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

Optimal. Leaf size=244 \[ -\frac{(-1)^{3/4} a^{3/2} (12 A-11 i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{4 d}-\frac{(2+2 i) a^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}+\frac{a (5 B+4 i A) \sqrt{a+i a \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}+\frac{i a B \sqrt{a+i a \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)} \]

[Out]

-((-1)^(3/4)*a^(3/2)*(12*A - (11*I)*B)*ArcTan[((-1)^(3/4)*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]])/(4*d) - ((2 + 2*I)*a^(3/2)*(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 + ((I/2)*a*B*Sqrt[a + I*a*
Tan[c + d*x]])/(d*Cot[c + d*x]^(3/2)) + (a*((4*I)*A + 5*B)*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*Sqrt[Cot[c + d*x]]
)

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Rubi [A]  time = 0.857799, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {4241, 3594, 3597, 3601, 3544, 205, 3599, 63, 217, 203} \[ -\frac{(-1)^{3/4} a^{3/2} (12 A-11 i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{4 d}-\frac{(2+2 i) a^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}+\frac{a (5 B+4 i A) \sqrt{a+i a \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}+\frac{i a B \sqrt{a+i a \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-1)^(3/4)*a^(3/2)*(12*A - (11*I)*B)*ArcTan[((-1)^(3/4)*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]])/(4*d) - ((2 + 2*I)*a^(3/2)*(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 + ((I/2)*a*B*Sqrt[a + I*a*
Tan[c + d*x]])/(d*Cot[c + d*x]^(3/2)) + (a*((4*I)*A + 5*B)*Sqrt[a + I*a*Tan[c + d*x]])/(4*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 3594

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*B*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1))/(d*f
*(m + n)), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n)
+ B*(a*c*(m - 1) - b*d*(n + 1)) - (B*(b*c - a*d)*(m - 1) - d*(A*b + a*B)*(m + n))*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] && GtQ[m, 1] &&  !LtQ[n, -1]

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{(a+i a \tan (c+d x))^{3/2} (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)} (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\\ &=\frac{i a B \sqrt{a+i a \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{1}{2} \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)} \left (\frac{1}{2} a (4 A-3 i B)+\frac{1}{2} a (4 i A+5 B) \tan (c+d x)\right ) \, dx\\ &=\frac{i a B \sqrt{a+i a \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a (4 i A+5 B) \sqrt{a+i a \tan (c+d x)}}{4 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{1}{4} a^2 (4 i A+5 B)+\frac{1}{4} a^2 (12 A-11 i B) \tan (c+d x)\right )}{\sqrt{\tan (c+d x)}} \, dx}{2 a}\\ &=\frac{i a B \sqrt{a+i a \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a (4 i A+5 B) \sqrt{a+i a \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}-\left (2 a (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{1}{8} \left ((12 i A+11 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\\ &=\frac{i a B \sqrt{a+i a \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a (4 i A+5 B) \sqrt{a+i a \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}+\frac{\left (4 i a^3 (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 (12 i A+11 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 )}{8 d}\\ &=-\frac{(2-2 i) a^{3/2} (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{i a B \sqrt{a+i a \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a (4 i A+5 B) \sqrt{a+i a \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}+\frac{\left (a^2 (12 i A+11 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 )}{4 d}\\ &=-\frac{(2-2 i) a^{3/2} (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{i a B \sqrt{a+i a \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a (4 i A+5 B) \sqrt{a+i a \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}+\frac{\left (a^2 (12 i A+11 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 )}{4 d}\\ &=-\frac{\sqrt [4]{-1} a^{3/2} (12 i A+11 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)}}{4 d}-\frac{(2-2 i) a^{3/2} (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{i a B \sqrt{a+i a \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a (4 i A+5 B) \sqrt{a+i a \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 6.21254, size = 441, normalized size = 1.81 \[ \frac{\cos ^2(c+d x) \sqrt{\cot (c+d x)} (\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left (4 (\sin (c)+i \cos (c)) \tan (c+d x) (4 A+2 B \tan (c+d x)-5 i B)-\sqrt{2} (\cos (2 c+d x)-i \sin (2 c+d x)) \sqrt{i \sin ^2(c+d x) (\cot (c+d x)+i)} \left (\sqrt{2} (12 A-11 i B) \log \left (\frac{2 e^{\frac{5 i c}{2}} \left (2 i \sqrt{-1+e^{2 i (c+d x)}}-i \sqrt{2} e^{i (c+d x)}+\sqrt{2}\right )}{(12 A-11 i B) \left (e^{i (c+d x)}-i\right )}\right )+\sqrt{2} (-12 A+11 i B) \log \left (\frac{2 e^{\frac{5 i c}{2}} \left (2 \sqrt{-1+e^{2 i (c+d x)}}+\sqrt{2} e^{i (c+d x)}-i \sqrt{2}\right )}{(11 B+12 i A) \left (e^{i (c+d x)}+i\right )}\right )+32 (A-i B) \log \left ((\cos (c)-i \sin (c)) \left (i \sin (c+d x)+\cos (c+d x)+\sqrt{i \sin (2 (c+d x))+\cos (2 (c+d x))-1}\right )\right )\right )\right )}{16 d (A \cos (c+d x)+B \sin (c+d x))} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[c + d*x]^2*Sqrt[Cot[c + d*x]]*(Cos[d*x] - I*Sin[d*x])*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x])*(
-(Sqrt[2]*(Sqrt[2]*(12*A - (11*I)*B)*Log[(2*E^(((5*I)/2)*c)*(Sqrt[2] - I*Sqrt[2]*E^(I*(c + d*x)) + (2*I)*Sqrt[
-1 + E^((2*I)*(c + d*x))]))/((12*A - (11*I)*B)*(-I + E^(I*(c + d*x))))] + Sqrt[2]*(-12*A + (11*I)*B)*Log[(2*E^
(((5*I)/2)*c)*((-I)*Sqrt[2] + Sqrt[2]*E^(I*(c + d*x)) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]))/(((12*I)*A + 11*B)*
(I + E^(I*(c + d*x))))] + 32*(A - I*B)*Log[(Cos[c] - I*Sin[c])*(Cos[c + d*x] + I*Sin[c + d*x] + Sqrt[-1 + Cos[
2*(c + d*x)] + I*Sin[2*(c + d*x)]])])*Sqrt[I*(I + Cot[c + d*x])*Sin[c + d*x]^2]*(Cos[2*c + d*x] - I*Sin[2*c +
d*x])) + 4*(I*Cos[c] + Sin[c])*Tan[c + d*x]*(4*A - (5*I)*B + 2*B*Tan[c + d*x])))/(16*d*(A*Cos[c + d*x] + B*Sin
[c + d*x]))

