\(\int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx\) [102]

Optimal result

Integrand size = 36, antiderivative size = 217 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {(3 i A-2 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d} \]

[Out]

(3*I*A-2*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d+1/4*(I*A+B)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/
2)*2^(1/2)/a^(1/2))/a^(3/2)/d*2^(1/2)+1/6*(13*A+7*I*B)*cot(d*x+c)/a/d/(a+I*a*tan(d*x+c))^(1/2)-1/2*(7*A+3*I*B)
*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a^2/d+1/3*(A+I*B)*cot(d*x+c)/d/(a+I*a*tan(d*x+c))^(3/2)

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3677, 3679, 3681, 3561, 212, 3680, 65, 214} \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {(-2 B+3 i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {(B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]

[In]

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

[Out]

(((3*I)*A - 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((I*A + B)*ArcTanh[Sqrt[a + I*a*Ta
n[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) + ((A + I*B)*Cot[c + d*x])/(3*d*(a + I*a*Tan[c + d*x])^(
3/2)) + ((13*A + (7*I)*B)*Cot[c + d*x])/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - ((7*A + (3*I)*B)*Cot[c + d*x]*Sqr
t[a + I*a*Tan[c + d*x]])/(2*a^2*d)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

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

Rule 214

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

Rule 3561

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 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]

Rule 3679

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*d - B*c)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(f*
(n + 1)*(c^2 + d^2))), x] - Dist[1/(a*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n
+ 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c*m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3680

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b*(B/f), Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x
]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && EqQ[A*b + a*B,
0]

Rule 3681

Int[(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*tan[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(A*b + a*B)/(b*c + a*d), Int[(a + b*Tan[e + f*x])^m, x], x] - Dist[(B*c
- A*d)/(b*c + a*d), Int[(a + b*Tan[e + f*x])^m*((a - b*Tan[e + f*x])/(c + d*Tan[e + f*x])), x], x] /; FreeQ[{
a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {\int \frac {\cot ^2(c+d x) \left (a (4 A+i B)-\frac {5}{2} a (i A-B) \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2} \\ & = \frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (\frac {3}{2} a^2 (7 A+3 i B)-\frac {3}{4} a^2 (13 i A-7 B) \tan (c+d x)\right ) \, dx}{3 a^4} \\ & = \frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{2} a^3 (3 i A-2 B)-\frac {3}{4} a^3 (7 A+3 i B) \tan (c+d x)\right ) \, dx}{3 a^5} \\ & = \frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {(3 i A-2 B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{2 a^3}-\frac {(A-i B) \int \sqrt {a+i a \tan (c+d x)} \, dx}{4 a^2} \\ & = \frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {(3 i A-2 B) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{2 a d}+\frac {(i A+B) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{2 a d} \\ & = \frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {(3 A+2 i B) \text {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{a^2 d} \\ & = \frac {(3 i A-2 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(7 A+3 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.58 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.77 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {12 i (3 A+2 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )+3 \sqrt {2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )-\frac {2 \sqrt {a} \left (-3 (7 A+3 i B)+(29 i A-11 B) \cot (c+d x)+6 A \cot ^2(c+d x)\right )}{(i+\cot (c+d x)) \sqrt {a+i a \tan (c+d x)}}}{12 a^{3/2} d} \]

[In]

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

[Out]

((12*I)*(3*A + (2*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] + 3*Sqrt[2]*(I*A + B)*ArcTanh[Sqrt[a + I*a
*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])] - (2*Sqrt[a]*(-3*(7*A + (3*I)*B) + ((29*I)*A - 11*B)*Cot[c + d*x] + 6*A*Cot[
c + d*x]^2))/((I + Cot[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]]))/(12*a^(3/2)*d)

