\(\int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx\) [529]

Optimal result

Integrand size = 36, antiderivative size = 319 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((2+7 i) A-(23+2 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {((9+5 i) A-(25-21 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {5 (i A-5 B)}{8 a^2 d \sqrt {\cot (c+d x)}}+\frac {3 A+7 i B}{8 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))}+\frac {i A-B}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((7+2 i) A+(2+23 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((7+2 i) A+(2+23 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^2 d} \]

[Out]

(1/32-1/32*I)*((2+7*I)*A-(23+2*I)*B)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+1/32*((9+5*I)*A+(-25+21
*I)*B)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+(-1/64+1/64*I)*((7+2*I)*A+(2+23*I)*B)*ln(1+cot(d*x+c)-
2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+(1/64-1/64*I)*((7+2*I)*A+(2+23*I)*B)*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c
)^(1/2))/a^2/d*2^(1/2)+5/8*(I*A-5*B)/a^2/d/cot(d*x+c)^(1/2)+1/8*(3*A+7*I*B)/a^2/d/(I+cot(d*x+c))/cot(d*x+c)^(1
/2)+1/4*(I*A-B)/d/(I*a+a*cot(d*x+c))^2/cot(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3662, 3677, 3610, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((2+7 i) A-(23+2 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {((9+5 i) A-(25-21 i) B) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{16 \sqrt {2} a^2 d}+\frac {5 (-5 B+i A)}{8 a^2 d \sqrt {\cot (c+d x)}}+\frac {3 A+7 i B}{8 a^2 d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((7+2 i) A+(2+23 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((7+2 i) A+(2+23 i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^2 d}+\frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2} \]

[In]

Int[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2),x]

[Out]

((-1/16 + I/16)*((2 + 7*I)*A - (23 + 2*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d) + (((9 +
5*I)*A - (25 - 21*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^2*d) + (5*(I*A - 5*B))/(8*a^2*d*
Sqrt[Cot[c + d*x]]) + (3*A + (7*I)*B)/(8*a^2*d*Sqrt[Cot[c + d*x]]*(I + Cot[c + d*x])) + (I*A - B)/(4*d*Sqrt[Co
t[c + d*x]]*(I*a + a*Cot[c + d*x])^2) - ((1/32 - I/32)*((7 + 2*I)*A + (2 + 23*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c
+ d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d) + ((1/32 - I/32)*((7 + 2*I)*A + (2 + 23*I)*B)*Log[1 + Sqrt[2]*Sqrt[C
ot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

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 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3662

Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.)
+ (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[g^(m + n), Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d
+ c*Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&  !IntegerQ[p] && IntegerQ[m] && IntegerQ
[n]

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}& = \int \frac {B+A \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))^2} \, dx \\ & = \frac {i A-B}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}+\frac {\int \frac {-\frac {1}{2} a (A+9 i B)-\frac {5}{2} a (i A-B) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))} \, dx}{4 a^2} \\ & = \frac {3 A+7 i B}{8 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))}+\frac {i A-B}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}+\frac {\int \frac {\frac {5}{2} a^2 (i A-5 B)-\frac {3}{2} a^2 (3 A+7 i B) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx}{8 a^4} \\ & = \frac {5 (i A-5 B)}{8 a^2 d \sqrt {\cot (c+d x)}}+\frac {3 A+7 i B}{8 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))}+\frac {i A-B}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}+\frac {\int \frac {-\frac {3}{2} a^2 (3 A+7 i B)-\frac {5}{2} a^2 (i A-5 B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{8 a^4} \\ & = \frac {5 (i A-5 B)}{8 a^2 d \sqrt {\cot (c+d x)}}+\frac {3 A+7 i B}{8 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))}+\frac {i A-B}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}+\frac {\text {Subst}\left (\int \frac {\frac {3}{2} a^2 (3 A+7 i B)+\frac {5}{2} a^2 (i A-5 B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 a^4 d} \\ & = \frac {5 (i A-5 B)}{8 a^2 d \sqrt {\cot (c+d x)}}+\frac {3 A+7 i B}{8 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))}+\frac {i A-B}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}+\frac {((9+5 i) A-(25-21 i) B) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{16 a^2 d}+\frac {((9-5 i) A+(25+21 i) B) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{16 a^2 d} \\ & = \frac {5 (i A-5 B)}{8 a^2 d \sqrt {\cot (c+d x)}}+\frac {3 A+7 i B}{8 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))}+\frac {i A-B}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}+\frac {((9+5 i) A-(25-21 i) B) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^2 d}+\frac {((9+5 i) A-(25-21 i) B) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^2 d}-\frac {((9-5 i) A+(25+21 i) B) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 \sqrt {2} a^2 d}-\frac {((9-5 i) A+(25+21 i) B) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 \sqrt {2} a^2 d} \\ & = \frac {5 (i A-5 B)}{8 a^2 d \sqrt {\cot (c+d x)}}+\frac {3 A+7 i B}{8 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))}+\frac {i A-B}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac {((9-5 i) A+(25+21 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {((9-5 i) A+(25+21 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {((9+5 i) A-(25-21 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {((9+5 i) A-(25-21 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d} \\ & = -\frac {((9+5 i) A-(25-21 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {((9+5 i) A-(25-21 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {5 (i A-5 B)}{8 a^2 d \sqrt {\cot (c+d x)}}+\frac {3 A+7 i B}{8 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))}+\frac {i A-B}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac {((9-5 i) A+(25+21 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {((9-5 i) A+(25+21 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.94 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.67 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-2 \sqrt [4]{-1} (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+(-1)^{3/4} (7 A+23 i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+\sqrt {\tan (c+d x)} \left (5 i (A+5 i B)-(7 A+43 i B) \tan (c+d x)+16 B \tan ^2(c+d x)\right )\right )}{8 a^2 d (-i+\tan (c+d x))^2} \]

