3.549 \(\int \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=205 \[ -\frac{2 a^2 (5 B+8 i A) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}+\frac{2 a^2 (38 A-35 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{(4+4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d} \]

[Out]

((-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[Co
t[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*a^2*(38*A - (35*I)*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(1
5*d) - (2*a^2*((8*I)*A + 5*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(15*d) - (2*a*A*Cot[c + d*x]^(5/2
)*(a + I*a*Tan[c + d*x])^(3/2))/(5*d)

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Rubi [A]  time = 0.735829, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4241, 3593, 3598, 12, 3544, 205} \[ -\frac{2 a^2 (5 B+8 i A) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}+\frac{2 a^2 (38 A-35 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{(4+4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

((-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[Co
t[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*a^2*(38*A - (35*I)*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(1
5*d) - (2*a^2*((8*I)*A + 5*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(15*d) - (2*a*A*Cot[c + d*x]^(5/2
)*(a + I*a*Tan[c + d*x])^(3/2))/(5*d)

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 3593

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

Rule 3598

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rubi steps

\begin{align*} \int \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{7}{2}}(c+d x)} \, dx\\ &=-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac{1}{5} \left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a+i a \tan (c+d x))^{3/2} \left (\frac{1}{2} a (8 i A+5 B)-\frac{1}{2} a (2 A-5 i B) \tan (c+d x)\right )}{\tan ^{\frac{5}{2}}(c+d x)} \, dx\\ &=-\frac{2 a^2 (8 i A+5 B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac{1}{15} \left (4 \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 (38 A-35 i B)-\frac{1}{4} a^2 (22 i A+25 B) \tan (c+d x)\right )}{\tan ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (38 A-35 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{2 a^2 (8 i A+5 B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac{\left (8 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int -\frac{15 a^3 (i A+B) \sqrt{a+i a \tan (c+d x)}}{2 \sqrt{\tan (c+d x)}} \, dx}{15 a}\\ &=\frac{2 a^2 (38 A-35 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{2 a^2 (8 i A+5 B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}-\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{2 a^2 (38 A-35 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{2 a^2 (8 i A+5 B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\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{(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{2 a^2 (38 A-35 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{2 a^2 (8 i A+5 B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{2 a A \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\\ \end{align*}

Mathematica [A]  time = 7.53274, size = 306, normalized size = 1.49 \[ \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left (-4 \sqrt{2} (A-i B) e^{-3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left (1+e^{2 i (c+d x)}\right )}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )-\frac{(\cos (2 c)-i \sin (2 c)) \sqrt{\cot (c+d x)} \csc ^2(c+d x) \sqrt{\sec (c+d x)} ((5 B+11 i A) \sin (2 (c+d x))+(41 A-35 i B) \cos (2 (c+d x))-35 (A-i B))}{15 (\cos (d x)+i \sin (d x))^2}\right )}{d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

(((-4*Sqrt[2]*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I
*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]
])/E^((3*I)*(c + d*x)) - (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^2*Sqrt[Sec[c + d*x]]*(Cos[2*c] - I*Sin[2*c])*(-35*(A
- I*B) + (41*A - (35*I)*B)*Cos[2*(c + d*x)] + ((11*I)*A + 5*B)*Sin[2*(c + d*x)]))/(15*(Cos[d*x] + I*Sin[d*x])
^2))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]
))

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Maple [B]  time = 0.799, size = 2246, normalized size = 11. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x)

[Out]

