\(\int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx\) [65]

Optimal result

Integrand size = 34, antiderivative size = 220 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=-\frac {5 (13 A+3 i B) x}{16 a^4}-\frac {5 (13 A+3 i B) \cot (c+d x)}{16 a^4 d}-\frac {(4 i A-B) \log (\sin (c+d x))}{a^4 d}+\frac {(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {(4 A+i B) \cot (c+d x)}{2 a^4 d (1+i \tan (c+d x))}+\frac {(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3} \]

[Out]

-5/16*(13*A+3*I*B)*x/a^4-5/16*(13*A+3*I*B)*cot(d*x+c)/a^4/d-(4*I*A-B)*ln(sin(d*x+c))/a^4/d+1/48*(31*A+9*I*B)*c
ot(d*x+c)/a^4/d/(1+I*tan(d*x+c))^2+1/2*(4*A+I*B)*cot(d*x+c)/a^4/d/(1+I*tan(d*x+c))+1/8*(A+I*B)*cot(d*x+c)/d/(a
+I*a*tan(d*x+c))^4+1/24*(7*A+3*I*B)*cot(d*x+c)/a/d/(a+I*a*tan(d*x+c))^3

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3677, 3610, 3612, 3556} \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=-\frac {5 (13 A+3 i B) \cot (c+d x)}{16 a^4 d}-\frac {(-B+4 i A) \log (\sin (c+d x))}{a^4 d}+\frac {(4 A+i B) \cot (c+d x)}{2 a^4 d (1+i \tan (c+d x))}+\frac {(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}-\frac {5 x (13 A+3 i B)}{16 a^4}+\frac {(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4} \]

[In]

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

[Out]

(-5*(13*A + (3*I)*B)*x)/(16*a^4) - (5*(13*A + (3*I)*B)*Cot[c + d*x])/(16*a^4*d) - (((4*I)*A - B)*Log[Sin[c + d
*x]])/(a^4*d) + ((31*A + (9*I)*B)*Cot[c + d*x])/(48*a^4*d*(1 + I*Tan[c + d*x])^2) + ((4*A + I*B)*Cot[c + d*x])
/(2*a^4*d*(1 + I*Tan[c + d*x])) + ((A + I*B)*Cot[c + d*x])/(8*d*(a + I*a*Tan[c + d*x])^4) + ((7*A + (3*I)*B)*C
ot[c + d*x])/(24*a*d*(a + I*a*Tan[c + d*x])^3)

Rule 3556

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

Rule 3610

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 3612

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 3677

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]

Rubi steps \begin{align*} \text {integral}& = \frac {(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {\int \frac {\cot ^2(c+d x) (a (9 A+i B)-5 a (i A-B) \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx}{8 a^2} \\ & = \frac {(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {\int \frac {\cot ^2(c+d x) \left (4 a^2 (17 A+3 i B)-8 a^2 (7 i A-3 B) \tan (c+d x)\right )}{(a+i a \tan (c+d x))^2} \, dx}{48 a^4} \\ & = \frac {(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {\int \frac {\cot ^2(c+d x) \left (12 a^3 (33 A+7 i B)-12 a^3 (31 i A-9 B) \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{192 a^6} \\ & = \frac {(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {(4 A+i B) \cot (c+d x)}{2 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\int \cot ^2(c+d x) \left (120 a^4 (13 A+3 i B)-384 a^4 (4 i A-B) \tan (c+d x)\right ) \, dx}{384 a^8} \\ & = -\frac {5 (13 A+3 i B) \cot (c+d x)}{16 a^4 d}+\frac {(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {(4 A+i B) \cot (c+d x)}{2 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\int \cot (c+d x) \left (-384 a^4 (4 i A-B)-120 a^4 (13 A+3 i B) \tan (c+d x)\right ) \, dx}{384 a^8} \\ & = -\frac {5 (13 A+3 i B) x}{16 a^4}-\frac {5 (13 A+3 i B) \cot (c+d x)}{16 a^4 d}+\frac {(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {(4 A+i B) \cot (c+d x)}{2 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {(4 i A-B) \int \cot (c+d x) \, dx}{a^4} \\ & = -\frac {5 (13 A+3 i B) x}{16 a^4}-\frac {5 (13 A+3 i B) \cot (c+d x)}{16 a^4 d}-\frac {(4 i A-B) \log (\sin (c+d x))}{a^4 d}+\frac {(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {(4 A+i B) \cot (c+d x)}{2 d \left (a^4+i a^4 \tan (c+d x)\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 3.58 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=\frac {\frac {6 (A+i B) \cot ^5(c+d x)}{(i+\cot (c+d x))^4}+\frac {2 (7 A+3 i B) \cot ^4(c+d x)}{(i+\cot (c+d x))^3}+\frac {(31 A+9 i B) \cot ^3(c+d x)}{(i+\cot (c+d x))^2}+\frac {24 (4 A+i B) \cot ^2(c+d x)}{i+\cot (c+d x)}-15 (13 A+3 i B) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(c+d x)\right )+48 (-4 i A+B) (\log (\cos (c+d x))+\log (\tan (c+d x)))}{48 a^4 d} \]

