\(\int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx\) [520]

Optimal result

Integrand size = 36, antiderivative size = 268 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\frac {((-5-3 i) A+(3-i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}+\frac {((5+3 i) A-(3-i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{2 a d}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) ((4+i) A+(1+2 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d}+\frac {((5-3 i) A+(3+i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d} \]

[Out]

1/2*(A+I*B)*cot(d*x+c)^(3/2)/d/(I*a+a*cot(d*x+c))-1/8*((-5-3*I)*A+(3-I)*B)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))
/a/d*2^(1/2)+1/8*((5+3*I)*A+(-3+I)*B)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/a/d*2^(1/2)+(-1/16+1/16*I)*((4+I)*A+(
1+2*I)*B)*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/a/d*2^(1/2)+1/16*((5-3*I)*A+(3+I)*B)*ln(1+cot(d*x+c)+2^(1/
2)*cot(d*x+c)^(1/2))/a/d*2^(1/2)-1/2*(5*A+I*B)*cot(d*x+c)^(1/2)/a/d

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\frac {((3-i) B-(5+3 i) A) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}+\frac {((5+3 i) A-(3-i) B) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{4 \sqrt {2} a d}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{2 a d}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) ((4+i) A+(1+2 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d}+\frac {((5-3 i) A+(3+i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{8 \sqrt {2} a d} \]

[In]

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

[Out]

(((-5 - 3*I)*A + (3 - I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(4*Sqrt[2]*a*d) + (((5 + 3*I)*A - (3 - I)*
B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(4*Sqrt[2]*a*d) - ((5*A + I*B)*Sqrt[Cot[c + d*x]])/(2*a*d) + ((A +
I*B)*Cot[c + d*x]^(3/2))/(2*d*(I*a + a*Cot[c + d*x])) - ((1/8 - I/8)*((4 + I)*A + (1 + 2*I)*B)*Log[1 - Sqrt[2]
*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d) + (((5 - 3*I)*A + (3 + I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d
*x]] + Cot[c + d*x]])/(8*Sqrt[2]*a*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 {3}{2}}(c+d x) (B+A \cot (c+d x))}{i a+a \cot (c+d x)} \, dx \\ & = \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {\int \sqrt {\cot (c+d x)} \left (-\frac {3}{2} a (i A-B)+\frac {1}{2} a (5 A+i B) \cot (c+d x)\right ) \, dx}{2 a^2} \\ & = -\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{2 a d}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {\int \frac {-\frac {1}{2} a (5 A+i B)-\frac {3}{2} a (i A-B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{2 a^2} \\ & = -\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{2 a d}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} a (5 A+i B)+\frac {3}{2} a (i A-B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 d} \\ & = -\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{2 a d}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {((5+3 i) A-(3-i) B) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 a d}+\frac {((5-3 i) A+(3+i) B) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 a d} \\ & = -\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{2 a d}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {((5+3 i) A-(3-i) B) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{8 a d}+\frac {((5+3 i) A-(3-i) B) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{8 a d}-\frac {((5-3 i) A+(3+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 )}{8 \sqrt {2} a d}-\frac {((5-3 i) A+(3+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 )}{8 \sqrt {2} a d} \\ & = -\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{2 a d}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}-\frac {((5-3 i) A+(3+i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}+\frac {((5-3 i) A+(3+i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}+\frac {((5+3 i) A-(3-i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}-\frac {((5+3 i) A-(3-i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d} \\ & = -\frac {((5+3 i) A-(3-i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}+\frac {((5+3 i) A-(3-i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{2 a d}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}-\frac {((5-3 i) A+(3+i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}+\frac {((5-3 i) A+(3+i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.38 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.59 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=-\frac {\sqrt {\cot (c+d x)} \left (-4 i A+5 A \tan (c+d x)+i B \tan (c+d x)+\sqrt [4]{-1} (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)} (-i+\tan (c+d x))+2 \sqrt [4]{-1} (-2 i A+B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)} (-i+\tan (c+d x))\right )}{2 a d (-i+\tan (c+d x))} \]

[In]

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

[Out]

-1/2*(Sqrt[Cot[c + d*x]]*((-4*I)*A + 5*A*Tan[c + d*x] + I*B*Tan[c + d*x] + (-1)^(1/4)*(I*A + B)*ArcTan[(-1)^(3
/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]]*(-I + Tan[c + d*x]) + 2*(-1)^(1/4)*((-2*I)*A + B)*ArcTanh[(-1)^(3/4
)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]]*(-I + Tan[c + d*x])))/(a*d*(-I + Tan[c + d*x]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (222 ) = 444\).

