3.83 \(\int \frac{\cot ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=159 \[ -\frac{65 \cot (c+d x)}{16 a^4 d}-\frac{4 i \log (\sin (c+d x))}{a^4 d}+\frac{2 \cot (c+d x)}{a^4 d (1+i \tan (c+d x))}+\frac{31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}-\frac{65 x}{16 a^4}+\frac{7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4} \]

[Out]

(-65*x)/(16*a^4) - (65*Cot[c + d*x])/(16*a^4*d) - ((4*I)*Log[Sin[c + d*x]])/(a^4*d) + (31*Cot[c + d*x])/(48*a^
4*d*(1 + I*Tan[c + d*x])^2) + (2*Cot[c + d*x])/(a^4*d*(1 + I*Tan[c + d*x])) + Cot[c + d*x]/(8*d*(a + I*a*Tan[c
+ d*x])^4) + (7*Cot[c + d*x])/(24*a*d*(a + I*a*Tan[c + d*x])^3)

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Rubi [A]  time = 0.437061, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3559, 3596, 3529, 3531, 3475} \[ -\frac{65 \cot (c+d x)}{16 a^4 d}-\frac{4 i \log (\sin (c+d x))}{a^4 d}+\frac{2 \cot (c+d x)}{a^4 d (1+i \tan (c+d x))}+\frac{31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}-\frac{65 x}{16 a^4}+\frac{7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^4,x]

[Out]

(-65*x)/(16*a^4) - (65*Cot[c + d*x])/(16*a^4*d) - ((4*I)*Log[Sin[c + d*x]])/(a^4*d) + (31*Cot[c + d*x])/(48*a^
4*d*(1 + I*Tan[c + d*x])^2) + (2*Cot[c + d*x])/(a^4*d*(1 + I*Tan[c + d*x])) + Cot[c + d*x]/(8*d*(a + I*a*Tan[c
+ d*x])^4) + (7*Cot[c + d*x])/(24*a*d*(a + I*a*Tan[c + d*x])^3)

Rule 3559

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(a*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d))
, Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan
[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2
+ d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3596

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((a*A + b*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2
*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] &&  !GtQ[n,
0]

Rule 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3531

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cot ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=\frac{\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{\int \frac{\cot ^2(c+d x) (9 a-5 i a \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx}{8 a^2}\\ &=\frac{\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot ^2(c+d x) \left (68 a^2-56 i a^2 \tan (c+d x)\right )}{(a+i a \tan (c+d x))^2} \, dx}{48 a^4}\\ &=\frac{31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot ^2(c+d x) \left (396 a^3-372 i a^3 \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{192 a^6}\\ &=\frac{31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{2 \cot (c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\int \cot ^2(c+d x) \left (1560 a^4-1536 i a^4 \tan (c+d x)\right ) \, dx}{384 a^8}\\ &=-\frac{65 \cot (c+d x)}{16 a^4 d}+\frac{31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{2 \cot (c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\int \cot (c+d x) \left (-1536 i a^4-1560 a^4 \tan (c+d x)\right ) \, dx}{384 a^8}\\ &=-\frac{65 x}{16 a^4}-\frac{65 \cot (c+d x)}{16 a^4 d}+\frac{31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{2 \cot (c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{(4 i) \int \cot (c+d x) \, dx}{a^4}\\ &=-\frac{65 x}{16 a^4}-\frac{65 \cot (c+d x)}{16 a^4 d}-\frac{4 i \log (\sin (c+d x))}{a^4 d}+\frac{31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{2 \cot (c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 2.57, size = 444, normalized size = 2.79 \[ \frac{i \csc (c) \sec ^4(c+d x) (\cos (d x)+i \sin (d x))^4 \left (1536 d x \cos ^3(c)+4608 i d x \sin (c) \cos ^2(c)-64 \cos (c) \left (24 i d x \sin (4 c)+24 d x \cos (4 c)+\sin ^2(c) (72 d x-\sin (6 d x)-i \cos (6 d x))\right )+1536 i \sin (c) (\cos (4 c)+i \sin (4 c)) \tan ^{-1}(\tan (d x))+i \left (-1536 d x \sin ^3(c)+1560 i d x \sin (4 c) \sin (c)+864 i \sin (2 c) \sin (c) \sin (2 d x)+180 \sin (c) \sin (4 d x)-3 i \sin (4 c) \sin (c) \sin (8 d x)-768 \sin (4 c) \sin (c) \log \left (\sin ^2(c+d x)\right )+1560 d x \sin (c) \cos (4 c)+864 i \sin (c) \cos (2 c) \cos (2 d x)+180 i \sin (c) \cos (4 d x)+32 i \sin (c) \cos (2 c) \cos (6 d x)+3 i \sin (c) \cos (4 c) \cos (8 d x)-864 \sin (2 c) \sin (c) \cos (2 d x)+3 \sin (4 c) \sin (c) \cos (8 d x)+864 \sin (c) \cos (2 c) \sin (2 d x)+32 \sin (c) \cos (2 c) \sin (6 d x)+3 \sin (c) \cos (4 c) \sin (8 d x)-192 i \cos (4 c-d x) \csc (c+d x)+192 i \cos (4 c+d x) \csc (c+d x)+192 \sin (4 c-d x) \csc (c+d x)-192 \sin (4 c+d x) \csc (c+d x)+768 i \sin (c) \cos (4 c) \log \left (\sin ^2(c+d x)\right )\right )\right )}{384 a^4 d (\tan (c+d x)-i)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^4,x]

