\(\int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx\) [95]

Optimal result

Integrand size = 36, antiderivative size = 167 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {(i A-2 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d} \]

[Out]

(I*A-2*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)+1/2*(I*A+B)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)
*2^(1/2)/a^(1/2))/d*2^(1/2)/a^(1/2)+(A+I*B)*cot(d*x+c)/d/(a+I*a*tan(d*x+c))^(1/2)-(2*A+I*B)*cot(d*x+c)*(a+I*a*
tan(d*x+c))^(1/2)/a/d

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {(-2 B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {(B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}+\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}} \]

[In]

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

[Out]

((I*A - 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + ((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c
+ d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + ((A + I*B)*Cot[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]]) - ((
2*A + I*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a*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)}{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 (a (2 A+i B)-\frac {3}{2} a (i A-B) \tan (c+d x)\right ) \, dx}{a^2} \\ & = \frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {1}{2} a^2 (i A-2 B)-\frac {1}{2} a^2 (2 A+i B) \tan (c+d x)\right ) \, dx}{a^3} \\ & = \frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}-\frac {(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^2}-\frac {(A-i B) \int \sqrt {a+i a \tan (c+d x)} \, dx}{2 a} \\ & = \frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}-\frac {(i A-2 B) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{2 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 )}{d} \\ & = \frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d}-\frac {(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 d} \\ & = \frac {(i A-2 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {(A+i B) \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 A+i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.64 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.78 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {i (A+2 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {-2 i A+B-A \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}} \]

[In]

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

[Out]

(I*(A + (2*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + ((I*A + B)*ArcTanh[Sqrt[a + I*a*Ta
n[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + ((-2*I)*A + B - A*Cot[c + d*x])/(d*Sqrt[a + I*a*Tan[c +
d*x]])

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {2 i a^{2} \left (-\frac {i B +A}{2 a^{2} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {\left (i B -A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4 a^{\frac {5}{2}}}+\frac {\frac {i A \sqrt {a +i a \tan \left (d x +c \right )}}{2 a \tan \left (d x +c \right )}+\frac {\left (2 i B +A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{2 \sqrt {a}}}{a^{2}}\right )}{d}\)	\(139\)
default	\(\frac {2 i a^{2} \left (-\frac {i B +A}{2 a^{2} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {\left (i B -A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4 a^{\frac {5}{2}}}+\frac {\frac {i A \sqrt {a +i a \tan \left (d x +c \right )}}{2 a \tan \left (d x +c \right )}+\frac {\left (2 i B +A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{2 \sqrt {a}}}{a^{2}}\right )}{d}\)	\(139\)

[In]

int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*I/d*a^2*(-1/2/a^2*(A+I*B)/(a+I*a*tan(d*x+c))^(1/2)-1/4*(-A+I*B)/a^(5/2)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c
))^(1/2)*2^(1/2)/a^(1/2))+1/a^2*(1/2*I*A*(a+I*a*tan(d*x+c))^(1/2)/a/tan(d*x+c)+1/2*(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. 742 vs. \(2 (135) = 270\).

Time = 0.27 (sec) , antiderivative size = 742, normalized size of antiderivative = 4.44 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {\sqrt {2} {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a d^{2}}} \log \left (-\frac {4 \, {\left ({\left (-i \, A - B\right )} a e^{\left (i \, d x + i \, c\right )} + {\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a 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 d^{2}}}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - \sqrt {2} {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {-\frac {A^{2} - 2 i \, A B - B^{2}}{a d^{2}}} \log \left (-\frac {4 \, {\left ({\left (-i \, A - B\right )} a e^{\left (i \, d x + i \, c\right )} - {\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a 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 d^{2}}}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {-\frac {A^{2} + 4 i \, A B - 4 \, B^{2}}{a d^{2}}} \log \left (-\frac {16 \, {\left (3 \, {\left (i \, A - 2 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (i \, A - 2 \, B\right )} a^{2} + 2 \, \sqrt {2} {\left (a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{2} 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 {A^{2} + 4 i \, A B - 4 \, B^{2}}{a d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{-i \, A + 2 \, B}\right ) + {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {-\frac {A^{2} + 4 i \, A B - 4 \, B^{2}}{a d^{2}}} \log \left (-\frac {16 \, {\left (3 \, {\left (i \, A - 2 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (i \, A - 2 \, B\right )} a^{2} - 2 \, \sqrt {2} {\left (a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{2} 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 {A^{2} + 4 i \, A B - 4 \, B^{2}}{a d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{-i \, A + 2 \, B}\right ) + 2 \, \sqrt {2} {\left ({\left (3 i \, A - B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, A 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}}}{4 \, {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )}} \]

