3.553 \(\int \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx\)
Optimal. Leaf size=292 \[ -\frac{a^2 (2 A-3 i B) \sqrt{a+i a \tan (c+d x)}}{4 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{(-1)^{3/4} a^{5/2} (46 A-45 i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{8 d}+\frac{a^2 (19 B+18 i A) \sqrt{a+i a \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}-\frac{(4+4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}+\frac{i a B (a+i a \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(c+d x)} \]
[Out]
-((-1)^(3/4)*a^(5/2)*(46*A - (45*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]])/(8*d) - ((4 + 4*I)*a^(5/2)*(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 - (a^2*(2*A - (3*I)*B)*Sqr
t[a + I*a*Tan[c + d*x]])/(4*d*Cot[c + d*x]^(3/2)) + (a^2*((18*I)*A + 19*B)*Sqrt[a + I*a*Tan[c + d*x]])/(8*d*Sq
rt[Cot[c + d*x]]) + ((I/3)*a*B*(a + I*a*Tan[c + d*x])^(3/2))/(d*Cot[c + d*x]^(3/2))
________________________________________________________________________________________
Rubi [A] time = 1.08641, antiderivative size = 292, normalized size of antiderivative = 1.,
number of steps used = 11, 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} \[ -\frac{a^2 (2 A-3 i B) \sqrt{a+i a \tan (c+d x)}}{4 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{(-1)^{3/4} a^{5/2} (46 A-45 i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{8 d}+\frac{a^2 (19 B+18 i A) \sqrt{a+i a \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}-\frac{(4+4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}+\frac{i a B (a+i a \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]
[Out]
-((-1)^(3/4)*a^(5/2)*(46*A - (45*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]])/(8*d) - ((4 + 4*I)*a^(5/2)*(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 - (a^2*(2*A - (3*I)*B)*Sqr
t[a + I*a*Tan[c + d*x]])/(4*d*Cot[c + d*x]^(3/2)) + (a^2*((18*I)*A + 19*B)*Sqrt[a + I*a*Tan[c + d*x]])/(8*d*Sq
rt[Cot[c + d*x]]) + ((I/3)*a*B*(a + I*a*Tan[c + d*x])^(3/2))/(d*Cot[c + d*x]^(3/2))
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))^{5/2} (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))^{5/2} (A+B \tan (c+d x)) \, dx\\ &=\frac{i a B (a+i a \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{1}{3} \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} \left (\frac{3}{2} a (2 A-i B)+\frac{3}{2} a (2 i A+3 B) \tan (c+d x)\right ) \, dx\\ &=-\frac{a^2 (2 A-3 i B) \sqrt{a+i a \tan (c+d x)}}{4 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{i a B (a+i a \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{1}{6} \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{3}{4} a^2 (14 A-13 i B)+\frac{3}{4} a^2 (18 i A+19 B) \tan (c+d x)\right ) \, dx\\ &=-\frac{a^2 (2 A-3 i B) \sqrt{a+i a \tan (c+d x)}}{4 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a^2 (18 i A+19 B) \sqrt{a+i a \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{i a B (a+i a \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(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{3}{8} a^3 (18 i A+19 B)+\frac{3}{8} a^3 (46 A-45 i B) \tan (c+d x)\right )}{\sqrt{\tan (c+d x)}} \, dx}{6 a}\\ &=-\frac{a^2 (2 A-3 i B) \sqrt{a+i a \tan (c+d x)}}{4 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a^2 (18 i A+19 B) \sqrt{a+i a \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{i a B (a+i a \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\left (4 a^2 (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}{16} \left (a (46 i A+45 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{a^2 (2 A-3 i B) \sqrt{a+i a \tan (c+d x)}}{4 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a^2 (18 i A+19 B) \sqrt{a+i a \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{i a B (a+i a \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (8 i a^4 (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^3 (46 i A+45 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 )}{16 d}\\ &=-\frac{(4-4 i) a^{5/2} (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{a^2 (2 A-3 i B) \sqrt{a+i a \tan (c+d x)}}{4 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a^2 (18 i A+19 B) \sqrt{a+i a \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{i a B (a+i a \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (a^3 (46 i A+45 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 )}{8 d}\\ &=-\frac{(4-4 i) a^{5/2} (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{a^2 (2 A-3 i B) \sqrt{a+i a \tan (c+d x)}}{4 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a^2 (18 i A+19 B) \sqrt{a+i a \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{i a B (a+i a \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (a^3 (46 i A+45 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 )}{8 d}\\ &=-\frac{\sqrt [4]{-1} a^{5/2} (46 i A+45 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)}}{8 d}-\frac{(4-4 i) a^{5/2} (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{a^2 (2 A-3 i B) \sqrt{a+i a \tan (c+d x)}}{4 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{a^2 (18 i A+19 B) \sqrt{a+i a \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{i a B (a+i a \tan (c+d x))^{3/2}}{3 d \cot ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 9.61155, size = 484, normalized size = 1.66 \[ \frac{\cos ^3(c+d x) \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left (\frac{2}{3} (\cos (2 c)-i \sin (2 c)) \tan (c+d x) \sec ^2(c+d x) ((-12 A+26 i B) \sin (2 (c+d x))+(65 B+54 i A) \cos (2 (c+d x))+54 i A+49 B)-\sqrt{2} (\cos (3 c+d x)-i \sin (3 c+d x)) \sqrt{i \sin ^2(c+d x) (\cot (c+d x)+i)} \left (\sqrt{2} (46 A-45 i B) \log \left (\frac{2 e^{\frac{7 i c}{2}} \left (2 i \sqrt{-1+e^{2 i (c+d x)}}-i \sqrt{2} e^{i (c+d x)}+\sqrt{2}\right )}{(46 A-45 i B) \left (e^{i (c+d x)}-i\right )}\right )+\sqrt{2} (-46 A+45 i B) \log \left (\frac{2 e^{\frac{7 i c}{2}} \left (2 \sqrt{-1+e^{2 i (c+d x)}}+\sqrt{2} e^{i (c+d x)}-i \sqrt{2}\right )}{(45 B+46 i A) \left (e^{i (c+d x)}+i\right )}\right )+128 (A-i B) \log \left ((\cos (c)-i \sin (c)) \left (i \sin (c+d x)+\cos (c+d x)+\sqrt{i \sin (2 (c+d x))+\cos (2 (c+d x))-1}\right )\right )\right )\right )}{32 d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]],x]
[Out]
(Cos[c + d*x]^3*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x])*(-(Sqrt[2]*(Sqrt[2]*(46*A
- (45*I)*B)*Log[(2*E^(((7*I)/2)*c)*(Sqrt[2] - I*Sqrt[2]*E^(I*(c + d*x)) + (2*I)*Sqrt[-1 + E^((2*I)*(c + d*x))
]))/((46*A - (45*I)*B)*(-I + E^(I*(c + d*x))))] + Sqrt[2]*(-46*A + (45*I)*B)*Log[(2*E^(((7*I)/2)*c)*((-I)*Sqrt
[2] + Sqrt[2]*E^(I*(c + d*x)) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]))/(((46*I)*A + 45*B)*(I + E^(I*(c + d*x))))]
+ 128*(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[3*c + d*x] - I*Sin[3*c + d*x])) + (2*Sec[c + d*x
]^2*(Cos[2*c] - I*Sin[2*c])*((54*I)*A + 49*B + ((54*I)*A + 65*B)*Cos[2*(c + d*x)] + (-12*A + (26*I)*B)*Sin[2*(
c + d*x)])*Tan[c + d*x])/3))/(32*d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))
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Maple [B] time = 0.623, size = 4851, normalized size = 16.6 \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))^(5/2)*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x)
[Out]
1/96/d*2^(1/2)*a^2*(-24*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^3+52*B*2^(1/2)*((cos(d*x+c)-1)/
sin(d*x+c))^(1/2)*cos(d*x+c)^3-276*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^3+138*A*ln((
(cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)*2^(1/2)*cos(d*x+c)^3+132*I*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos
(d*x+c)^3*sin(d*x+c)-384*A*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-108*A*2^(1/2)*((co
s(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)+24*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c
)*sin(d*x+c)+384*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^3-138*I*A*2^(1/2)*cos(d*x+c)
^3*sin(d*x+c)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)-384*B*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1
)*cos(d*x+c)^3-384*B*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^3-192*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)*si
n(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))*cos(d*x+c)^3+16*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-130*B*2^(1/2)*(
(cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)-68*B*cos(d*x+c)*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c)
)^(1/2)*2^(1/2)+384*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^3+192*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)*s
in(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))*cos(d*x+c)^3+138*I*A*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)*ln(((cos(d*x+c)-1)/si
n(d*x+c))^(1/2)-1)+182*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^3*sin(d*x+c)+135*I*B*2^(1/2)*c
