\(\int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx\) [530]

Optimal result

Integrand size = 36, antiderivative size = 367 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((1+29 i) A-(6+i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {((30+28 i) A-(7-5 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^3 d}-\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^3 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^3 d} \]

[Out]

1/6*(A+I*B)*cot(d*x+c)^(7/2)/d/(I*a+a*cot(d*x+c))^3+1/12*(5*A+2*I*B)*cot(d*x+c)^(5/2)/a/d/(I*a+a*cot(d*x+c))^2
+7/24*(4*A+I*B)*cot(d*x+c)^(3/2)/d/(I*a^3+a^3*cot(d*x+c))+(1/32-1/32*I)*((1+29*I)*A-(6+I)*B)*arctan(-1+2^(1/2)
*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+1/32*((30+28*I)*A+(-7+5*I)*B)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2
)+(-1/64+1/64*I)*((29+I)*A+(1+6*I)*B)*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+(1/64-1/64*I)*((
29+I)*A+(1+6*I)*B)*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)-5/8*(6*A+I*B)*cot(d*x+c)^(1/2)/a^3/
d

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3662, 3676, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((1+29 i) A-(6+i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {((30+28 i) A-(7-5 i) B) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{16 \sqrt {2} a^3 d}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3 \cot (c+d x)+i a^3\right )}-\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^3 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (a \cot (c+d x)+i a)^2} \]

[In]

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

[Out]

((-1/16 + I/16)*((1 + 29*I)*A - (6 + I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + (((30 + 2
8*I)*A - (7 - 5*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d) - (5*(6*A + I*B)*Sqrt[Cot[c +
d*x]])/(8*a^3*d) + ((A + I*B)*Cot[c + d*x]^(7/2))/(6*d*(I*a + a*Cot[c + d*x])^3) + ((5*A + (2*I)*B)*Cot[c + d
*x]^(5/2))/(12*a*d*(I*a + a*Cot[c + d*x])^2) + (7*(4*A + I*B)*Cot[c + d*x]^(3/2))/(24*d*(I*a^3 + a^3*Cot[c + d
*x])) - ((1/32 - I/32)*((29 + I)*A + (1 + 6*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]
*a^3*d) + ((1/32 - I/32)*((29 + I)*A + (1 + 6*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[
2]*a^3*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 3609

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

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 3676

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*b - a*B))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(2*a*f
*m)), x] + Dist[1/(2*a^2*m), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[A*(a*c*m + b*d
*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a*A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A
, B}, 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 {\cot ^{\frac {7}{2}}(c+d x) (B+A \cot (c+d x))}{(i a+a \cot (c+d x))^3} \, dx \\ & = \frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {\int \frac {\cot ^{\frac {5}{2}}(c+d x) \left (-\frac {7}{2} a (i A-B)+\frac {1}{2} a (13 A+i B) \cot (c+d x)\right )}{(i a+a \cot (c+d x))^2} \, dx}{6 a^2} \\ & = \frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {\int \frac {\cot ^{\frac {3}{2}}(c+d x) \left (-5 a^2 (5 i A-2 B)+a^2 (31 A+4 i B) \cot (c+d x)\right )}{i a+a \cot (c+d x)} \, dx}{24 a^4} \\ & = \frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {\int \sqrt {\cot (c+d x)} \left (-21 a^3 (4 i A-B)+15 a^3 (6 A+i B) \cot (c+d x)\right ) \, dx}{48 a^6} \\ & = -\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {\int \frac {-15 a^3 (6 A+i B)-21 a^3 (4 i A-B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{48 a^6} \\ & = -\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {15 a^3 (6 A+i B)+21 a^3 (4 i A-B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{24 a^6 d} \\ & = -\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {((30+28 i) A-(7-5 i) B) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{16 a^3 d}+\frac {((30-28 i) A+(7+5 i) B) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{16 a^3 d} \\ & = -\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {((30+28 i) A-(7-5 i) B) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^3 d}+\frac {((30+28 i) A-(7-5 i) B) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^3 d}-\frac {((30-28 i) A+(7+5 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^3 d}-\frac {((30-28 i) A+(7+5 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^3 d} \\ & = -\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac {((30-28 i) A+(7+5 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}+\frac {((30-28 i) A+(7+5 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}+\frac {((30+28 i) A-(7-5 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^3 d}-\frac {((30+28 i) A-(7-5 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^3 d} \\ & = -\frac {((30+28 i) A-(7-5 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^3 d}+\frac {((30+28 i) A-(7-5 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^3 d}-\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac {((30-28 i) A+(7+5 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}+\frac {((30-28 i) A+(7+5 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.00 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.68 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\frac {\sqrt {\cot (c+d x)} \left (-48 i A+3 \sqrt [4]{-1} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^3(c+d x) (\cos (3 (c+d x))+i \sin (3 (c+d x))) \sqrt {\tan (c+d x)}-3 \sqrt [4]{-1} (29 A+6 i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^3(c+d x) (\cos (3 (c+d x))+i \sin (3 (c+d x))) \sqrt {\tan (c+d x)}+204 A \tan (c+d x)+27 i B \tan (c+d x)+242 i A \tan ^2(c+d x)-38 B \tan ^2(c+d x)-90 A \tan ^3(c+d x)-15 i B \tan ^3(c+d x)\right )}{24 a^3 d (-i+\tan (c+d x))^3} \]

[In]

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

[Out]

