3.547 \(\int \cot ^{\frac {11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=297 \[ \frac {(4+4 i) a^{5/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 a^2 (3 B+4 i A) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}+\frac {2 a^2 (46 A-45 i B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {8 a^2 (60 B+59 i A) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{315 d}-\frac {8 a^2 (197 A-195 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{315 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d} \]

[Out]

(4+4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(
d*x+c)^(1/2)/d+8/315*a^2*(59*I*A+60*B)*cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^(1/2)/d+2/105*a^2*(46*A-45*I*B)*cot
(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^(1/2)/d-2/21*a^2*(4*I*A+3*B)*cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(1/2)/d-8/31
5*a^2*(197*A-195*I*B)*cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^(1/2)/d-2/9*a*A*cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^
(3/2)/d

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Rubi [A]  time = 1.13, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {2 a^2 (3 B+4 i A) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}+\frac {2 a^2 (46 A-45 i B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {8 a^2 (60 B+59 i A) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{315 d}-\frac {8 a^2 (197 A-195 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {(4+4 i) a^{5/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^(11/2)*(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[Cot
[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (8*a^2*(197*A - (195*I)*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(
315*d) + (8*a^2*((59*I)*A + 60*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(315*d) + (2*a^2*(46*A - (45*
I)*B)*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a^2*((4*I)*A + 3*B)*Cot[c + d*x]^(7/2)*Sqrt[
a + I*a*Tan[c + d*x]])/(21*d) - (2*a*A*Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2))/(9*d)

Rule 12

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

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 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 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 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]

Rubi steps

\begin {align*} \int \cot ^{\frac {11}{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 {11}{2}}(c+d x)} \, dx\\ &=-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac {1}{9} \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 {3}{2} a (4 i A+3 B)-\frac {3}{2} a (2 A-3 i B) \tan (c+d x)\right )}{\tan ^{\frac {9}{2}}(c+d x)} \, dx\\ &=-\frac {2 a^2 (4 i A+3 B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac {1}{63} \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 {3}{4} a^2 (46 A-45 i B)-\frac {3}{4} a^2 (38 i A+39 B) \tan (c+d x)\right )}{\tan ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (46 A-45 i B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (4 i A+3 B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{2} a^3 (59 i A+60 B)+\frac {3}{2} a^3 (46 A-45 i B) \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx}{315 a}\\ &=\frac {8 a^2 (59 i A+60 B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {2 a^2 (46 A-45 i B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (4 i A+3 B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac {\left (16 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)} \left (\frac {3}{4} a^4 (197 A-195 i B)+\frac {3}{2} a^4 (59 i A+60 B) \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{945 a^2}\\ &=-\frac {8 a^2 (197 A-195 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {8 a^2 (59 i A+60 B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {2 a^2 (46 A-45 i B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (4 i A+3 B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac {\left (32 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {945 a^5 (i A+B) \sqrt {a+i a \tan (c+d x)}}{8 \sqrt {\tan (c+d x)}} \, dx}{945 a^3}\\ &=-\frac {8 a^2 (197 A-195 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {8 a^2 (59 i A+60 B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {2 a^2 (46 A-45 i B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (4 i A+3 B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 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 {8 a^2 (197 A-195 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {8 a^2 (59 i A+60 B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {2 a^2 (46 A-45 i B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (4 i A+3 B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 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 {8 a^2 (197 A-195 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {8 a^2 (59 i A+60 B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {2 a^2 (46 A-45 i B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (4 i A+3 B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}\\ \end {align*}

