3.64 \(\int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=162 \[ \frac{15 A+i B}{16 a^4 d (1+i \tan (c+d x))}+\frac{7 A+i B}{16 a^4 d (1+i \tan (c+d x))^2}-\frac{x (-B+15 i A)}{16 a^4}+\frac{A \log (\sin (c+d x))}{a^4 d}+\frac{A+i B}{8 d (a+i a \tan (c+d x))^4}+\frac{3 A+i B}{12 a d (a+i a \tan (c+d x))^3} \]

[Out]

-(((15*I)*A - B)*x)/(16*a^4) + (A*Log[Sin[c + d*x]])/(a^4*d) + (7*A + I*B)/(16*a^4*d*(1 + I*Tan[c + d*x])^2) +
(15*A + I*B)/(16*a^4*d*(1 + I*Tan[c + d*x])) + (A + I*B)/(8*d*(a + I*a*Tan[c + d*x])^4) + (3*A + I*B)/(12*a*d
*(a + I*a*Tan[c + d*x])^3)

________________________________________________________________________________________

Rubi [A]  time = 0.494243, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {3596, 3531, 3475} \[ \frac{15 A+i B}{16 a^4 d (1+i \tan (c+d x))}+\frac{7 A+i B}{16 a^4 d (1+i \tan (c+d x))^2}-\frac{x (-B+15 i A)}{16 a^4}+\frac{A \log (\sin (c+d x))}{a^4 d}+\frac{A+i B}{8 d (a+i a \tan (c+d x))^4}+\frac{3 A+i B}{12 a d (a+i a \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((15*I)*A - B)*x)/(16*a^4) + (A*Log[Sin[c + d*x]])/(a^4*d) + (7*A + I*B)/(16*a^4*d*(1 + I*Tan[c + d*x])^2) +
(15*A + I*B)/(16*a^4*d*(1 + I*Tan[c + d*x])) + (A + I*B)/(8*d*(a + I*a*Tan[c + d*x])^4) + (3*A + I*B)/(12*a*d
*(a + I*a*Tan[c + d*x])^3)

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 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 (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx &=\frac{A+i B}{8 d (a+i a \tan (c+d x))^4}+\frac{\int \frac{\cot (c+d x) (8 a A-4 a (i A-B) \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx}{8 a^2}\\ &=\frac{A+i B}{8 d (a+i a \tan (c+d x))^4}+\frac{3 A+i B}{12 a d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot (c+d x) \left (48 a^2 A-12 a^2 (3 i A-B) \tan (c+d x)\right )}{(a+i a \tan (c+d x))^2} \, dx}{48 a^4}\\ &=\frac{7 A+i B}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{A+i B}{8 d (a+i a \tan (c+d x))^4}+\frac{3 A+i B}{12 a d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot (c+d x) \left (192 a^3 A-24 a^3 (7 i A-B) \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{192 a^6}\\ &=\frac{7 A+i B}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{A+i B}{8 d (a+i a \tan (c+d x))^4}+\frac{3 A+i B}{12 a d (a+i a \tan (c+d x))^3}+\frac{15 A+i B}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\int \cot (c+d x) \left (384 a^4 A-24 a^4 (15 i A-B) \tan (c+d x)\right ) \, dx}{384 a^8}\\ &=-\frac{(15 i A-B) x}{16 a^4}+\frac{7 A+i B}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{A+i B}{8 d (a+i a \tan (c+d x))^4}+\frac{3 A+i B}{12 a d (a+i a \tan (c+d x))^3}+\frac{15 A+i B}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{A \int \cot (c+d x) \, dx}{a^4}\\ &=-\frac{(15 i A-B) x}{16 a^4}+\frac{A \log (\sin (c+d x))}{a^4 d}+\frac{7 A+i B}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{A+i B}{8 d (a+i a \tan (c+d x))^4}+\frac{3 A+i B}{12 a d (a+i a \tan (c+d x))^3}+\frac{15 A+i B}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.12394, size = 193, normalized size = 1.19 \[ \frac{\sec ^4(c+d x) (16 (21 A+4 i B) \cos (2 (c+d x))+3 \cos (4 (c+d x)) (128 A \log (\sin (c+d x))-120 i A d x+A+8 B d x+i B)+288 i A \sin (2 (c+d x))+360 A d x \sin (4 (c+d x))-3 i A \sin (4 (c+d x))+384 i A \sin (4 (c+d x)) \log (\sin (c+d x))+96 A-32 B \sin (2 (c+d x))+3 B \sin (4 (c+d x))+24 i B d x \sin (4 (c+d x))+36 i B)}{384 a^4 d (\tan (c+d x)-i)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c + d*x]^4*(96*A + (36*I)*B + 16*(21*A + (4*I)*B)*Cos[2*(c + d*x)] + 3*Cos[4*(c + d*x)]*(A + I*B - (120*I
)*A*d*x + 8*B*d*x + 128*A*Log[Sin[c + d*x]]) + (288*I)*A*Sin[2*(c + d*x)] - 32*B*Sin[2*(c + d*x)] - (3*I)*A*Si
n[4*(c + d*x)] + 3*B*Sin[4*(c + d*x)] + 360*A*d*x*Sin[4*(c + d*x)] + (24*I)*B*d*x*Sin[4*(c + d*x)] + (384*I)*A
*Log[Sin[c + d*x]]*Sin[4*(c + d*x)]))/(384*a^4*d*(-I + Tan[c + d*x])^4)