[In]

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

[Out]

((6*(A + I*B)*Cot[c + d*x]^5)/(I + Cot[c + d*x])^4 + (2*(7*A + (3*I)*B)*Cot[c + d*x]^4)/(I + Cot[c + d*x])^3 +
((31*A + (9*I)*B)*Cot[c + d*x]^3)/(I + Cot[c + d*x])^2 + (24*(4*A + I*B)*Cot[c + d*x]^2)/(I + Cot[c + d*x]) -
15*(13*A + (3*I)*B)*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 48*((-4*I)*A + B)*(Log[Co
s[c + d*x]] + Log[Tan[c + d*x]]))/(48*a^4*d)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.14

method	result	size
risch	\(-\frac {i {\mathrm e}^{-6 i \left (d x +c \right )} A}{12 d \,a^{4}}-\frac {129 x A}{16 a^{4}}+\frac {13 \,{\mathrm e}^{-2 i \left (d x +c \right )} B}{16 d \,a^{4}}-\frac {15 i {\mathrm e}^{-4 i \left (d x +c \right )} A}{32 d \,a^{4}}+\frac {{\mathrm e}^{-4 i \left (d x +c \right )} B}{4 d \,a^{4}}-\frac {2 i B c}{d \,a^{4}}+\frac {{\mathrm e}^{-6 i \left (d x +c \right )} B}{16 d \,a^{4}}-\frac {i {\mathrm e}^{-8 i \left (d x +c \right )} A}{128 d \,a^{4}}+\frac {{\mathrm e}^{-8 i \left (d x +c \right )} B}{128 d \,a^{4}}-\frac {2 i A}{a^{4} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {9 i {\mathrm e}^{-2 i \left (d x +c \right )} A}{4 d \,a^{4}}-\frac {4 i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A}{a^{4} d}-\frac {8 A c}{a^{4} d}-\frac {31 i x B}{16 a^{4}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{a^{4} d}\)	\(251\)
derivativedivides	\(-\frac {15 i B}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}+\frac {B}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {B \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{4}}+\frac {17 i A}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {i B}{4 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {65 A \arctan \left (\tan \left (d x +c \right )\right )}{16 d \,a^{4}}-\frac {7 B}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {i A}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {49 A}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}+\frac {2 i A \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \,a^{4}}+\frac {5 A}{12 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {4 i A \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}-\frac {15 i B \arctan \left (\tan \left (d x +c \right )\right )}{16 d \,a^{4}}+\frac {B \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}-\frac {A}{d \,a^{4} \tan \left (d x +c \right )}\)	\(289\)
default	\(-\frac {15 i B}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}+\frac {B}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {B \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{4}}+\frac {17 i A}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {i B}{4 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {65 A \arctan \left (\tan \left (d x +c \right )\right )}{16 d \,a^{4}}-\frac {7 B}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {i A}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {49 A}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}+\frac {2 i A \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \,a^{4}}+\frac {5 A}{12 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {4 i A \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}-\frac {15 i B \arctan \left (\tan \left (d x +c \right )\right )}{16 d \,a^{4}}+\frac {B \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}-\frac {A}{d \,a^{4} \tan \left (d x +c \right )}\)	\(289\)

[In]

int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

-1/12*I/d/a^4*exp(-6*I*(d*x+c))*A-129/16*x/a^4*A+13/16/d/a^4*exp(-2*I*(d*x+c))*B-15/32*I/d/a^4*exp(-4*I*(d*x+c
))*A+1/4/d/a^4*exp(-4*I*(d*x+c))*B-2*I/a^4/d*B*c+1/16/d/a^4*exp(-6*I*(d*x+c))*B-1/128*I/d/a^4*exp(-8*I*(d*x+c)
)*A+1/128/d/a^4*exp(-8*I*(d*x+c))*B-2*I*A/a^4/d/(exp(2*I*(d*x+c))-1)-9/4*I/d/a^4*exp(-2*I*(d*x+c))*A-4*I/a^4/d
*ln(exp(2*I*(d*x+c))-1)*A-8/a^4/d*A*c-31/16*I*x/a^4*B+1/a^4/d*ln(exp(2*I*(d*x+c))-1)*B

