right )}\right )-i \text {Li}_2\left (e^{i \left (b x+\tan ^{-1}(\tan (a))\right )}\right )+\left (\tan ^{-1}(\tan (a))+b x\right ) \left (\log \left (1-e^{i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )-\log \left (1+e^{i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )\right )\right )}{\sqrt {\sec ^2(a)}}\right )}{2 b^3} \]

Warning: Unable to verify antiderivative.

[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

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fricas [B]  time = 0.50, size = 448, normalized size = 3.22 \[ -\frac {2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} + i \, d^{2} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - i \, d^{2} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + i \, d^{2} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - i \, d^{2} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + {\left (b d^{2} x + b c d\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + {\left (b d^{2} x + b c d\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - {\left (b c d - a d^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) - {\left (b c d - a d^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) - \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) - {\left (b d^{2} x + a d^{2}\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - {\left (b d^{2} x + a d^{2}\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 2 \, d^{2}}{b^{3} \sin \left (b x + a\right )} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{2} \cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}\,{d x} \]

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)

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maple [B]  time = 0.13, size = 332, normalized size = 2.39 \[ \frac {i \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +2 i b \,d^{2} x +b^{2} c^{2}+2 i b c d -2 d^{2}\right ) {\mathrm e}^{i \left (b x +a \right )}}{2 b^{3}}-\frac {i \left (d^{2} x^{2} b^{2}+2 b^{2} c d x -2 i b \,d^{2} x +b^{2} c^{2}-2 i b c d -2 d^{2}\right ) {\mathrm e}^{-i \left (b x +a \right )}}{2 b^{3}}-\frac {2 i {\mathrm e}^{i \left (b x +a \right )} \left (d^{2} x^{2}+2 c d x +c^{2}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}-\frac {4 d c \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {2 d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{2}}-\frac {2 d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) a}{b^{3}}+\frac {2 i d^{2} \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{3}}-\frac {2 i d^{2} \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {4 d^{2} a \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}} \]

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

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maxima [B]  time = 1.66, size = 3284, normalized size = 23.63 \[ \text {result too large to display} \]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (a+b\,x\right )\,{\mathrm {cot}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{2} \cos {\left (a + b x \right )} \cot ^{2}{\left (a + b x \right )}\, dx \]

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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