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {2 i a^{2} \left (-\frac {3 i B +5 A}{4 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {i B +A}{6 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\left (i B -A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {7}{2}}}+\frac {\frac {i A \sqrt {a +i a \tan \left (d x +c \right )}}{2 a \tan \left (d x +c \right )}+\frac {\left (2 i B +3 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{2 \sqrt {a}}}{a^{3}}\right )}{d}\)	\(168\)
default	\(\frac {2 i a^{2} \left (-\frac {3 i B +5 A}{4 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {i B +A}{6 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\left (i B -A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {7}{2}}}+\frac {\frac {i A \sqrt {a +i a \tan \left (d x +c \right )}}{2 a \tan \left (d x +c \right )}+\frac {\left (2 i B +3 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{2 \sqrt {a}}}{a^{3}}\right )}{d}\)	\(168\)

[In]

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

[Out]

2*I/d*a^2*(-1/4/a^3*(5*A+3*I*B)/(a+I*a*tan(d*x+c))^(1/2)-1/6/a^2*(A+I*B)/(a+I*a*tan(d*x+c))^(3/2)-1/8*(-A+I*B)
/a^(7/2)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))+1/a^3*(1/2*I*A*(a+I*a*tan(d*x+c))^(1/2)
/a/tan(d*x+c)+1/2*(3*A+2*I*B)/a^(1/2)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(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. 814 vs. \(2 (170) = 340\).

Time = 0.28 (sec) , antiderivative size = 814, normalized size of antiderivative = 3.75 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {3 \, \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a^{3} d^{2}}} \log \left (-\frac {4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a^{3} d^{2}}} + {\left (-i \, A - B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - 3 \, \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a^{3} d^{2}}} \log \left (\frac {4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a^{3} d^{2}}} - {\left (-i \, A - B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - 3 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {9 \, A^{2} + 12 i \, A B - 4 \, B^{2}}{a^{3} d^{2}}} \log \left (-\frac {16 \, {\left (3 \, {\left (3 i \, A - 2 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (3 i \, A - 2 \, B\right )} a^{2} + 2 \, \sqrt {2} {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {9 \, A^{2} + 12 i \, A B - 4 \, B^{2}}{a^{3} d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{-3 i \, A + 2 \, B}\right ) + 3 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {9 \, A^{2} + 12 i \, A B - 4 \, B^{2}}{a^{3} d^{2}}} \log \left (-\frac {16 \, {\left (3 \, {\left (3 i \, A - 2 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (3 i \, A - 2 \, B\right )} a^{2} - 2 \, \sqrt {2} {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {9 \, A^{2} + 12 i \, A B - 4 \, B^{2}}{a^{3} d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{-3 i \, A + 2 \, B}\right ) + \sqrt {2} {\left (2 \, {\left (14 i \, A - 5 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - {\left (-13 i \, A + B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (-8 i \, A + 5 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, A + B\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{12 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \]

[In]

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

[Out]

-1/12*(3*sqrt(1/2)*(a^2*d*e^(5*I*d*x + 5*I*c) - a^2*d*e^(3*I*d*x + 3*I*c))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a^3*d^
2))*log(-4*(sqrt(2)*sqrt(1/2)*(a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2
- 2*I*A*B - B^2)/(a^3*d^2)) + (-I*A - B)*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/(I*A + B)) - 3*sqrt(1/2)*(a^2*d*
e^(5*I*d*x + 5*I*c) - a^2*d*e^(3*I*d*x + 3*I*c))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a^3*d^2))*log(4*(sqrt(2)*sqrt(1/
2)*(a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a^3*d^2)
) - (-I*A - B)*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/(I*A + B)) - 3*(a^2*d*e^(5*I*d*x + 5*I*c) - a^2*d*e^(3*I*d*
x + 3*I*c))*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2))*log(-16*(3*(3*I*A - 2*B)*a^2*e^(2*I*d*x + 2*I*c) + (3*
I*A - 2*B)*a^2 + 2*sqrt(2)*(a^3*d*e^(3*I*d*x + 3*I*c) + a^3*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1
))*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2)))*e^(-2*I*d*x - 2*I*c)/(-3*I*A + 2*B)) + 3*(a^2*d*e^(5*I*d*x + 5
*I*c) - a^2*d*e^(3*I*d*x + 3*I*c))*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2))*log(-16*(3*(3*I*A - 2*B)*a^2*e^
(2*I*d*x + 2*I*c) + (3*I*A - 2*B)*a^2 - 2*sqrt(2)*(a^3*d*e^(3*I*d*x + 3*I*c) + a^3*d*e^(I*d*x + I*c))*sqrt(a/(
e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2)))*e^(-2*I*d*x - 2*I*c)/(-3*I*A + 2*B)) +
sqrt(2)*(2*(14*I*A - 5*B)*e^(6*I*d*x + 6*I*c) - (-13*I*A + B)*e^(4*I*d*x + 4*I*c) + 2*(-8*I*A + 5*B)*e^(2*I*d*
x + 2*I*c) - I*A + B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))/(a^2*d*e^(5*I*d*x + 5*I*c) - a^2*d*e^(3*I*d*x + 3*I*c
))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.99 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {i \, a {\left (\frac {4 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} {\left (7 \, A + 3 i \, B\right )} - {\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (13 \, A + 7 i \, B\right )} a - 2 \, {\left (A + i \, B\right )} a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3}} + \frac {3 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {5}{2}}} + \frac {12 \, {\left (3 \, A + 2 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )}}{24 \, d} \]