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=-\frac {24 \, {\left (129 \, A + 31 i \, B\right )} d x e^{\left (10 i \, d x + 10 i \, c\right )} - 24 \, {\left ({\left (129 \, A + 31 i \, B\right )} d x - 68 i \, A + 13 \, B\right )} e^{\left (8 i \, d x + 8 i \, c\right )} + 36 \, {\left (-19 i \, A + 6 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 4 \, {\left (-37 i \, A + 18 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (29 i \, A - 21 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 384 \, {\left ({\left (4 i \, A - B\right )} e^{\left (10 i \, d x + 10 i \, c\right )} + {\left (-4 i \, A + B\right )} e^{\left (8 i \, d x + 8 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 3 i \, A + 3 \, B}{384 \, {\left (a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} - a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )}\right )}} \]

[In]

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

[Out]

-1/384*(24*(129*A + 31*I*B)*d*x*e^(10*I*d*x + 10*I*c) - 24*((129*A + 31*I*B)*d*x - 68*I*A + 13*B)*e^(8*I*d*x +
8*I*c) + 36*(-19*I*A + 6*B)*e^(6*I*d*x + 6*I*c) + 4*(-37*I*A + 18*B)*e^(4*I*d*x + 4*I*c) - (29*I*A - 21*B)*e^
(2*I*d*x + 2*I*c) + 384*((4*I*A - B)*e^(10*I*d*x + 10*I*c) + (-4*I*A + B)*e^(8*I*d*x + 8*I*c))*log(e^(2*I*d*x
+ 2*I*c) - 1) - 3*I*A + 3*B)/(a^4*d*e^(10*I*d*x + 10*I*c) - a^4*d*e^(8*I*d*x + 8*I*c))

Sympy [A] (verification not implemented)

Time = 0.91 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.85 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=- \frac {2 i A}{a^{4} d e^{2 i c} e^{2 i d x} - a^{4} d} + \begin {cases} \frac {\left (\left (- 24576 i A a^{12} d^{3} e^{12 i c} + 24576 B a^{12} d^{3} e^{12 i c}\right ) e^{- 8 i d x} + \left (- 262144 i A a^{12} d^{3} e^{14 i c} + 196608 B a^{12} d^{3} e^{14 i c}\right ) e^{- 6 i d x} + \left (- 1474560 i A a^{12} d^{3} e^{16 i c} + 786432 B a^{12} d^{3} e^{16 i c}\right ) e^{- 4 i d x} + \left (- 7077888 i A a^{12} d^{3} e^{18 i c} + 2555904 B a^{12} d^{3} e^{18 i c}\right ) e^{- 2 i d x}\right ) e^{- 20 i c}}{3145728 a^{16} d^{4}} & \text {for}\: a^{16} d^{4} e^{20 i c} \neq 0 \\x \left (- \frac {- 129 A - 31 i B}{16 a^{4}} + \frac {\left (- 129 A e^{8 i c} - 72 A e^{6 i c} - 30 A e^{4 i c} - 8 A e^{2 i c} - A - 31 i B e^{8 i c} - 26 i B e^{6 i c} - 16 i B e^{4 i c} - 6 i B e^{2 i c} - i B\right ) e^{- 8 i c}}{16 a^{4}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (- 129 A - 31 i B\right )}{16 a^{4}} - \frac {i \left (4 A + i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{4} d} \]

[In]

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

[Out]