________________________________________________________________________________________

Maple [A]  time = 0.122, size = 259, normalized size = 1.6 \begin{align*}{\frac{A}{8\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}+{\frac{{\frac{i}{8}}B}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}-{\frac{{\frac{i}{32}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) B}{{a}^{4}d}}-{\frac{31\,\ln \left ( \tan \left ( dx+c \right ) -i \right ) A}{32\,{a}^{4}d}}-{\frac{{\frac{15\,i}{16}}A}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{B}{16\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{B}{12\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{{\frac{i}{4}}A}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{7\,A}{16\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{{\frac{i}{16}}B}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{A\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{32\,{a}^{4}d}}+{\frac{{\frac{i}{32}}B\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{{a}^{4}d}}+{\frac{A\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{4}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/d/a^4/(tan(d*x+c)-I)^4*A+1/8*I/d/a^4/(tan(d*x+c)-I)^4*B-1/32*I/d/a^4*ln(tan(d*x+c)-I)*B-31/32/d/a^4*ln(tan
(d*x+c)-I)*A-15/16*I/d/a^4/(tan(d*x+c)-I)*A+1/16/d/a^4/(tan(d*x+c)-I)*B-1/12/d/a^4/(tan(d*x+c)-I)^3*B+1/4*I/d/
a^4/(tan(d*x+c)-I)^3*A-7/16/d/a^4/(tan(d*x+c)-I)^2*A-1/16*I/d/a^4/(tan(d*x+c)-I)^2*B-1/32/d/a^4*A*ln(tan(d*x+c
)+I)+1/32*I/d/a^4*B*ln(tan(d*x+c)+I)+1/d/a^4*A*ln(tan(d*x+c))

________________________________________________________________________________________

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)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 1.53233, size = 362, normalized size = 2.23 \begin{align*} \frac{{\left ({\left (-744 i \, A + 24 \, B\right )} d x e^{\left (8 i \, d x + 8 i \, c\right )} + 384 \, A e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) + 24 \,{\left (13 \, A + 2 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 12 \,{\left (8 \, A + 3 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \,{\left (3 \, A + 2 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 3 \, A + 3 i \, B\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{384 \, a^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*((-744*I*A + 24*B)*d*x*e^(8*I*d*x + 8*I*c) + 384*A*e^(8*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) - 1) + 24
*(13*A + 2*I*B)*e^(6*I*d*x + 6*I*c) + 12*(8*A + 3*I*B)*e^(4*I*d*x + 4*I*c) + 8*(3*A + 2*I*B)*e^(2*I*d*x + 2*I*
c) + 3*A + 3*I*B)*e^(-8*I*d*x - 8*I*c)/(a^4*d)