[Out]

((I/384)*Csc[c]*Sec[c + d*x]^4*(Cos[d*x] + I*Sin[d*x])^4*(1536*d*x*Cos[c]^3 + (4608*I)*d*x*Cos[c]^2*Sin[c] + (
1536*I)*ArcTan[Tan[d*x]]*Sin[c]*(Cos[4*c] + I*Sin[4*c]) - 64*Cos[c]*(24*d*x*Cos[4*c] + (24*I)*d*x*Sin[4*c] + S
in[c]^2*(72*d*x - I*Cos[6*d*x] - Sin[6*d*x])) + I*((-192*I)*Cos[4*c - d*x]*Csc[c + d*x] + (192*I)*Cos[4*c + d*
x]*Csc[c + d*x] + 1560*d*x*Cos[4*c]*Sin[c] + (864*I)*Cos[2*c]*Cos[2*d*x]*Sin[c] + (180*I)*Cos[4*d*x]*Sin[c] +
(32*I)*Cos[2*c]*Cos[6*d*x]*Sin[c] + (3*I)*Cos[4*c]*Cos[8*d*x]*Sin[c] + (768*I)*Cos[4*c]*Log[Sin[c + d*x]^2]*Si
n[c] - 1536*d*x*Sin[c]^3 - 864*Cos[2*d*x]*Sin[c]*Sin[2*c] + (1560*I)*d*x*Sin[c]*Sin[4*c] + 3*Cos[8*d*x]*Sin[c]
*Sin[4*c] - 768*Log[Sin[c + d*x]^2]*Sin[c]*Sin[4*c] + 864*Cos[2*c]*Sin[c]*Sin[2*d*x] + (864*I)*Sin[c]*Sin[2*c]
*Sin[2*d*x] + 180*Sin[c]*Sin[4*d*x] + 32*Cos[2*c]*Sin[c]*Sin[6*d*x] + 3*Cos[4*c]*Sin[c]*Sin[8*d*x] - (3*I)*Sin
[c]*Sin[4*c]*Sin[8*d*x] + 192*Csc[c + d*x]*Sin[4*c - d*x] - 192*Csc[c + d*x]*Sin[4*c + d*x])))/(a^4*d*(-I + Ta
n[c + d*x])^4)

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Maple [A]  time = 0.082, size = 150, normalized size = 0.9 \begin{align*}{\frac{{\frac{17\,i}{16}}}{d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{{\frac{i}{8}}}{d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}+{\frac{{\frac{129\,i}{32}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{d{a}^{4}}}+{\frac{5}{12\,d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{49}{16\,d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{{\frac{i}{32}}\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{d{a}^{4}}}-{\frac{1}{d{a}^{4}\tan \left ( dx+c \right ) }}-{\frac{4\,i\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2/(a+I*a*tan(d*x+c))^4,x)

[Out]