-1/15/d*a^2*2^(1/2)*(-5*B*2^(1/2)*cos(d*x+c)*sin(d*x+c)+60*I*A*cos(d*x+c)^2*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x
+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+60*I*B*cos(d*x+c)^2*sin(d*x+c)*((cos(d*x+c)-1)/
sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+52*A*cos(d*x+c)^3*2^(1/2)-49*A*cos(d*x+c
)*2^(1/2)-41*A*2^(1/2)*cos(d*x+c)^2-35*I*B*2^(1/2)+38*A*2^(1/2)+60*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*sin(d*x
+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+30*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*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))+60*I*B*cos(d*x+c)^2*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1
/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+30*I*B*cos(d*x+c)^2*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x
+c))^(1/2)*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))-35*B*2^(1/2)*sin(d*x+c)+40*B*cos(d*x+c)^2*sin(
d*x+c)*2^(1/2)+30*I*A*cos(d*x+c)^2*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))*((cos(d*x+c
)-1)/sin(d*x+c))^(1/2)+60*I*A*cos(d*x+c)^2*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)
/sin(d*x+c))^(1/2)*2^(1/2)+1)+52*I*A*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)-11*I*A*cos(d*x+c)*sin(d*x+c)*2^(1/2)-30*I
*A*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))*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-60*I*A*si
n(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-60*I*A*sin(d*x+
c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-60*I*B*sin(d*x+c)*((c
os(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-30*I*B*sin(d*x+c)*((cos(d*x
+c)-1)/sin(d*x+c))^(1/2)*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))-60*I*B*sin(d*x+c)*((cos(d*x+c)-1
)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+35*I*B*cos(d*x+c)^2*2^(1/2)-38*I*A*sin
(d*x+c)*2^(1/2)+40*I*B*cos(d*x+c)*2^(1/2)-40*I*B*cos(d*x+c)^3*2^(1/2)-60*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*s
in(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-60*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*sin(d*x+c
)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-30*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*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))+60*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*sin(d*x+c)*arctan(((cos(
d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-60*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*arctan((
(cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-60*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*arc
tan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-30*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*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))+60*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin
(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+60*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^
2*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+30*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*
x+c)^2*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)))*(cos(d*x+c)/sin(d*x+c))^(7/2)*(a*(I*si
n(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*sin(d*x+c)/(I*sin(d*x+c)+cos(d*x+c)-1)/cos(d*x+c)^3