________________________________________________________________________________________

Sympy [A]  time = 23.5548, size = 360, normalized size = 2.22 \begin{align*} \frac{A \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{4} d} + \begin{cases} \frac{\left (\left (24576 A a^{12} d^{3} e^{12 i c} + 24576 i B a^{12} d^{3} e^{12 i c}\right ) e^{- 8 i d x} + \left (196608 A a^{12} d^{3} e^{14 i c} + 131072 i B a^{12} d^{3} e^{14 i c}\right ) e^{- 6 i d x} + \left (786432 A a^{12} d^{3} e^{16 i c} + 294912 i B a^{12} d^{3} e^{16 i c}\right ) e^{- 4 i d x} + \left (2555904 A a^{12} d^{3} e^{18 i c} + 393216 i B a^{12} d^{3} e^{18 i c}\right ) e^{- 2 i d x}\right ) e^{- 20 i c}}{3145728 a^{16} d^{4}} & \text{for}\: 3145728 a^{16} d^{4} e^{20 i c} \neq 0 \\x \left (\frac{31 i A - B}{16 a^{4}} - \frac{\left (31 i A e^{8 i c} + 26 i A e^{6 i c} + 16 i A e^{4 i c} + 6 i A e^{2 i c} + i A - B e^{8 i c} - 4 B e^{6 i c} - 6 B e^{4 i c} - 4 B e^{2 i c} - B\right ) e^{- 8 i c}}{16 a^{4}}\right ) & \text{otherwise} \end{cases} + \frac{x \left (- 31 i A + B\right )}{16 a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*log(exp(2*I*d*x) - exp(-2*I*c))/(a**4*d) + Piecewise((((24576*A*a**12*d**3*exp(12*I*c) + 24576*I*B*a**12*d**
3*exp(12*I*c))*exp(-8*I*d*x) + (196608*A*a**12*d**3*exp(14*I*c) + 131072*I*B*a**12*d**3*exp(14*I*c))*exp(-6*I*
d*x) + (786432*A*a**12*d**3*exp(16*I*c) + 294912*I*B*a**12*d**3*exp(16*I*c))*exp(-4*I*d*x) + (2555904*A*a**12*
d**3*exp(18*I*c) + 393216*I*B*a**12*d**3*exp(18*I*c))*exp(-2*I*d*x))*exp(-20*I*c)/(3145728*a**16*d**4), Ne(314
5728*a**16*d**4*exp(20*I*c), 0)), (x*((31*I*A - B)/(16*a**4) - (31*I*A*exp(8*I*c) + 26*I*A*exp(6*I*c) + 16*I*A
*exp(4*I*c) + 6*I*A*exp(2*I*c) + I*A - B*exp(8*I*c) - 4*B*exp(6*I*c) - 6*B*exp(4*I*c) - 4*B*exp(2*I*c) - B)*ex
p(-8*I*c)/(16*a**4)), True)) + x*(-31*I*A + B)/(16*a**4)

________________________________________________________________________________________

Giac [A]  time = 1.34165, size = 224, normalized size = 1.38 \begin{align*} -\frac{\frac{12 \,{\left (A - i \, B\right )} \log \left (\tan \left (d x + c\right ) + i\right )}{a^{4}} + \frac{12 \,{\left (31 \, A + i \, B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a^{4}} - \frac{384 \, A \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac{775 \, A \tan \left (d x + c\right )^{4} + 25 i \, B \tan \left (d x + c\right )^{4} - 3460 i \, A \tan \left (d x + c\right )^{3} + 124 \, B \tan \left (d x + c\right )^{3} - 5898 \, A \tan \left (d x + c\right )^{2} - 246 i \, B \tan \left (d x + c\right )^{2} + 4612 i \, A \tan \left (d x + c\right ) - 252 \, B \tan \left (d x + c\right ) + 1447 \, A + 153 i \, B}{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)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/384*(12*(A - I*B)*log(tan(d*x + c) + I)/a^4 + 12*(31*A + I*B)*log(tan(d*x + c) - I)/a^4 - 384*A*log(abs(tan
(d*x + c)))/a^4 - (775*A*tan(d*x + c)^4 + 25*I*B*tan(d*x + c)^4 - 3460*I*A*tan(d*x + c)^3 + 124*B*tan(d*x + c)
^3 - 5898*A*tan(d*x + c)^2 - 246*I*B*tan(d*x + c)^2 + 4612*I*A*tan(d*x + c) - 252*B*tan(d*x + c) + 1447*A + 15
3*I*B)/(a^4*(tan(d*x + c) - I)^4))/d