3.74 \(\int \frac{\cot ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=133 \[ -\frac{25 \cot (c+d x)}{8 a^3 d}-\frac{3 i \log (\sin (c+d x))}{a^3 d}+\frac{3 \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{25 x}{8 a^3}+\frac{11 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3} \]

[Out]

(-25*x)/(8*a^3) - (25*Cot[c + d*x])/(8*a^3*d) - ((3*I)*Log[Sin[c + d*x]])/(a^3*d) + Cot[c + d*x]/(6*d*(a + I*a
*Tan[c + d*x])^3) + (11*Cot[c + d*x])/(24*a*d*(a + I*a*Tan[c + d*x])^2) + (3*Cot[c + d*x])/(2*d*(a^3 + I*a^3*T
an[c + d*x]))

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Rubi [A]  time = 0.320908, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{25 \cot (c+d x)}{8 a^3 d}-\frac{3 i \log (\sin (c+d x))}{a^3 d}+\frac{3 \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{25 x}{8 a^3}+\frac{11 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(-25*x)/(8*a^3) - (25*Cot[c + d*x])/(8*a^3*d) - ((3*I)*Log[Sin[c + d*x]])/(a^3*d) + Cot[c + d*x]/(6*d*(a + I*a
*Tan[c + d*x])^3) + (11*Cot[c + d*x])/(24*a*d*(a + I*a*Tan[c + d*x])^2) + (3*Cot[c + d*x])/(2*d*(a^3 + I*a^3*T
an[c + d*x]))

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))^3} \, dx &=\frac{\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot ^2(c+d x) (7 a-4 i a \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{\int \frac{\cot ^2(c+d x) \left (39 a^2-33 i a^2 \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=\frac{\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{3 \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\int \cot ^2(c+d x) \left (150 a^3-144 i a^3 \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac{25 \cot (c+d x)}{8 a^3 d}+\frac{\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{3 \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\int \cot (c+d x) \left (-144 i a^3-150 a^3 \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac{25 x}{8 a^3}-\frac{25 \cot (c+d x)}{8 a^3 d}+\frac{\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{3 \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{(3 i) \int \cot (c+d x) \, dx}{a^3}\\ &=-\frac{25 x}{8 a^3}-\frac{25 \cot (c+d x)}{8 a^3 d}-\frac{3 i \log (\sin (c+d x))}{a^3 d}+\frac{\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{3 \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 3.94513, size = 379, normalized size = 2.85 \[ \frac{\sec ^3(c+d x) (\cos (d x)+i \sin (d x))^3 \left (288 d x \sin (c)+300 d x \sin (3 c)+138 \sin (c) \sin (2 d x)-21 \sin (c) \sin (4 d x)-2 \sin (3 c) \sin (6 d x)-300 i d x \cos (3 c)+2 \cos (3 c) \cos (6 d x)+144 i \sin (3 c) \log \left (\sin ^2(c+d x)\right )-2 i \cos (3 c) \sin (6 d x)+138 i \sin (c) \cos (2 d x)-21 i \sin (c) \cos (4 d x)-2 i \sin (3 c) \cos (6 d x)+288 d x \cos (3 c) \cot (c)+288 i d x \sin (3 c) \cot (c)-48 i \csc (c) \sin (3 c-d x) \csc (c+d x)+48 i \csc (c) \sin (3 c+d x) \csc (c+d x)+144 \cos (3 c) \log \left (\sin ^2(c+d x)\right )+288 (\sin (3 c)-i \cos (3 c)) \tan ^{-1}(\tan (d x))-3 \cos (c) (96 d x \cot (c)+192 i d x+46 i \sin (2 d x)+7 i \sin (4 d x)-46 \cos (2 d x)-7 \cos (4 d x))-24 \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \cos (3 c-d x) \csc (c+d x)+24 \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \cos (3 c+d x) \csc (c+d x)\right )}{96 a^3 d (\tan (c+d x)-i)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^3*((-300*I)*d*x*Cos[3*c] + 2*Cos[3*c]*Cos[6*d*x] + 288*d*x*Cos[3*c]*Co
t[c] + 144*Cos[3*c]*Log[Sin[c + d*x]^2] - 24*Cos[3*c - d*x]*Csc[c/2]*Csc[c + d*x]*Sec[c/2] + 24*Cos[3*c + d*x]
*Csc[c/2]*Csc[c + d*x]*Sec[c/2] + 288*d*x*Sin[c] + (138*I)*Cos[2*d*x]*Sin[c] - (21*I)*Cos[4*d*x]*Sin[c] + 300*
d*x*Sin[3*c] - (2*I)*Cos[6*d*x]*Sin[3*c] + (288*I)*d*x*Cot[c]*Sin[3*c] + (144*I)*Log[Sin[c + d*x]^2]*Sin[3*c]
+ 288*ArcTan[Tan[d*x]]*((-I)*Cos[3*c] + Sin[3*c]) + 138*Sin[c]*Sin[2*d*x] - 3*Cos[c]*((192*I)*d*x - 46*Cos[2*d
*x] - 7*Cos[4*d*x] + 96*d*x*Cot[c] + (46*I)*Sin[2*d*x] + (7*I)*Sin[4*d*x]) - 21*Sin[c]*Sin[4*d*x] - (2*I)*Cos[
3*c]*Sin[6*d*x] - 2*Sin[3*c]*Sin[6*d*x] - (48*I)*Csc[c]*Csc[c + d*x]*Sin[3*c - d*x] + (48*I)*Csc[c]*Csc[c + d*
x]*Sin[3*c + d*x]))/(96*a^3*d*(-I + Tan[c + d*x])^3)