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Mathematica [A]  time = 12.09, size = 354, normalized size = 1.19 \[ \frac {(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left (4 \sqrt {2} (A-i B) e^{-3 i (c+d x)} \sqrt {-1+e^{2 i (c+d x)}} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {\frac {i \left (1+e^{2 i (c+d x)}\right )}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left (\frac {e^{i (c+d x)}}{\sqrt {-1+e^{2 i (c+d x)}}}\right )+\frac {(\cos (2 c)-i \sin (2 c)) \sqrt {\cot (c+d x)} \csc ^4(c+d x) \sqrt {\sec (c+d x)} (12 (251 A-260 i B) \cos (2 (c+d x))+(-961 A+915 i B) \cos (4 (c+d x))+282 i A \sin (2 (c+d x))-331 i A \sin (4 (c+d x))-2331 A+390 B \sin (2 (c+d x))-285 B \sin (4 (c+d x))+2205 i B)}{1260 (\cos (d x)+i \sin (d x))^2}\right )}{d \sec ^{\frac {7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^(11/2)*(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]^4*Sqrt[Sec[c + d*x]]*(Cos[2*c] - I*Sin[2*c])*(-2331*A
+ (2205*I)*B + 12*(251*A - (260*I)*B)*Cos[2*(c + d*x)] + (-961*A + (915*I)*B)*Cos[4*(c + d*x)] + (282*I)*A*Si
n[2*(c + d*x)] + 390*B*Sin[2*(c + d*x)] - (331*I)*A*Sin[4*(c + d*x)] - 285*B*Sin[4*(c + d*x)]))/(1260*(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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fricas [B]  time = 0.72, size = 627, normalized size = 2.11 \[ -\frac {16 \, \sqrt {2} {\left (2 \, {\left (323 \, A - 300 i \, B\right )} a^{2} e^{\left (9 i \, d x + 9 i \, c\right )} - 27 \, {\left (61 \, A - 65 i \, B\right )} a^{2} e^{\left (7 i \, d x + 7 i \, c\right )} + 63 \, {\left (37 \, A - 35 i \, B\right )} a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} - 1365 \, {\left (A - i \, B\right )} a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + 315 \, {\left (A - i \, B\right )} a^{2} 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 {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} - 315 \, \sqrt {\frac {{\left (128 i \, A^{2} + 256 \, A B - 128 i \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (-\frac {{\left (16 \, {\left (A - i \, B\right )} a^{3} e^{\left (i \, d x + i \, c\right )} - \sqrt {2} \sqrt {\frac {{\left (128 i \, A^{2} + 256 \, A B - 128 i \, B^{2}\right )} a^{5}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (4 i \, A + 4 \, B\right )} a^{2}}\right ) + 315 \, \sqrt {\frac {{\left (128 i \, A^{2} + 256 \, A B - 128 i \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (-\frac {{\left (16 \, {\left (A - i \, B\right )} a^{3} e^{\left (i \, d x + i \, c\right )} - \sqrt {2} \sqrt {\frac {{\left (128 i \, A^{2} + 256 \, A B - 128 i \, B^{2}\right )} a^{5}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (4 i \, A + 4 \, B\right )} a^{2}}\right )}{1260 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/1260*(16*sqrt(2)*(2*(323*A - 300*I*B)*a^2*e^(9*I*d*x + 9*I*c) - 27*(61*A - 65*I*B)*a^2*e^(7*I*d*x + 7*I*c)
+ 63*(37*A - 35*I*B)*a^2*e^(5*I*d*x + 5*I*c) - 1365*(A - I*B)*a^2*e^(3*I*d*x + 3*I*c) + 315*(A - I*B)*a^2*e^(I
*d*x + I*c))*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)) - 3
15*sqrt((128*I*A^2 + 256*A*B - 128*I*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4
*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log(-(16*(A - I*B)*a^3*e^(I*d*x + I*c) - sqrt(2)*sqrt((128*I*A^
2 + 256*A*B - 128*I*B^2)*a^5/d^2)*(I*d*e^(2*I*d*x + 2*I*c) - I*d)*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)/((4*I*A + 4*B)*a^2)) + 315*sqrt((128*I*A^2
+ 256*A*B - 128*I*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) -
4*d*e^(2*I*d*x + 2*I*c) + d)*log(-(16*(A - I*B)*a^3*e^(I*d*x + I*c) - sqrt(2)*sqrt((128*I*A^2 + 256*A*B - 128*
I*B^2)*a^5/d^2)*(-I*d*e^(2*I*d*x + 2*I*c) + I*d)*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)/((4*I*A + 4*B)*a^2)))/(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d
*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 4.06, size = 3412, normalized size = 11.49 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x)

[Out]

