3.540 \(\int \frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx\)

Optimal. Leaf size=192 \[ -\frac{(-1)^{3/4} \sqrt{a} (2 A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{(1+i) \sqrt{a} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}+\frac{B \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.614432, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237, Rules used = {4241, 3597, 3601, 3544, 205, 3599, 63, 217, 203} \[ -\frac{(-1)^{3/4} \sqrt{a} (2 A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{(1+i) \sqrt{a} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}+\frac{B \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4241

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ
[u, x]

Rule 3597

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

Rule 3601

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

Rule 3544

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

Rule 205

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

Rule 3599

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 63

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx\\ &=\frac{B \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)} \left (-\frac{a B}{2}+\frac{1}{2} a (2 A-i B) \tan (c+d x)\right )}{\sqrt{\tan (c+d x)}} \, dx}{a}\\ &=\frac{B \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}-\left ((i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx+\frac{\left ((2 i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a-i a \tan (c+d x)) \sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx}{2 a}\\ &=\frac{B \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}+\frac{\left (2 i a^2 (i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-i a-2 a^2 x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}+\frac{\left (a (2 i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{a+i a x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac{(1-i) \sqrt{a} (i A+B) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{d}+\frac{B \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}+\frac{\left (a (2 i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+i a x^2}} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=-\frac{(1-i) \sqrt{a} (i A+B) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{d}+\frac{B \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}+\frac{\left (a (2 i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-i a x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}\\ &=-\frac{\sqrt [4]{-1} \sqrt{a} (2 i A+B) \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{d}-\frac{(1-i) \sqrt{a} (i A+B) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{d}+\frac{B \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}\\ \end{align*}