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Maple [A]  time = 0.079, size = 130, normalized size = 1. \begin{align*}{\frac{{\frac{5\,i}{8}}}{d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+{\frac{{\frac{49\,i}{16}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{d{a}^{3}}}+{\frac{1}{6\,d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{17}{8\,d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{{\frac{i}{16}}\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{d{a}^{3}}}-{\frac{1}{d{a}^{3}\tan \left ( dx+c \right ) }}-{\frac{3\,i\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{3}}} \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))^3,x)

[Out]

5/8*I/d/a^3/(tan(d*x+c)-I)^2+49/16*I/d/a^3*ln(tan(d*x+c)-I)+1/6/d/a^3/(tan(d*x+c)-I)^3-17/8/a^3/d/(tan(d*x+c)-
I)-1/16*I/d/a^3*ln(tan(d*x+c)+I)-1/d/a^3/tan(d*x+c)-3*I/d/a^3*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))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.28203, size = 381, normalized size = 2.86 \begin{align*} -\frac{588 \, d x e^{\left (8 i \, d x + 8 i \, c\right )} -{\left (588 \, d x - 330 i\right )} e^{\left (6 i \, d x + 6 i \, c\right )} -{\left (-288 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 288 i \, e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 117 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 19 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i}{96 \,{\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} - a^{3} d e^{\left (6 i \, d x + 6 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))^3,x, algorithm="fricas")

[Out]

-1/96*(588*d*x*e^(8*I*d*x + 8*I*c) - (588*d*x - 330*I)*e^(6*I*d*x + 6*I*c) - (-288*I*e^(8*I*d*x + 8*I*c) + 288
*I*e^(6*I*d*x + 6*I*c))*log(e^(2*I*d*x + 2*I*c) - 1) - 117*I*e^(4*I*d*x + 4*I*c) - 19*I*e^(2*I*d*x + 2*I*c) -
2*I)/(a^3*d*e^(8*I*d*x + 8*I*c) - a^3*d*e^(6*I*d*x + 6*I*c))

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Sympy [A]  time = 10.4019, size = 168, normalized size = 1.26 \begin{align*} - \frac{\left (\begin{cases} 49 x e^{6 i c} + \frac{23 i e^{4 i c} e^{- 2 i d x}}{2 d} + \frac{7 i e^{2 i c} e^{- 4 i d x}}{4 d} + \frac{i e^{- 6 i d x}}{6 d} & \text{for}\: d \neq 0 \\x \left (49 e^{6 i c} + 23 e^{4 i c} + 7 e^{2 i c} + 1\right ) & \text{otherwise} \end{cases}\right ) e^{- 6 i c}}{8 a^{3}} - \frac{3 i \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{3} d} - \frac{2 i e^{- 2 i c}}{a^{3} 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))**3,x)

[Out]

-Piecewise((49*x*exp(6*I*c) + 23*I*exp(4*I*c)*exp(-2*I*d*x)/(2*d) + 7*I*exp(2*I*c)*exp(-4*I*d*x)/(4*d) + I*exp
(-6*I*d*x)/(6*d), Ne(d, 0)), (x*(49*exp(6*I*c) + 23*exp(4*I*c) + 7*exp(2*I*c) + 1), True))*exp(-6*I*c)/(8*a**3
) - 3*I*log(exp(2*I*d*x) - exp(-2*I*c))/(a**3*d) - 2*I*exp(-2*I*c)/(a**3*d*(exp(2*I*d*x) - exp(-2*I*c)))

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Giac [A]  time = 1.40308, size = 162, normalized size = 1.22 \begin{align*} -\frac{-\frac{294 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{6 i \, \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} + \frac{288 i \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} + \frac{96 \,{\left (-3 i \, \tan \left (d x + c\right ) + 1\right )}}{a^{3} \tan \left (d x + c\right )} + \frac{539 \, \tan \left (d x + c\right )^{3} - 1821 i \, \tan \left (d x + c\right )^{2} - 2085 \, \tan \left (d x + c\right ) + 819 i}{a^{3}{\left (i \, \tan \left (d x + c\right ) + 1\right )}^{3}}}{96 \, 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))^3,x, algorithm="giac")

[Out]

-1/96*(-294*I*log(I*tan(d*x + c) + 1)/a^3 + 6*I*log(I*tan(d*x + c) - 1)/a^3 + 288*I*log(abs(tan(d*x + c)))/a^3
+ 96*(-3*I*tan(d*x + c) + 1)/(a^3*tan(d*x + c)) + (539*tan(d*x + c)^3 - 1821*I*tan(d*x + c)^2 - 2085*tan(d*x
+ c) + 819*I)/(a^3*(I*tan(d*x + c) + 1)^3))/d