-1/315/d*(1260*I*A*cos(d*x+c)^4*sin(d*x+c)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*((-1+cos(d*x+c
))/sin(d*x+c))^(1/2)+1292*A*2^(1/2)*cos(d*x+c)^5-961*A*2^(1/2)*cos(d*x+c)^4-2281*A*2^(1/2)*cos(d*x+c)^3+1714*A
*2^(1/2)*cos(d*x+c)^2+1024*A*2^(1/2)*cos(d*x+c)+780*B*sin(d*x+c)*2^(1/2)-788*A*2^(1/2)+630*I*A*cos(d*x+c)^4*si
n(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x
+c)-sin(d*x+c)+1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))-1260*A*sin(
d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)-630*A*sin(d*x+c
)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin
(d*x+c)-1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))-1260*A*sin(d*x+c)*
((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)+1260*B*sin(d*x+c)*((-1
+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)+1260*B*sin(d*x+c)*((-1+cos
(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)+630*B*sin(d*x+c)*((-1+cos(d*x+
c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(2^(
1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))+1200*B*2^(1/2)*cos(d*x+c)^4*sin(d
*x+c)-1695*I*B*cos(d*x+c)^2*2^(1/2)-1260*A*cos(d*x+c)^4*sin(d*x+c)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))
^(1/2)-1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-630*A*cos(d*x+c)^4*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*
ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(2^(1/2)*((-1+cos(d*x+c))/
sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))-1260*A*cos(d*x+c)^4*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x
+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)+1260*B*cos(d*x+c)^4*sin(d*x+c)*arctan(2^(1/2)*
((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+630*B*cos(d*x+c)^4*sin(d*x+c)*((-1+co
s(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1
)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))+1260*B*cos(d*x+c)^4*sin(d*x
+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)+1292*I*A*cos(d*x+c
)^4*sin(d*x+c)*2^(1/2)-331*I*A*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)-1950*I*A*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)-285*B*
cos(d*x+c)^3*sin(d*x+c)*2^(1/2)+788*I*A*sin(d*x+c)*2^(1/2)-1020*I*B*cos(d*x+c)*2^(1/2)-1200*I*B*cos(d*x+c)^5*2
^(1/2)+915*I*B*cos(d*x+c)^4*2^(1/2)+2220*I*B*cos(d*x+c)^3*2^(1/2)+780*I*B*2^(1/2)+240*B*2^(1/2)*cos(d*x+c)*sin
(d*x+c)-1935*B*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+236*I*A*cos(d*x+c)*sin(d*x+c)*2^(1/2)+1260*I*A*sin(d*x+c)*arcta
n(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+630*I*A*sin(d*x+c)*((-1+cos
(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)
/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))+1260*I*A*sin(d*x+c)*((-1+cos
(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)+1260*I*B*sin(d*x+c)*arctan(2^(
1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+630*I*B*sin(d*x+c)*((-1+cos(d*x+
c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(2^(
1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))+1260*I*B*sin(d*x+c)*((-1+cos(d*x+
c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)+1260*I*A*cos(d*x+c)^4*sin(d*x+c)*((
-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)+1260*I*B*cos(d*x+c)^4*si
n(d*x+c)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+630*I*B*cos(d
*x+c)^4*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+
c)+cos(d*x+c)+sin(d*x+c)-1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))+1
260*I*B*cos(d*x+c)^4*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))
^(1/2)+1)-2520*I*A*cos(d*x+c)^2*sin(d*x+c)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1)*((-1+cos(d*x+c
))/sin(d*x+c))^(1/2)-1260*I*A*cos(d*x+c)^2*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos
(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin
(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))-2520*I*A*cos(d*x+c)^2*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2
^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)-2520*I*B*cos(d*x+c)^2*sin(d*x+c)*arctan(2^(1/2)*((-1+cos(d*x+c))/
sin(d*x+c))^(1/2)-1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-1260*I*B*cos(d*x+c)^2*sin(d*x+c)*((-1+cos(d*x+c))/sin(
d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(2^(1/2)*((-
1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))-2520*I*B*cos(d*x+c)^2*sin(d*x+c)*((-1+cos
(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)-1260*B*cos(d*x+c)^2*sin(d*x+c)
*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(
d*x+c)+1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))+2520*A*cos(d*x+c)^2
*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+1)+1260*A*cos
(d*x+c)^2*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*ln(-(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*
x+c)+cos(d*x+c)+sin(d*x+c)-1)/(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))
+2520*A*cos(d*x+c)^2*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))
^(1/2)-1)-2520*B*cos(d*x+c)^2*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d*x+c))/si
n(d*x+c))^(1/2)+1)-2520*B*cos(d*x+c)^2*sin(d*x+c)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*arctan(2^(1/2)*((-1+cos(d
*x+c))/sin(d*x+c))^(1/2)-1))*(cos(d*x+c)/sin(d*x+c))^(11/2)*(a*(I*sin(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)^5*2^(1/2)*a^2