(Sqrt[Cot[c + d*x]]*((-48*I)*A + 3*(-1)^(1/4)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sec[c + d*x]^3*(
Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)])*Sqrt[Tan[c + d*x]] - 3*(-1)^(1/4)*(29*A + (6*I)*B)*ArcTanh[(-1)^(3/4)*S
qrt[Tan[c + d*x]]]*Sec[c + d*x]^3*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)])*Sqrt[Tan[c + d*x]] + 204*A*Tan[c + d
*x] + (27*I)*B*Tan[c + d*x] + (242*I)*A*Tan[c + d*x]^2 - 38*B*Tan[c + d*x]^2 - 90*A*Tan[c + d*x]^3 - (15*I)*B*
Tan[c + d*x]^3))/(24*a^3*d*(-I + Tan[c + d*x])^3)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.49

method	result	size
derivativedivides	\(\frac {-2 A \sqrt {\cot \left (d x +c \right )}+\frac {4 \left (-\frac {i A}{16}-\frac {B}{16}\right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}+\frac {i \left (\frac {i \left (20 i A -9 B \right ) \cot \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {98 i A}{3}+\frac {38 B}{3}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+\left (5 i B +14 A \right ) \sqrt {\cot \left (d x +c \right )}}{\left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {2 \left (6 i B +29 A \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{\sqrt {2}+i \sqrt {2}}\right )}{8}}{a^{3} d}\)	\(180\)
default	\(\frac {-2 A \sqrt {\cot \left (d x +c \right )}+\frac {4 \left (-\frac {i A}{16}-\frac {B}{16}\right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}+\frac {i \left (\frac {i \left (20 i A -9 B \right ) \cot \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {98 i A}{3}+\frac {38 B}{3}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+\left (5 i B +14 A \right ) \sqrt {\cot \left (d x +c \right )}}{\left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {2 \left (6 i B +29 A \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{\sqrt {2}+i \sqrt {2}}\right )}{8}}{a^{3} d}\)	\(180\)

[In]

int(cot(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/a^3/d*(-2*A*cot(d*x+c)^(1/2)+4*(-1/16*I*A-1/16*B)/(2^(1/2)-I*2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(2^(1/2)-I*2
^(1/2)))+1/8*I*((I*(20*I*A-9*B)*cot(d*x+c)^(5/2)+(-98/3*I*A+38/3*B)*cot(d*x+c)^(3/2)+(14*A+5*I*B)*cot(d*x+c)^(
1/2))/(I+cot(d*x+c))^3+2*(29*A+6*I*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. 688 vs. \(2 (276) = 552\).

Time = 0.28 (sec) , antiderivative size = 688, normalized size of antiderivative = 1.87 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {{\left (3 \, a^{3} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} 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^{6} 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 ) - 3 \, a^{3} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} 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^{6} 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 ) - 3 \, a^{3} d \sqrt {\frac {-841 i \, A^{2} + 348 \, A B + 36 i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left ({\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} 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 {-841 i \, A^{2} + 348 \, A B + 36 i \, B^{2}}{a^{6} d^{2}}} + 29 i \, A - 6 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) + 3 \, a^{3} d \sqrt {\frac {-841 i \, A^{2} + 348 \, A B + 36 i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {{\left ({\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} 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 {-841 i \, A^{2} + 348 \, A B + 36 i \, B^{2}}{a^{6} d^{2}}} - 29 i \, A + 6 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) + 2 \, {\left (2 \, {\left (73 \, A + 10 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - {\left (41 \, A + 14 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (8 \, 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}}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \]

[In]

integrate(cot(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/96*(3*a^3*d*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(-2*((I*a^3*d*e^(2*I*d*x + 2*I*c
) - I*a^3*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^6*d^2
)) + (A - I*B)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)) - 3*a^3*d*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^
6*d^2))*e^(6*I*d*x + 6*I*c)*log(-2*((-I*a^3*d*e^(2*I*d*x + 2*I*c) + I*a^3*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^6*d^2)) + (A - I*B)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*
x - 2*I*c)/(I*A + B)) - 3*a^3*d*sqrt((-841*I*A^2 + 348*A*B + 36*I*B^2)/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(1/8*
((a^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((-841*I*
A^2 + 348*A*B + 36*I*B^2)/(a^6*d^2)) + 29*I*A - 6*B)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) + 3*a^3*d*sqrt((-841*I*A^2
+ 348*A*B + 36*I*B^2)/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(-1/8*((a^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*sqrt((I*e^(
2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((-841*I*A^2 + 348*A*B + 36*I*B^2)/(a^6*d^2)) - 29*I*A +
6*B)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) + 2*(2*(73*A + 10*I*B)*e^(6*I*d*x + 6*I*c) - (41*A + 14*I*B)*e^(4*I*d*x + 4
*I*c) - (8*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
)))*e^(-6*I*d*x - 6*I*c)/(a^3*d)

Sympy [F]

\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\frac {i \left (\int \frac {A \cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx + \int \frac {B \tan {\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx\right )}{a^{3}} \]

[In]

integrate(cot(d*x+c)**(3/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**3,x)

[Out]

I*(Integral(A*cot(c + d*x)**(3/2)/(tan(c + d*x)**3 - 3*I*tan(c + d*x)**2 - 3*tan(c + d*x) + I), x) + Integral(
B*tan(c + d*x)*cot(c + d*x)**(3/2)/(tan(c + d*x)**3 - 3*I*tan(c + d*x)**2 - 3*tan(c + d*x) + I), x))/a**3

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(cot(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac {3}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(cot(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*cot(d*x + c)^(3/2)/(I*a*tan(d*x + c) + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]

[In]

int((cot(c + d*x)^(3/2)*(A + B*tan(c + d*x)))/(a + a*tan(c + d*x)*1i)^3,x)

[Out]

int((cot(c + d*x)^(3/2)*(A + B*tan(c + d*x)))/(a + a*tan(c + d*x)*1i)^3, x)