-2*I*A/(a**4*d*exp(2*I*c)*exp(2*I*d*x) - a**4*d) + Piecewise((((-24576*I*A*a**12*d**3*exp(12*I*c) + 24576*B*a*
*12*d**3*exp(12*I*c))*exp(-8*I*d*x) + (-262144*I*A*a**12*d**3*exp(14*I*c) + 196608*B*a**12*d**3*exp(14*I*c))*e
xp(-6*I*d*x) + (-1474560*I*A*a**12*d**3*exp(16*I*c) + 786432*B*a**12*d**3*exp(16*I*c))*exp(-4*I*d*x) + (-70778
88*I*A*a**12*d**3*exp(18*I*c) + 2555904*B*a**12*d**3*exp(18*I*c))*exp(-2*I*d*x))*exp(-20*I*c)/(3145728*a**16*d
**4), Ne(a**16*d**4*exp(20*I*c), 0)), (x*(-(-129*A - 31*I*B)/(16*a**4) + (-129*A*exp(8*I*c) - 72*A*exp(6*I*c)
- 30*A*exp(4*I*c) - 8*A*exp(2*I*c) - A - 31*I*B*exp(8*I*c) - 26*I*B*exp(6*I*c) - 16*I*B*exp(4*I*c) - 6*I*B*exp
(2*I*c) - I*B)*exp(-8*I*c)/(16*a**4)), True)) + x*(-129*A - 31*I*B)/(16*a**4) - I*(4*A + I*B)*log(exp(2*I*d*x)
- exp(-2*I*c))/(a**4*d)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 1.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.93 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=\frac {\frac {12 \, {\left (-i \, A - B\right )} \log \left (\tan \left (d x + c\right ) + i\right )}{a^{4}} - \frac {12 \, {\left (-129 i \, A + 31 \, B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a^{4}} - \frac {384 \, {\left (4 i \, A - B\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}} - \frac {384 \, {\left (-4 i \, A \tan \left (d x + c\right ) + B \tan \left (d x + c\right ) + A\right )}}{a^{4} \tan \left (d x + c\right )} - \frac {3225 i \, A \tan \left (d x + c\right )^{4} - 775 \, B \tan \left (d x + c\right )^{4} + 14076 \, A \tan \left (d x + c\right )^{3} + 3460 i \, B \tan \left (d x + c\right )^{3} - 23286 i \, A \tan \left (d x + c\right )^{2} + 5898 \, B \tan \left (d x + c\right )^{2} - 17404 \, A \tan \left (d x + c\right ) - 4612 i \, B \tan \left (d x + c\right ) + 5017 i \, A - 1447 \, B}{a^{4} {\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{384 \, d} \]

[In]

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

[Out]

1/384*(12*(-I*A - B)*log(tan(d*x + c) + I)/a^4 - 12*(-129*I*A + 31*B)*log(tan(d*x + c) - I)/a^4 - 384*(4*I*A -
B)*log(tan(d*x + c))/a^4 - 384*(-4*I*A*tan(d*x + c) + B*tan(d*x + c) + A)/(a^4*tan(d*x + c)) - (3225*I*A*tan(
d*x + c)^4 - 775*B*tan(d*x + c)^4 + 14076*A*tan(d*x + c)^3 + 3460*I*B*tan(d*x + c)^3 - 23286*I*A*tan(d*x + c)^
2 + 5898*B*tan(d*x + c)^2 - 17404*A*tan(d*x + c) - 4612*I*B*tan(d*x + c) + 5017*I*A - 1447*B)/(a^4*(tan(d*x +
c) - I)^4))/d

Mupad [B] (verification not implemented)

Time = 8.15 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.03 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=-\frac {\frac {A}{a^4}+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {65\,A}{16\,a^4}+\frac {B\,15{}\mathrm {i}}{16\,a^4}\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {851\,A}{48\,a^4}+\frac {B\,63{}\mathrm {i}}{16\,a^4}\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-\frac {13\,B}{4\,a^4}+\frac {A\,57{}\mathrm {i}}{4\,a^4}\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (-\frac {7\,B}{4\,a^4}+\frac {A\,26{}\mathrm {i}}{3\,a^4}\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^5-{\mathrm {tan}\left (c+d\,x\right )}^4\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (c+d\,x\right )}^3+{\mathrm {tan}\left (c+d\,x\right )}^2\,4{}\mathrm {i}+\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-B+A\,4{}\mathrm {i}\right )}{a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{32\,a^4\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-31\,B+A\,129{}\mathrm {i}\right )}{32\,a^4\,d} \]

[In]

int((cot(c + d*x)^2*(A + B*tan(c + d*x)))/(a + a*tan(c + d*x)*1i)^4,x)

[Out]

(log(tan(c + d*x) - 1i)*(A*129i - 31*B))/(32*a^4*d) - (log(tan(c + d*x))*(A*4i - B))/(a^4*d) - (log(tan(c + d*
x) + 1i)*(A*1i + B))/(32*a^4*d) - (tan(c + d*x)^4*((65*A)/(16*a^4) + (B*15i)/(16*a^4)) - tan(c + d*x)^3*((A*57
i)/(4*a^4) - (13*B)/(4*a^4)) - tan(c + d*x)^2*((851*A)/(48*a^4) + (B*63i)/(16*a^4)) + A/a^4 + tan(c + d*x)*((A
*26i)/(3*a^4) - (7*B)/(4*a^4)))/(d*(tan(c + d*x) + tan(c + d*x)^2*4i - 6*tan(c + d*x)^3 - tan(c + d*x)^4*4i +
tan(c + d*x)^5))