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Maxima [B]  time = 3.58871, size = 2059, normalized size = 10.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/225*(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*(((-(900*I + 900)*A + (900*I -
900)*B)*a^2*cos(3*d*x + 3*c) + ((930*I + 930)*A - (750*I - 750)*B)*a^2*cos(d*x + c) + (-(900*I - 900)*A - (900
*I + 900)*B)*a^2*sin(3*d*x + 3*c) + ((930*I - 930)*A + (750*I + 750)*B)*a^2*sin(d*x + c))*cos(3/2*arctan2(sin(
2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((900*I - 900)*A + (900*I + 900)*B)*a^2*cos(3*d*x + 3*c) + (-(930*I -
930)*A - (750*I + 750)*B)*a^2*cos(d*x + c) + (-(900*I + 900)*A + (900*I - 900)*B)*a^2*sin(3*d*x + 3*c) + ((930
*I + 930)*A - (750*I - 750)*B)*a^2*sin(d*x + c))*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqr
t(a) + ((((900*I - 900)*A + (900*I + 900)*B)*a^2*cos(2*d*x + 2*c)^2 + ((900*I - 900)*A + (900*I + 900)*B)*a^2*
sin(2*d*x + 2*c)^2 + (-(1800*I - 1800)*A - (1800*I + 1800)*B)*a^2*cos(2*d*x + 2*c) + ((900*I - 900)*A + (900*I
+ 900)*B)*a^2)*arctan2(2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arc
tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 2*sin(d*x + c), 2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2
*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 2*cos(d*x + c)) + (((4
50*I + 450)*A - (450*I - 450)*B)*a^2*cos(2*d*x + 2*c)^2 + ((450*I + 450)*A - (450*I - 450)*B)*a^2*sin(2*d*x +
2*c)^2 + (-(900*I + 900)*A + (900*I - 900)*B)*a^2*cos(2*d*x + 2*c) + ((450*I + 450)*A - (450*I - 450)*B)*a^2)*
log(4*cos(d*x + c)^2 + 4*sin(d*x + c)^2 + 4*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c)
+ 1)*(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c) - 1))^2) + 8*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*(cos(d*x + c)*co
s(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + sin(d*x + c)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*
d*x + 2*c) - 1)))))*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*sqrt(a) + (((-(90
0*I + 900)*A + (900*I - 900)*B)*a^2*cos(5*d*x + 5*c) + ((750*I + 750)*A - (1650*I - 1650)*B)*a^2*cos(3*d*x + 3
*c) + (-(210*I + 210)*A + (750*I - 750)*B)*a^2*cos(d*x + c) + (-(900*I - 900)*A - (900*I + 900)*B)*a^2*sin(5*d
*x + 5*c) + ((750*I - 750)*A + (1650*I + 1650)*B)*a^2*sin(3*d*x + 3*c) + (-(210*I - 210)*A - (750*I + 750)*B)*
a^2*sin(d*x + c))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((240*I + 240)*A - (600*I - 600)
*B)*a^2*cos(d*x + c) + ((240*I - 240)*A + (600*I + 600)*B)*a^2*sin(d*x + c) + (((240*I + 240)*A - (600*I - 600
)*B)*a^2*cos(d*x + c) + ((240*I - 240)*A + (600*I + 600)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (((240*I +
240)*A - (600*I - 600)*B)*a^2*cos(d*x + c) + ((240*I - 240)*A + (600*I + 600)*B)*a^2*sin(d*x + c))*sin(2*d*x +
2*c)^2 + ((-(480*I + 480)*A + (1200*I - 1200)*B)*a^2*cos(d*x + c) + (-(480*I - 480)*A - (1200*I + 1200)*B)*a^
2*sin(d*x + c))*cos(2*d*x + 2*c))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((900*I - 900)*A
+ (900*I + 900)*B)*a^2*cos(5*d*x + 5*c) + (-(750*I - 750)*A - (1650*I + 1650)*B)*a^2*cos(3*d*x + 3*c) + ((210
*I - 210)*A + (750*I + 750)*B)*a^2*cos(d*x + c) + (-(900*I + 900)*A + (900*I - 900)*B)*a^2*sin(5*d*x + 5*c) +
((750*I + 750)*A - (1650*I - 1650)*B)*a^2*sin(3*d*x + 3*c) + (-(210*I + 210)*A + (750*I - 750)*B)*a^2*sin(d*x
+ c))*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + ((-(240*I - 240)*A - (600*I + 600)*B)*a^2*cos
(d*x + c) + ((240*I + 240)*A - (600*I - 600)*B)*a^2*sin(d*x + c) + ((-(240*I - 240)*A - (600*I + 600)*B)*a^2*c
os(d*x + c) + ((240*I + 240)*A - (600*I - 600)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(240*I - 240)*A -
(600*I + 600)*B)*a^2*cos(d*x + c) + ((240*I + 240)*A - (600*I - 600)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 +
(((480*I - 480)*A + (1200*I + 1200)*B)*a^2*cos(d*x + c) + (-(480*I + 480)*A + (1200*I - 1200)*B)*a^2*sin(d*x
+ c))*cos(2*d*x + 2*c))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a))/((cos(2*d*x + 2*c)^
2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(5/4)*d)

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Fricas [B]  time = 1.48638, size = 1469, normalized size = 7.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/30*(8*sqrt(2)*(2*(13*A - 10*I*B)*a^2*e^(4*I*d*x + 4*I*c) - 35*(A - I*B)*a^2*e^(2*I*d*x + 2*I*c) + 15*(A - 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) - 15*sqrt((32*I*A^2 + 64*A*B - 32*I*B^2)*a^5/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)*((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)) + 15*sqrt((32*I*A^2 + 64*A*B -
32*I*B^2)*a^5/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)*((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^(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(cot(d*x+c)**(7/2)*(a+I*a*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \tan \left (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cot \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^(5/2)*cot(d*x + c)^(7/2), x)