os(d*x+c)^3*sin(d*x+c)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)+270*I*B*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)*arctan(
((cos(d*x+c)-1)/sin(d*x+c))^(1/2))-135*I*B*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2
)-1)-384*A*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-192*A*cos(d*x+c)^4*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))+384*B*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+3
84*B*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+192*B*cos(d*x+c)^4*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))-135*B*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^3+270*B*arcta
n(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))*2^(1/2)*cos(d*x+c)^3+135*B*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)*2^(1/2
)*cos(d*x+c)^3+16*B*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-198*B*cos(d*x+c)^2*((cos(d*x+c)-1)/si
n(d*x+c))^(1/2)*2^(1/2)+24*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)-138*A*ln(((cos(d*x+c)-1)/sin
(d*x+c))^(1/2)+1)*2^(1/2)*cos(d*x+c)^3-52*B*2^(1/2)*cos(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-108*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)+130*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)-24*I*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)*sin(d*x+c)-68*I*B*2^(1/2)
*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)*sin(d*x+c)+276*I*A*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)*arctan(((cos(
d*x+c)-1)/sin(d*x+c))^(1/2))-270*B*2^(1/2)*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))-135*B*2^(1/2
)*cos(d*x+c)^4*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)+384*A*cos(d*x+c)^3*sin(d*x+c)*arctan(((cos(d*x+c)-1)/si
n(d*x+c))^(1/2)*2^(1/2)-1)+384*A*cos(d*x+c)^3*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+1
92*A*cos(d*x+c)^3*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))+138*A*2^(1/2)*cos(d*x+c)^4*l
n(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)+276*A*2^(1/2)*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))-13
8*A*2^(1/2)*cos(d*x+c)^4*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)+132*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/
2)*cos(d*x+c)^4+182*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^4-132*A*2^(1/2)*((cos(d*x+c)-1)/sin
(d*x+c))^(1/2)*cos(d*x+c)^2-384*B*cos(d*x+c)^3*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-
384*B*cos(d*x+c)^3*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-192*B*cos(d*x+c)^3*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))+384*I*A*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))
^(1/2)*2^(1/2)-1)+384*I*A*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+192*I*A*cos(d*x+c)^
4*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))+384*I*B*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))
^(1/2)*2^(1/2)-1)+384*I*B*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+192*I*B*cos(d*x+c)^
4*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))-384*I*A*cos(d*x+c)^3*arctan(((cos(d*x+c)-1)/sin(d*x+c))
^(1/2)*2^(1/2)-1)-384*I*A*cos(d*x+c)^3*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-192*I*A*cos(d*x+c)^
3*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))-384*I*B*cos(d*x+c)^3*arctan(((cos(d*x+c)-1)/sin(d*x+c))
^(1/2)*2^(1/2)-1)-384*I*B*cos(d*x+c)^3*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-192*I*B*cos(d*x+c)^
3*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))-16*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+135*B*
2^(1/2)*cos(d*x+c)^4*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)-132*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*c
os(d*x+c)^3*sin(d*x+c)-192*I*A*cos(d*x+c)^3*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))+52
*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^3+135*I*B*2^(1/2)*cos(d*x+c)^3*ln(((cos(d*x+c)-1)/si
n(d*x+c))^(1/2)+1)+270*I*B*2^(1/2)*cos(d*x+c)^3*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))-135*I*B*2^(1/2)*cos(
d*x+c)^3*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)-384*I*B*cos(d*x+c)^3*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*
x+c))^(1/2)*2^(1/2)-1)-384*I*B*cos(d*x+c)^3*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-192
*I*B*cos(d*x+c)^3*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))-132*I*A*2^(1/2)*((cos(d*x+c)
-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2+198*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2-24*I*A*2^(1/