Mathematica [F]  time = 8.87314, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+i a \tan (c+d x)} (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.609, size = 3889, normalized size = 20.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/d*2^(1/2)*(B*cos(d*x+c)*sin(d*x+c)*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)-2*A*cos(d*x+c)^2*ln(-((
(cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2
)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))+4*B*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/
2)+1)+4*B*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+2*B*cos(d*x+c)^2*ln(-(((cos(d*x+c)-
1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*si
n(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))+4*A*cos(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+4*A*cos(
d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+2*A*cos(d*x+c)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/
2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c
)-sin(d*x+c)+1))-4*B*cos(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-4*B*cos(d*x+c)*arctan(((co
s(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-2*B*cos(d*x+c)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x
+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))-2
*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+I*B*cos(d*x+c)^2*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)+
2*I*B*cos(d*x+c)*2^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))-I*B*cos(d*x+c)*2^(1/2)*ln(((cos(d*x+c)-1)/s
in(d*x+c))^(1/2)-1)+2*I*B*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+2*I*A*cos(d*x+c)^2*2^(1/2)*ln((
(cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)-4*I*A*cos(d*x+c)^2*2^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))-2*I*A
*cos(d*x+c)^2*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)-2*I*B*cos(d*x+c)^2*((cos(d*x+c)-1)/sin(d*x+c))^(
1/2)*2^(1/2)-I*B*cos(d*x+c)^2*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)-2*I*B*cos(d*x+c)^2*2^(1/2)*arcta
n(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))-4*I*A*cos(d*x+c)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1
/2)+1)-4*I*A*cos(d*x+c)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-2*I*A*cos(d*x+c)*sin(d*
x+c)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d
*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))-2*I*A*cos(d*x+c)*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+
c))^(1/2)+1)+4*I*A*cos(d*x+c)*2^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))-4*A*cos(d*x+c)^2*arctan(((cos(
d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+I*B*cos(d*x+c)*sin(d*x+c)*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+
1)-4*A*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-2*I*A*cos(d*x+c)*sin(d*x+c)*2^(1/2)*ln
(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)+4*A*cos(d*x+c)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/
2)+1)+4*A*cos(d*x+c)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+2*A*cos(d*x+c)*sin(d*x+c)*
ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((cos(d*x+c)-1)/sin(d*x+c)
)^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))-2*A*cos(d*x+c)*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/
2)+1)-4*A*cos(d*x+c)*2^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))+2*A*cos(d*x+c)*2^(1/2)*ln(((cos(d*x+c)-
1)/sin(d*x+c))^(1/2)-1)-4*B*cos(d*x+c)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-4*B*cos(
d*x+c)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-2*B*cos(d*x+c)*sin(d*x+c)*ln(-(((cos(d*x
+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2
)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))-B*cos(d*x+c)*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)+2*B*cos(d*
x+c)*2^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))+B*cos(d*x+c)*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/
2)-1)-2*B*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+4*I*A*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d
*x+c))^(1/2)*2^(1/2)+1)+4*I*A*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+2*I*A*cos(d*x+c
)^2*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*
x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))+4*I*B*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))
^(1/2)*2^(1/2)+1)+4*I*B*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+2*I*B*cos(d*x+c)^2*ln
(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((cos(d*x+c)-1)/sin(d*x+c))^
(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))-4*I*A*cos(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2
^(1/2)+1)-4*I*A*cos(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-2*I*A*cos(d*x+c)*ln(-(((cos(d*x
+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2
)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))-4*I*B*cos(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-4*
I*B*cos(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-2*I*B*cos(d*x+c)*ln(-(((cos(d*x+c)-1)/sin(d
*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)
-cos(d*x+c)-sin(d*x+c)+1))+2*I*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+2*I*A*cos(d*x+c)*2^(1/2)*ln(((cos(d
*x+c)-1)/sin(d*x+c))^(1/2)-1)-4*I*B*cos(d*x+c)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-
4*I*B*cos(d*x+c)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-2*I*B*cos(d*x+c)*sin(d*x+c)*ln
(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((cos(d*x+c)-1)/sin(d*x+c))^
(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))-2*A*cos(d*x+c)*sin(d*x+c)*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*
x+c))^(1/2)+1)-4*A*cos(d*x+c)*sin(d*x+c)*2^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))+2*A*cos(d*x+c)*sin(
d*x+c)*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)+I*B*cos(d*x+c)*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(
1/2)+1)+2*B*cos(d*x+c)*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-B*cos(d*x+c)*sin(d*x+c)*2^(1/2)*ln
(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)+2*B*cos(d*x+c)*sin(d*x+c)*2^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/
2))+4*I*A*cos(d*x+c)*sin(d*x+c)*2^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))+2*I*A*cos(d*x+c)*sin(d*x+c)*
2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)+2*I*B*cos(d*x+c)*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*
2^(1/2)+2*I*B*cos(d*x+c)*sin(d*x+c)*2^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))-I*B*cos(d*x+c)*sin(d*x+c
)*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)+2*B*cos(d*x+c)^2*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+2
*A*cos(d*x+c)^2*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)+4*A*cos(d*x+c)^2*2^(1/2)*arctan(((cos(d*x+c)-1
)/sin(d*x+c))^(1/2))-2*A*cos(d*x+c)^2*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)+B*cos(d*x+c)^2*2^(1/2)*l
n(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)-2*B*cos(d*x+c)^2*2^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))-B*co
s(d*x+c)^2*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1))*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)/(I*
cos(d*x+c)+I*sin(d*x+c)-1+I+cos(d*x+c)-sin(d*x+c))/((cos(d*x+c)-1)/sin(d*x+c))^(1/2)/(cos(d*x+c)/sin(d*x+c))^(
1/2)/sin(d*x+c)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (d x + c\right ) + A\right )} \sqrt{i \, a \tan \left (d x + c\right ) + a}}{\sqrt{\cot \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.55832, size = 2071, normalized size = 10.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(2)*(-2*I*B*e^(2*I*d*x + 2*I*c) + 2*I*B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c
) + I)/(e^(2*I*d*x + 2*I*c) - 1))*e^(I*d*x + I*c) + (d*e^(2*I*d*x + 2*I*c) + d)*sqrt((4*I*A^2 + 4*A*B - I*B^2)
*a/d^2)*log((sqrt(2)*((2*I*A + B)*e^(2*I*d*x + 2*I*c) - 2*I*A - B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e
^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*e^(I*d*x + I*c) + 2*I*d*sqrt((4*I*A^2 + 4*A*B - I*B^2)*a/d^
2)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(2*I*A + B)) - (d*e^(2*I*d*x + 2*I*c) + d)*sqrt((4*I*A^2 + 4*A*B
- I*B^2)*a/d^2)*log((sqrt(2)*((2*I*A + B)*e^(2*I*d*x + 2*I*c) - 2*I*A - B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*s
qrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*e^(I*d*x + I*c) - 2*I*d*sqrt((4*I*A^2 + 4*A*B - I*B
^2)*a/d^2)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(2*I*A + B)) - (d*e^(2*I*d*x + 2*I*c) + d)*sqrt((2*I*A^2
+ 4*A*B - 2*I*B^2)*a/d^2)*log((sqrt(2)*((I*A + B)*e^(2*I*d*x + 2*I*c) - I*A - B)*sqrt(a/(e^(2*I*d*x + 2*I*c) +
1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*e^(I*d*x + I*c) + I*d*sqrt((2*I*A^2 + 4*A*B -
2*I*B^2)*a/d^2)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)) + (d*e^(2*I*d*x + 2*I*c) + d)*sqrt((2*I*
A^2 + 4*A*B - 2*I*B^2)*a/d^2)*log((sqrt(2)*((I*A + B)*e^(2*I*d*x + 2*I*c) - I*A - B)*sqrt(a/(e^(2*I*d*x + 2*I*
c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*e^(I*d*x + I*c) - I*d*sqrt((2*I*A^2 + 4*A
*B - 2*I*B^2)*a/d^2)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)))/(d*e^(2*I*d*x + 2*I*c) + d)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \left (i \tan{\left (c + d x \right )} + 1\right )} \left (A + B \tan{\left (c + d x \right )}\right )}{\sqrt{\cot{\left (c + d x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (d x + c\right ) + A\right )} \sqrt{i \, a \tan \left (d x + c\right ) + a}}{\sqrt{\cot \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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