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Maple [B]  time = 0.571, size = 4490, normalized size = 18.4 \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))^(3/2)*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x)

[Out]

1/16/d*2^(1/2)*a*(-32*B*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^2*sin(d*x+c)-16*B*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))*cos(d*x+c)^2*sin(d*x+c)+8*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c)
)^(1/2)*cos(d*x+c)^3+14*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^3+24*A*arctan(((cos(d*x+c)-1)/s
in(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^3-12*A*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^3+8*I*A
*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)-4*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(
1/2)*sin(d*x+c)-8*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)-8*A*2^(1/2)*((cos(d*x+c)
-1)/sin(d*x+c))^(1/2)*cos(d*x+c)*sin(d*x+c)-12*A*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^2*
sin(d*x+c)-24*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+12*A*ln(((cos(d*x+c)
-1)/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)-11*B*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)*2^(1/2)*
cos(d*x+c)^2*sin(d*x+c)+22*B*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+11*B*ln
(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+11*I*B*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1
/2)+1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)-32*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^3-3
2*B*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-32*B*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/
sin(d*x+c))^(1/2)*2^(1/2)-1)-16*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))+32*B*arcta
n(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^3+32*B*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1
/2)+1)*cos(d*x+c)^3+16*B*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))*cos(d*x+c)^3-4*B*((cos(d*x+c)-1)
/sin(d*x+c))^(1/2)*2^(1/2)+14*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)+12*I*A*ln(((
cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^3-24*I*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))*2^(1/
2)*cos(d*x+c)^3-12*I*A*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^3+8*I*A*2^(1/2)*((cos(d*x+c)
-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^3-11*I*B*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^3-22*I*B*
arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^3+11*I*B*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)*
2^(1/2)*cos(d*x+c)^3-14*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^3-32*I*A*arctan(((cos(d*x+c)-
1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^2*sin(d*x+c)-32*I*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2
)+1)*cos(d*x+c)^2*sin(d*x+c)-16*I*A*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))*cos(d*x+c)^2*sin(d*x+
c)-12*I*A*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^2+24*I*A*arctan(((cos(d*x+c)-1)/sin(d*x+c
))^(1/2))*2^(1/2)*cos(d*x+c)^2+12*I*A*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2-32*I*B*arct
an(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^2*sin(d*x+c)-32*I*B*arctan(((cos(d*x+c)-1)/sin(d*x+
c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^2*sin(d*x+c)-16*I*B*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))*cos(d
*x+c)^2*sin(d*x+c)+11*I*B*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^2+22*I*B*arctan(((cos(d*x
+c)-1)/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^2-11*I*B*cos(d*x+c)^2*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2
)-1)+4*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2-8*I*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1
/2)*cos(d*x+c)+14*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)-10*B*cos(d*x+c)*sin(d*x+c)*((cos(d*
x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-32*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^3-16*A*l
n(-(((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))*cos(d*x+c)^3+32*A*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(
d*x+c))^(1/2)*2^(1/2)+1)+32*A*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+16*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))+11*B*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)*2^(1/2)*c
os(d*x+c)^3-22*B*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^3-11*B*ln(((cos(d*x+c)-1)/sin(d*
x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^3-4*B*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+4*B*cos(d*x+c)^2*
((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-12*A*cos(d*x+c)^2*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)-24
*A*cos(d*x+c)^2*2^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))+12*A*cos(d*x+c)^2*2^(1/2)*ln(((cos(d*x+c)-1)