17/16*I/d/a^4/(tan(d*x+c)-I)^2-1/8*I/d/a^4/(tan(d*x+c)-I)^4+129/32*I/d/a^4*ln(tan(d*x+c)-I)+5/12/d/a^4/(tan(d*
x+c)-I)^3-49/16/d/a^4/(tan(d*x+c)-I)-1/32*I/d/a^4*ln(tan(d*x+c)+I)-1/d/a^4/tan(d*x+c)-4*I/d/a^4*ln(tan(d*x+c))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2/(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.50117, size = 435, normalized size = 2.74 \begin{align*} -\frac{3096 \, d x e^{\left (10 i \, d x + 10 i \, c\right )} -{\left (3096 \, d x - 1632 i\right )} e^{\left (8 i \, d x + 8 i \, c\right )} -{\left (-1536 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 1536 i \, e^{\left (8 i \, d x + 8 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 684 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 148 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 29 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i}{384 \,{\left (a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} - a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2/(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/384*(3096*d*x*e^(10*I*d*x + 10*I*c) - (3096*d*x - 1632*I)*e^(8*I*d*x + 8*I*c) - (-1536*I*e^(10*I*d*x + 10*I
*c) + 1536*I*e^(8*I*d*x + 8*I*c))*log(e^(2*I*d*x + 2*I*c) - 1) - 684*I*e^(6*I*d*x + 6*I*c) - 148*I*e^(4*I*d*x
+ 4*I*c) - 29*I*e^(2*I*d*x + 2*I*c) - 3*I)/(a^4*d*e^(10*I*d*x + 10*I*c) - a^4*d*e^(8*I*d*x + 8*I*c))

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Sympy [A]  time = 19.7394, size = 197, normalized size = 1.24 \begin{align*} - \frac{\left (\begin{cases} 129 x e^{8 i c} + \frac{36 i e^{6 i c} e^{- 2 i d x}}{d} + \frac{15 i e^{4 i c} e^{- 4 i d x}}{2 d} + \frac{4 i e^{2 i c} e^{- 6 i d x}}{3 d} + \frac{i e^{- 8 i d x}}{8 d} & \text{for}\: d \neq 0 \\x \left (129 e^{8 i c} + 72 e^{6 i c} + 30 e^{4 i c} + 8 e^{2 i c} + 1\right ) & \text{otherwise} \end{cases}\right ) e^{- 8 i c}}{16 a^{4}} - \frac{4 i \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{4} d} - \frac{2 i e^{- 2 i c}}{a^{4} d \left (e^{2 i d x} - e^{- 2 i c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2/(a+I*a*tan(d*x+c))**4,x)

[Out]

-Piecewise((129*x*exp(8*I*c) + 36*I*exp(6*I*c)*exp(-2*I*d*x)/d + 15*I*exp(4*I*c)*exp(-4*I*d*x)/(2*d) + 4*I*exp
(2*I*c)*exp(-6*I*d*x)/(3*d) + I*exp(-8*I*d*x)/(8*d), Ne(d, 0)), (x*(129*exp(8*I*c) + 72*exp(6*I*c) + 30*exp(4*
I*c) + 8*exp(2*I*c) + 1), True))*exp(-8*I*c)/(16*a**4) - 4*I*log(exp(2*I*d*x) - exp(-2*I*c))/(a**4*d) - 2*I*ex
p(-2*I*c)/(a**4*d*(exp(2*I*d*x) - exp(-2*I*c)))

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Giac [A]  time = 1.34252, size = 174, normalized size = 1.09 \begin{align*} -\frac{\frac{1536 i \, \log \left (-i \, \tan \left (d x + c\right )\right )}{a^{4}} + \frac{12 i \, \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{4}} - \frac{1548 i \, \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{4}} + \frac{384 \,{\left (-4 i \, \tan \left (d x + c\right ) + 1\right )}}{a^{4} \tan \left (d x + c\right )} + \frac{3225 i \, \tan \left (d x + c\right )^{4} + 14076 \, \tan \left (d x + c\right )^{3} - 23286 i \, \tan \left (d x + c\right )^{2} - 17404 \, \tan \left (d x + c\right ) + 5017 i}{a^{4}{\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2/(a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/384*(1536*I*log(-I*tan(d*x + c))/a^4 + 12*I*log(I*tan(d*x + c) - 1)/a^4 - 1548*I*log(-I*tan(d*x + c) - 1)/a
^4 + 384*(-4*I*tan(d*x + c) + 1)/(a^4*tan(d*x + c)) + (3225*I*tan(d*x + c)^4 + 14076*tan(d*x + c)^3 - 23286*I*
tan(d*x + c)^2 - 17404*tan(d*x + c) + 5017*I)/(a^4*(tan(d*x + c) - I)^4))/d