Time = 0.36 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.91

method	result	size
derivativedivides	\(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (2 i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+i B \tan \left (d x +c \right ) \sqrt {2}+i B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-i A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+4 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-2 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-4 i A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+4 i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+2 i B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-4 i A \sqrt {2}+4 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+5 A \tan \left (d x +c \right ) \sqrt {2}+2 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )\right ) \sqrt {2}}{4 a d \left (-\tan \left (d x +c \right )+i\right )}\)	\(513\)
default	\(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (2 i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+i B \tan \left (d x +c \right ) \sqrt {2}+i B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-i A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+4 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-2 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-4 i A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+4 i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+2 i B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-4 i A \sqrt {2}+4 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+5 A \tan \left (d x +c \right ) \sqrt {2}+2 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )\right ) \sqrt {2}}{4 a d \left (-\tan \left (d x +c \right )+i\right )}\)	\(513\)

[In]

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

[Out]

1/4/a/d*(1/tan(d*x+c))^(3/2)*tan(d*x+c)*(2*I*B*tan(d*x+c)^(3/2)*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+I
*B*tan(d*x+c)*2^(1/2)+I*B*tan(d*x+c)^(1/2)*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-I*A*tan(d*x+c)^(1/2)*a
rctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+4*A*tan(d*x+c)^(3/2)*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+
A*tan(d*x+c)^(3/2)*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-2*B*tan(d*x+c)^(3/2)*arctan((1/2+1/2*I)*tan(d*
x+c)^(1/2)*2^(1/2))-B*tan(d*x+c)^(3/2)*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-I*B*tan(d*x+c)^(3/2)*arcta
n((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-4*I*A*tan(d*x+c)^(1/2)*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+4*
I*A*tan(d*x+c)^(3/2)*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+2*I*B*tan(d*x+c)^(1/2)*arctan((1/2+1/2*I)*ta
n(d*x+c)^(1/2)*2^(1/2))-I*A*tan(d*x+c)^(3/2)*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-4*I*A*2^(1/2)+4*A*ta
n(d*x+c)^(1/2)*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-A*tan(d*x+c)^(1/2)*arctan((1/2-1/2*I)*tan(d*x+c)^(
1/2)*2^(1/2))+5*A*tan(d*x+c)*2^(1/2)+2*B*tan(d*x+c)^(1/2)*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-B*tan(d
*x+c)^(1/2)*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2)))*2^(1/2)/(-tan(d*x+c)+I)

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. 623 vs. \(2 (201) = 402\).

Time = 0.26 (sec) , antiderivative size = 623, normalized size of antiderivative = 2.32 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=-\frac {{\left (a d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a 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^{2} 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 ) - a d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a 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^{2} 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 \, a d \sqrt {\frac {-4 i \, A^{2} + 4 \, A B + i \, B^{2}}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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 {-4 i \, A^{2} + 4 \, A B + i \, B^{2}}{a^{2} d^{2}}} + 2 i \, A - B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) + 2 \, a d \sqrt {\frac {-4 i \, A^{2} + 4 \, A B + i \, B^{2}}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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 {-4 i \, A^{2} + 4 \, A B + i \, B^{2}}{a^{2} d^{2}}} - 2 i \, A + B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) + 2 \, {\left ({\left (9 \, A + 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 (-2 i \, d x - 2 i \, c\right )}}{8 \, a d} \]

[In]

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

[Out]

-1/8*(a*d*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^2*d^2))*e^(2*I*d*x + 2*I*c)*log(-2*((I*a*d*e^(2*I*d*x + 2*I*c) - I*a
*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^2*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*d*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^2*d^2))*e^(2*
I*d*x + 2*I*c)*log(-2*((-I*a*d*e^(2*I*d*x + 2*I*c) + I*a*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^2*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*d*sqrt((-4*I*A^2 + 4*A*B + I*B^2)/(a^2*d^2))*e^(2*I*d*x + 2*I*c)*log(((a*d*e^(2*I*d*x + 2*I*c) - a*
d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((-4*I*A^2 + 4*A*B + I*B^2)/(a^2*d^2)) + 2*
I*A - B)*e^(-2*I*d*x - 2*I*c)/(a*d)) + 2*a*d*sqrt((-4*I*A^2 + 4*A*B + I*B^2)/(a^2*d^2))*e^(2*I*d*x + 2*I*c)*lo
g(-((a*d*e^(2*I*d*x + 2*I*c) - a*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((-4*I*A^2
+ 4*A*B + I*B^2)/(a^2*d^2)) - 2*I*A + B)*e^(-2*I*d*x - 2*I*c)/(a*d)) + 2*((9*A + 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^(-2*I*d*x - 2*I*c)/(a*d)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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