[In]

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

[Out]

-1/24*I*a*(4*(3*(I*a*tan(d*x + c) + a)^2*(7*A + 3*I*B) - (I*a*tan(d*x + c) + a)*(13*A + 7*I*B)*a - 2*(A + I*B)
*a^2)/((I*a*tan(d*x + c) + a)^(5/2)*a^2 - (I*a*tan(d*x + c) + a)^(3/2)*a^3) + 3*sqrt(2)*(A - I*B)*log(-(sqrt(2
)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(I*a*tan(d*x + c) + a)))/a^(5/2) + 12*(3*A + 2*
I*B)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x + c) + a) + sqrt(a)))/a^(5/2))/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.81 (sec) , antiderivative size = 3051, normalized size of antiderivative = 14.06 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2*atanh((3*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((33*B^2)/(64*a^3*d^2) - (73*A^2)/(64*a^3*d^2) - ((5041*A^4*a^6)/
d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*
a^6) - (A*B*47i)/(32*a^3*d^2))^(1/2)*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^
3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/((A^3*a^2*d*781i)/4 + (279*B^3*a^2*d)/4 - (A*B^2*a^2*d*1223i
)/4 - (1717*A^2*B*a^2*d)/4 + (A*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3
*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a) - (7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4
- (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a)) + (71*A^2*d^2*(a + a
*tan(c + d*x)*1i)^(1/2)*((33*B^2)/(64*a^3*d^2) - (73*A^2)/(64*a^3*d^2) - ((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d
^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*a^6) - (A*B*47i)/(32*
a^3*d^2))^(1/2))/((A^3*d*781i)/(4*a) + (279*B^3*d)/(4*a) - (A*B^2*d*1223i)/(4*a) - (1717*A^2*B*d)/(4*a) + (A*d
^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*1391
6i)/d^4)^(1/2)*13i)/(4*a^4) - (7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*
B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a^4)) - (31*B^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((33*
B^2)/(64*a^3*d^2) - (73*A^2)/(64*a^3*d^2) - ((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4
- (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*a^6) - (A*B*47i)/(32*a^3*d^2))^(1/2))/((A^3*d*781i
)/(4*a) + (279*B^3*d)/(4*a) - (A*B^2*d*1223i)/(4*a) - (1717*A^2*B*d)/(4*a) + (A*d^3*((5041*A^4*a^6)/d^4 + (961
*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a^4) -
(7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a
^6*13916i)/d^4)^(1/2))/(4*a^4)) + (A*B*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((33*B^2)/(64*a^3*d^2) - (73*A^2)/(64
*a^3*d^2) - ((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B
*a^6*13916i)/d^4)^(1/2)/(64*a^6) - (A*B*47i)/(32*a^3*d^2))^(1/2)*98i)/((A^3*d*781i)/(4*a) + (279*B^3*d)/(4*a)
- (A*B^2*d*1223i)/(4*a) - (1717*A^2*B*d)/(4*a) + (A*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B
^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a^4) - (7*B*d^3*((5041*A^4*a^6)/d^
4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a^