/sin(d*x+c))^(1/2)-1)-11*B*cos(d*x+c)^2*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)+22*B*cos(d*x+c)^2*2^(1
/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))+11*B*cos(d*x+c)^2*2^(1/2)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1
)-8*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)+12*A*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)*2^(1/2
)*cos(d*x+c)^3+32*I*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^3+32*I*A*arctan(((cos(d*x
+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^3+16*I*A*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))*c
os(d*x+c)^3+32*I*B*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^3+32*I*B*arctan(((cos(d*x+c)
-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^3+16*I*B*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))*cos(
d*x+c)^3-32*I*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^2-32*I*A*arctan(((cos(d*x+c)-1)
/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^2-16*I*A*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-co
s(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))*cos(d*x
+c)^2-32*I*B*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^2-32*I*B*arctan(((cos(d*x+c)-1)/si
n(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^2-16*I*B*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))*cos(d*x+c)
^2-4*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+32*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*co
s(d*x+c)^2*sin(d*x+c)+32*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^2*sin(d*x+c)+16*A*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))*cos(d*x+c)^2*sin(d*x+c)-32*B*arctan(((cos(d*x+c)-1)/sin(d*x
+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^2*sin(d*x+c)+22*I*B*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x
+c)^2*sin(d*x+c)-11*I*B*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+14*I*B*2^(1/2)
*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)-8*I*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos
(d*x+c)*sin(d*x+c)+10*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)*sin(d*x+c)-12*I*A*ln(((cos(d*x+
c)-1)/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+24*I*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))*2^
(1/2)*cos(d*x+c)^2*sin(d*x+c)+12*I*A*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)-1
4*B*2^(1/2)*cos(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2))*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)/(I*co
s(d*x+c)+I*sin(d*x+c)-1+I+cos(d*x+c)-sin(d*x+c))/cos(d*x+c)/(cos(d*x+c)/sin(d*x+c))^(1/2)/((cos(d*x+c)-1)/sin(
d*x+c))^(1/2)/sin(d*x+c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{\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))^(3/2)*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.55949, size = 2515, normalized size = 10.31 \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))^(3/2)*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

1/8*(2*sqrt(2)*((4*A - 7*I*B)*a*e^(4*I*d*x + 4*I*c) + 4*I*B*a*e^(2*I*d*x + 2*I*c) - (4*A - 3*I*B)*a)*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) + sqrt((1
44*I*A^2 + 264*A*B - 121*I*B^2)*a^3/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((
12*I*A + 11*B)*a*e^(2*I*d*x + 2*I*c) + (-12*I*A - 11*B)*a)*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*sqrt((144*I*A^2 + 264*A*B - 121*I*B^2)*a^3/d^
2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/((12*I*A + 11*B)*a)) - sqrt((144*I*A^2 + 264*A*B - 121*I*B^2)*a
^3/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((12*I*A + 11*B)*a*e^(2*I*d*x + 2*I
*c) + (-12*I*A - 11*B)*a)*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*sqrt((144*I*A^2 + 264*A*B - 121*I*B^2)*a^3/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I
*d*x - 2*I*c)/((12*I*A + 11*B)*a)) - 4*sqrt((8*I*A^2 + 16*A*B - 8*I*B^2)*a^3/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d
*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((2*I*A + 2*B)*a*e^(2*I*d*x + 2*I*c) + (-2*I*A - 2*B)*a)*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*sqrt((8*I*
A^2 + 16*A*B - 8*I*B^2)*a^3/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/((2*I*A + 2*B)*a)) + 4*sqrt((8*I*
A^2 + 16*A*B - 8*I*B^2)*a^3/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((2*I*A +
2*B)*a*e^(2*I*d*x + 2*I*c) + (-2*I*A - 2*B)*a)*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*sqrt((8*I*A^2 + 16*A*B - 8*I*B^2)*a^3/d^2)*d*e^(2*I*d*x + 2
*I*c))*e^(-2*I*d*x - 2*I*c)/((2*I*A + 2*B)*a)))/(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))**(3/2)*(A+B*tan(d*x+c))/cot(d*x+c)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{\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))^(3/2)*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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