2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)-52*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)-16
*I*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*sin(d*x+c)+182*B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(
d*x+c)^3*sin(d*x+c)-135*B*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)+270*B*2^(1/2
)*cos(d*x+c)^3*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))+135*B*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)*ln((
(cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)-138*A*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)
+1)-276*A*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))+138*A*2^(1/2)*cos(d*x+c)^3
*sin(d*x+c)*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)+132*I*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+
c)^4+138*I*A*2^(1/2)*cos(d*x+c)^4*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)-276*I*A*2^(1/2)*cos(d*x+c)^4*arctan(
((cos(d*x+c)-1)/sin(d*x+c))^(1/2))-138*I*A*2^(1/2)*cos(d*x+c)^4*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)-182*I*
B*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^4-135*I*B*2^(1/2)*cos(d*x+c)^4*ln(((cos(d*x+c)-1)/sin(d
*x+c))^(1/2)+1)-270*I*B*2^(1/2)*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2))+135*I*B*2^(1/2)*cos(d*x
+c)^4*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)+24*I*A*2^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^3-13
8*I*A*2^(1/2)*cos(d*x+c)^3*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+1)+276*I*A*2^(1/2)*cos(d*x+c)^3*arctan(((cos(d
*x+c)-1)/sin(d*x+c))^(1/2))+138*I*A*2^(1/2)*cos(d*x+c)^3*ln(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-1)-384*I*A*cos(d
*x+c)^3*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-384*I*A*cos(d*x+c)^3*sin(d*x+c)*arctan(
((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(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*s
in(d*x+c)-1+I+cos(d*x+c)-sin(d*x+c))/cos(d*x+c)^2/(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{5}{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))^(5/2)*(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)^(5/2)/sqrt(cot(d*x + c)), x)
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Fricas [B] time = 1.67243, size = 2843, normalized size = 9.74 \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))^(5/2)*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x, algorithm="fricas")
[Out]
1/48*(2*sqrt(2)*((66*A - 91*I*B)*a^2*e^(6*I*d*x + 6*I*c) + 7*(6*A - I*B)*a^2*e^(4*I*d*x + 4*I*c) - (66*A - 59*
I*B)*a^2*e^(2*I*d*x + 2*I*c) - 3*(14*A - 13*I*B)*a^2)*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) + 3*sqrt((2116*I*A^2 + 4140*A*B - 2025*I*B^2)*a^5/d^2)*(
d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((46*I*A + 45*B)*a
^2*e^(2*I*d*x + 2*I*c) + (-46*I*A - 45*B)*a^2)*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((2116*I*A^2 + 4140*A*B - 2025*I*B^2)*a^5/d^2)*d*e^(2
*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/((46*I*A + 45*B)*a^2)) - 3*sqrt((2116*I*A^2 + 4140*A*B - 2025*I*B^2)*a^5
/d^2)*(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((46*I*A +
45*B)*a^2*e^(2*I*d*x + 2*I*c) + (-46*I*A - 45*B)*a^2)*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((2116*I*A^2 + 4140*A*B - 2025*I*B^2)*a^5/d^2)
*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/((46*I*A + 45*B)*a^2)) - 24*sqrt((32*I*A^2 + 64*A*B - 32*I*B^2)*a
^5/d^2)*(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((4*I*A +
4*B)*a^2*e^(2*I*d*x + 2*I*c) + (-4*I*A - 4*B)*a^2)*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((32*I*A^2 + 64*A*B - 32*I*B^2)*a^5/d^2)*d*e^(2*I*
d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/((4*I*A + 4*B)*a^2)) + 24*sqrt((32*I*A^2 + 64*A*B - 32*I*B^2)*a^5/d^2)*(d*e
^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((4*I*A + 4*B)*a^2*e^
(2*I*d*x + 2*I*c) + (-4*I*A - 4*B)*a^2)*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((32*I*A^2 + 64*A*B - 32*I*B^2)*a^5/d^2)*d*e^(2*I*d*x + 2*I*c)
)*e^(-2*I*d*x - 2*I*c)/((4*I*A + 4*B)*a^2)))/(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*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))**(5/2)*(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{5}{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))^(5/2)*(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)^(5/2)/sqrt(cot(d*x + c)), x)