4)))*(-(d^2*(((73*A^2*a^3 - 33*B^2*a^3)/d^2 + (A*B*a^3*94i)/d^2)^2 + 128*a^6*(((A*B^3 + (3*A^3*B)/2)*1i)/d^4 -
((9*A^4)/4 + (11*A^2*B^2)/4 + B^4)/d^4))^(1/2) + 73*A^2*a^3 - 33*B^2*a^3 + A*B*a^3*94i)/(64*a^6*d^2))^(1/2) -
(((A*a + B*a*1i)*1i)/(3*d) + ((13*A + B*7i)*(a + a*tan(c + d*x)*1i)*1i)/(6*d) - ((7*A + B*3i)*(a + a*tan(c +
d*x)*1i)^2*1i)/(2*a*d))/(a*(a + a*tan(c + d*x)*1i)^(3/2) - (a + a*tan(c + d*x)*1i)^(5/2)) - 2*atanh((3*d^4*(a
+ a*tan(c + d*x)*1i)^(1/2)*(((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*607
6i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*a^6) - (73*A^2)/(64*a^3*d^2) + (33*B^2)/(64*a^3*d^2) - (A*B*47i)/(
32*a^3*d^2))^(1/2)*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 +
(A^3*B*a^6*13916i)/d^4)^(1/2))/((A^3*a^2*d*781i)/4 + (279*B^3*a^2*d)/4 - (A*B^2*a^2*d*1223i)/4 - (1717*A^2*B*
a^2*d)/4 - (A*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 +
(A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a) + (7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a
^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a)) - (71*A^2*d^2*(a + a*tan(c + d*x)*1i)^
(1/2)*(((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*
13916i)/d^4)^(1/2)/(64*a^6) - (73*A^2)/(64*a^3*d^2) + (33*B^2)/(64*a^3*d^2) - (A*B*47i)/(32*a^3*d^2))^(1/2))/(
(A^3*d*781i)/(4*a) + (279*B^3*d)/(4*a) - (A*B^2*d*1223i)/(4*a) - (1717*A^2*B*d)/(4*a) - (A*d^3*((5041*A^4*a^6)
/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i
)/(4*a^4) + (7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4
+ (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a^4)) + (31*B^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((5041*A^4*a^6)/d^4 +
(961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*a^6) -
(73*A^2)/(64*a^3*d^2) + (33*B^2)/(64*a^3*d^2) - (A*B*47i)/(32*a^3*d^2))^(1/2))/((A^3*d*781i)/(4*a) + (279*B^3
*d)/(4*a) - (A*B^2*d*1223i)/(4*a) - (1717*A^2*B*d)/(4*a) - (A*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (1
4006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a^4) + (7*B*d^3*((5041*A
^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1
/2))/(4*a^4)) - (A*B*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B
^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)/(64*a^6) - (73*A^2)/(64*a^3*d^2) + (33*B^2
)/(64*a^3*d^2) - (A*B*47i)/(32*a^3*d^2))^(1/2)*98i)/((A^3*d*781i)/(4*a) + (279*B^3*d)/(4*a) - (A*B^2*d*1223i)/
(4*a) - (1717*A^2*B*d)/(4*a) - (A*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B
^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2)*13i)/(4*a^4) + (7*B*d^3*((5041*A^4*a^6)/d^4 + (961*B^4*a^6)/
d^4 - (14006*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*6076i)/d^4 + (A^3*B*a^6*13916i)/d^4)^(1/2))/(4*a^4)))*((d^2*(((73*A
^2*a^3 - 33*B^2*a^3)/d^2 + (A*B*a^3*94i)/d^2)^2 + 128*a^6*(((A*B^3 + (3*A^3*B)/2)*1i)/d^4 - ((9*A^4)/4 + (11*A
^2*B^2)/4 + B^4)/d^4))^(1/2) - 73*A^2*a^3 + 33*B^2*a^3 - A*B*a^3*94i)/(64*a^6*d^2))^(1/2)