[In]

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

-1/4*(sqrt(2)*(a*d*e^(3*I*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a*d^2))*log(-4*((-I
*A - B)*a*e^(I*d*x + I*c) + (a*d*e^(2*I*d*x + 2*I*c) + a*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2 - 2*I
*A*B - B^2)/(a*d^2)))*e^(-I*d*x - I*c)/(I*A + B)) - sqrt(2)*(a*d*e^(3*I*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))*sq
rt(-(A^2 - 2*I*A*B - B^2)/(a*d^2))*log(-4*((-I*A - B)*a*e^(I*d*x + I*c) - (a*d*e^(2*I*d*x + 2*I*c) + a*d)*sqrt
(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-(A^2 - 2*I*A*B - B^2)/(a*d^2)))*e^(-I*d*x - I*c)/(I*A + B)) - (a*d*e^(3*I*
d*x + 3*I*c) - a*d*e^(I*d*x + I*c))*sqrt(-(A^2 + 4*I*A*B - 4*B^2)/(a*d^2))*log(-16*(3*(I*A - 2*B)*a^2*e^(2*I*d
*x + 2*I*c) + (I*A - 2*B)*a^2 + 2*sqrt(2)*(a^2*d*e^(3*I*d*x + 3*I*c) + a^2*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d
*x + 2*I*c) + 1))*sqrt(-(A^2 + 4*I*A*B - 4*B^2)/(a*d^2)))*e^(-2*I*d*x - 2*I*c)/(-I*A + 2*B)) + (a*d*e^(3*I*d*x
+ 3*I*c) - a*d*e^(I*d*x + I*c))*sqrt(-(A^2 + 4*I*A*B - 4*B^2)/(a*d^2))*log(-16*(3*(I*A - 2*B)*a^2*e^(2*I*d*x
+ 2*I*c) + (I*A - 2*B)*a^2 - 2*sqrt(2)*(a^2*d*e^(3*I*d*x + 3*I*c) + a^2*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x
+ 2*I*c) + 1))*sqrt(-(A^2 + 4*I*A*B - 4*B^2)/(a*d^2)))*e^(-2*I*d*x - 2*I*c)/(-I*A + 2*B)) + 2*sqrt(2)*((3*I*A
- B)*e^(4*I*d*x + 4*I*c) + 2*I*A*e^(2*I*d*x + 2*I*c) - I*A + B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))/(a*d*e^(3*I
*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))

Sympy [F]

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

[In]

integrate(cot(d*x+c)**2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.10 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {i \, a {\left (\frac {\sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left (A + 2 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {4 \, {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (2 \, A + i \, B\right )} - {\left (A + i \, B\right )} a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a - \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{2}}\right )}}{4 \, d} \]

[In]