[In]

Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2),x]

[Out]

-1/8*(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-2*(-1)^(1/4)*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sec
[c + d*x]^2*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + (-1)^(3/4)*(7*A + (23*I)*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[
c + d*x]]]*Sec[c + d*x]^2*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + Sqrt[Tan[c + d*x]]*((5*I)*(A + (5*I)*B) -
(7*A + (43*I)*B)*Tan[c + d*x] + 16*B*Tan[c + d*x]^2)))/(a^2*d*(-I + Tan[c + d*x])^2)

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.49

method	result	size
derivativedivides	\(\frac {-\frac {i \left (-i B +A \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}-\frac {2 B}{\sqrt {\cot \left (d x +c \right )}}+\frac {\left (\frac {5 i A}{2}-\frac {9 B}{2}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7 A}{2}-\frac {11 i B}{2}\right ) \sqrt {\cot \left (d x +c \right )}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {\left (7 i A -23 B \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{4 \sqrt {2}+4 i \sqrt {2}}}{a^{2} d}\)	\(157\)
default	\(\frac {-\frac {i \left (-i B +A \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}-\frac {2 B}{\sqrt {\cot \left (d x +c \right )}}+\frac {\left (\frac {5 i A}{2}-\frac {9 B}{2}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7 A}{2}-\frac {11 i B}{2}\right ) \sqrt {\cot \left (d x +c \right )}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {\left (7 i A -23 B \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{4 \sqrt {2}+4 i \sqrt {2}}}{a^{2} d}\)	\(157\)

[In]

int((A+B*tan(d*x+c))/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/a^2/d*(-1/2*I*(A-I*B)/(2^(1/2)-I*2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(2^(1/2)-I*2^(1/2)))-2*B/cot(d*x+c)^(1/2
)+1/4*((5/2*I*A-9/2*B)*cot(d*x+c)^(3/2)+(-7/2*A-11/2*I*B)*cot(d*x+c)^(1/2))/(I+cot(d*x+c))^2+1/4*(7*I*A-23*B)/
(2^(1/2)+I*2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(2^(1/2)+I*2^(1/2))))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 761 vs. \(2 (234) = 468\).

Time = 0.27 (sec) , antiderivative size = 761, normalized size of antiderivative = 2.39 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {2 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} \log \left (-\frac {2 \, {\left ({\left (i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} + {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) - 2 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} \log \left (-\frac {2 \, {\left ({\left (-i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} + {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) - {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} \log \left (\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} + 7 i \, A - 23 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) + {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} \log \left (-\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} - 7 i \, A + 23 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - 2 \, {\left (6 \, {\left (A + 7 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - {\left (A + 33 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, {\left (3 \, A + 5 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + A + i \, B\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{32 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]

[In]

integrate((A+B*tan(d*x+c))/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/32*(2*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^4*d^2))*log(-
2*((I*a^2*d*e^(2*I*d*x + 2*I*c) - I*a^2*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((I
*A^2 + 2*A*B - I*B^2)/(a^4*d^2)) + (A - I*B)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)) - 2*(a^2*d*e
^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^4*d^2))*log(-2*((-I*a^2*d*e^(2
*I*d*x + 2*I*c) + I*a^2*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((I*A^2 + 2*A*B - I
*B^2)/(a^4*d^2)) + (A - I*B)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)) - (a^2*d*e^(6*I*d*x + 6*I*c)
+ a^2*d*e^(4*I*d*x + 4*I*c))*sqrt((-49*I*A^2 + 322*A*B + 529*I*B^2)/(a^4*d^2))*log(1/8*((a^2*d*e^(2*I*d*x + 2
*I*c) - a^2*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((-49*I*A^2 + 322*A*B + 529*I*B
^2)/(a^4*d^2)) + 7*I*A - 23*B)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) + (a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x +
4*I*c))*sqrt((-49*I*A^2 + 322*A*B + 529*I*B^2)/(a^4*d^2))*log(-1/8*((a^2*d*e^(2*I*d*x + 2*I*c) - a^2*d)*sqrt(
(I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((-49*I*A^2 + 322*A*B + 529*I*B^2)/(a^4*d^2)) - 7*I
*A + 23*B)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) - 2*(6*(A + 7*I*B)*e^(6*I*d*x + 6*I*c) - (A + 33*I*B)*e^(4*I*d*x + 4*
I*c) - 2*(3*A + 5*I*B)*e^(2*I*d*x + 2*I*c) + A + I*B)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) -
1)))/(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate((A+B*tan(d*x+c))/cot(d*x+c)**(5/2)/(a+I*a*tan(d*x+c))**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((A+B*tan(d*x+c))/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int { \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((A+B*tan(d*x+c))/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)/((I*a*tan(d*x + c) + a)^2*cot(d*x + c)^(5/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]

[In]

int((A + B*tan(c + d*x))/(cot(c + d*x)^(5/2)*(a + a*tan(c + d*x)*1i)^2),x)

[Out]

int((A + B*tan(c + d*x))/(cot(c + d*x)^(5/2)*(a + a*tan(c + d*x)*1i)^2), x)