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.46 (sec) , antiderivative size = 2961, normalized size of antiderivative = 17.73 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2*atanh((3*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((9*B^2)/(16*a*d^2) - (3*A^2)/(16*a*d^2) - ((A^4*a^10)/d^4 + (49*
B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16*a^6) - (A*B*3
i)/(8*a*d^2))^(1/2)*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^
3*B*a^10*20i)/d^4)^(1/2))/((A^3*a^5*d*1i)/2 + (35*B^3*a^5*d)/2 + (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 -
(114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)*3i)/2 - (3*B*d^3*((A^4*a^10)/d^4
+ (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/2 - (A*B^2
*a^5*d*57i)/2 - (15*A^2*B*a^5*d)/2) + (A^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((9*B^2)/(16*a*d^2) - (3*A^2)
/(16*a*d^2) - ((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^
10*20i)/d^4)^(1/2)/(16*a^6) - (A*B*3i)/(8*a*d^2))^(1/2))/((A^3*a^2*d*1i)/2 + (35*B^3*a^2*d)/2 - (A*B^2*a^2*d*5
7i)/2 - (15*A^2*B*a^2*d)/2 + (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10
*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)*3i)/(2*a^3) - (3*B*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2
*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/(2*a^3)) - (7*B^2*a^2*d^2*(a + a*tan(c +
d*x)*1i)^(1/2)*((9*B^2)/(16*a*d^2) - (3*A^2)/(16*a*d^2) - ((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*
a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16*a^6) - (A*B*3i)/(8*a*d^2))^(1/2))/((A^3*a^
2*d*1i)/2 + (35*B^3*a^2*d)/2 - (A*B^2*a^2*d*57i)/2 - (15*A^2*B*a^2*d)/2 + (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^1
0)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)*3i)/(2*a^3) - (3*B*d^3*(
(A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1
/2))/(2*a^3)) + (A*B*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((9*B^2)/(16*a*d^2) - (3*A^2)/(16*a*d^2) - ((A^4*a^
10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16
*a^6) - (A*B*3i)/(8*a*d^2))^(1/2)*10i)/((A^3*a^2*d*1i)/2 + (35*B^3*a^2*d)/2 - (A*B^2*a^2*d*57i)/2 - (15*A^2*B*
a^2*d)/2 + (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*
B*a^10*20i)/d^4)^(1/2)*3i)/(2*a^3) - (3*B*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (
A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/(2*a^3)))*((9*B^2)/(16*a*d^2) - (3*A^2)/(16*a*d^2) - ((A^4
*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/
(16*a^6) - (A*B*3i)/(8*a*d^2))^(1/2) - 2*atanh((3*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*(((A^4*a^10)/d^4 + (49*B^4
*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16*a^6) - (3*A^2)/(
16*a*d^2) + (9*B^2)/(16*a*d^2) - (A*B*3i)/(8*a*d^2))^(1/2)*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*
a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/((A^3*a^5*d*1i)/2 + (35*B^3*a^5*d)/2 - (A*d^3
*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^
(1/2)*3i)/2 + (3*B*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 +
(A^3*B*a^10*20i)/d^4)^(1/2))/2 - (A*B^2*a^5*d*57i)/2 - (15*A^2*B*a^5*d)/2) - (A^2*a^2*d^2*(a + a*tan(c + d*x)*
1i)^(1/2)*(((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*
20i)/d^4)^(1/2)/(16*a^6) - (3*A^2)/(16*a*d^2) + (9*B^2)/(16*a*d^2) - (A*B*3i)/(8*a*d^2))^(1/2))/((A^3*a^2*d*1i
)/2 + (35*B^3*a^2*d)/2 - (A*B^2*a^2*d*57i)/2 - (15*A^2*B*a^2*d)/2 - (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4
- (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)*3i)/(2*a^3) + (3*B*d^3*((A^4*a
^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/(
2*a^3)) + (7*B^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^1
0)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16*a^6) - (3*A^2)/(16*a*d^2) + (9*B^2)/(16*a*d^2
) - (A*B*3i)/(8*a*d^2))^(1/2))/((A^3*a^2*d*1i)/2 + (35*B^3*a^2*d)/2 - (A*B^2*a^2*d*57i)/2 - (15*A^2*B*a^2*d)/2
- (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*2
0i)/d^4)^(1/2)*3i)/(2*a^3) + (3*B*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^
10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/(2*a^3)) - (A*B*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((A^4*a^10)
/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16*a^
6) - (3*A^2)/(16*a*d^2) + (9*B^2)/(16*a*d^2) - (A*B*3i)/(8*a*d^2))^(1/2)*10i)/((A^3*a^2*d*1i)/2 + (35*B^3*a^2*
d)/2 - (A*B^2*a^2*d*57i)/2 - (15*A^2*B*a^2*d)/2 - (A*d^3*((A^4*a^10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^
10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)*3i)/(2*a^3) + (3*B*d^3*((A^4*a^10)/d^4 + (49*B^4
*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2))/(2*a^3)))*(((A^4*a^
10)/d^4 + (49*B^4*a^10)/d^4 - (114*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*140i)/d^4 + (A^3*B*a^10*20i)/d^4)^(1/2)/(16
*a^6) - (3*A^2)/(16*a*d^2) + (9*B^2)/(16*a*d^2) - (A*B*3i)/(8*a*d^2))^(1/2) - (((A*a + B*a*1i)*1i)/d - ((2*A +
B*1i)*(a + a*tan(c + d*x)*1i)*1i)/d)/(a*(a + a*tan(c + d*x)*1i)^(1/2) - (a + a*tan(c + d*x)*1i)^(3/2))