________________________________________________________________________________________

maxima [B]  time = 11.32, size = 4459, normalized size = 15.01 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/1587600*(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*(((-(6350400*I + 6350400)*A
+ (6350400*I - 6350400)*B)*a^2*cos(7*d*x + 7*c) + ((21168000*I + 21168000)*A - (21168000*I - 21168000)*B)*a^2*
cos(5*d*x + 5*c) + (-(25824960*I + 25824960)*A + (25824960*I - 25824960)*B)*a^2*cos(3*d*x + 3*c) + ((11429460*
I + 11429460)*A - (11139660*I - 11139660)*B)*a^2*cos(d*x + c) + (-(6350400*I - 6350400)*A - (6350400*I + 63504
00)*B)*a^2*sin(7*d*x + 7*c) + ((21168000*I - 21168000)*A + (21168000*I + 21168000)*B)*a^2*sin(5*d*x + 5*c) + (
-(25824960*I - 25824960)*A - (25824960*I + 25824960)*B)*a^2*sin(3*d*x + 3*c) + ((11429460*I - 11429460)*A + (1
1139660*I + 11139660)*B)*a^2*sin(d*x + c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((12196
80*I + 1219680)*A - (756000*I - 756000)*B)*a^2*cos(d*x + c) + ((1219680*I - 1219680)*A + (756000*I + 756000)*B
)*a^2*sin(d*x + c) + (((1219680*I + 1219680)*A - (756000*I - 756000)*B)*a^2*cos(d*x + c) + ((1219680*I - 12196
80)*A + (756000*I + 756000)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (((1219680*I + 1219680)*A - (756000*I -
756000)*B)*a^2*cos(d*x + c) + ((1219680*I - 1219680)*A + (756000*I + 756000)*B)*a^2*sin(d*x + c))*sin(2*d*x +
2*c)^2 + ((-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(6350400*I + 6350400
)*A + (6350400*I - 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I + 12700800)*A - (12700800*I - 12700800)*B
)*a^2*cos(2*d*x + 2*c) + (-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2)*cos(3*d*x + 3*c) + ((-(2439
360*I + 2439360)*A + (1512000*I - 1512000)*B)*a^2*cos(d*x + c) + (-(2439360*I - 2439360)*A - (1512000*I + 1512
000)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + ((-(6350400*I - 6350400)*A - (6350400*I + 6350400)*B)*a^2*cos(2*d
*x + 2*c)^2 + (-(6350400*I - 6350400)*A - (6350400*I + 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I - 127
00800)*A + (12700800*I + 12700800)*B)*a^2*cos(2*d*x + 2*c) + (-(6350400*I - 6350400)*A - (6350400*I + 6350400)
*B)*a^2)*sin(3*d*x + 3*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((6350400*I - 6350400)*
A + (6350400*I + 6350400)*B)*a^2*cos(7*d*x + 7*c) + (-(21168000*I - 21168000)*A - (21168000*I + 21168000)*B)*a
^2*cos(5*d*x + 5*c) + ((25824960*I - 25824960)*A + (25824960*I + 25824960)*B)*a^2*cos(3*d*x + 3*c) + (-(114294
60*I - 11429460)*A - (11139660*I + 11139660)*B)*a^2*cos(d*x + c) + (-(6350400*I + 6350400)*A + (6350400*I - 63
50400)*B)*a^2*sin(7*d*x + 7*c) + ((21168000*I + 21168000)*A - (21168000*I - 21168000)*B)*a^2*sin(5*d*x + 5*c)
+ (-(25824960*I + 25824960)*A + (25824960*I - 25824960)*B)*a^2*sin(3*d*x + 3*c) + ((11429460*I + 11429460)*A -
(11139660*I - 11139660)*B)*a^2*sin(d*x + c))*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + ((-(1
219680*I - 1219680)*A - (756000*I + 756000)*B)*a^2*cos(d*x + c) + ((1219680*I + 1219680)*A - (756000*I - 75600
0)*B)*a^2*sin(d*x + c) + ((-(1219680*I - 1219680)*A - (756000*I + 756000)*B)*a^2*cos(d*x + c) + ((1219680*I +
1219680)*A - (756000*I - 756000)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(1219680*I - 1219680)*A - (75600
0*I + 756000)*B)*a^2*cos(d*x + c) + ((1219680*I + 1219680)*A - (756000*I - 756000)*B)*a^2*sin(d*x + c))*sin(2*
d*x + 2*c)^2 + (((6350400*I - 6350400)*A + (6350400*I + 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + ((6350400*I - 635
0400)*A + (6350400*I + 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + (-(12700800*I - 12700800)*A - (12700800*I + 127008
00)*B)*a^2*cos(2*d*x + 2*c) + ((6350400*I - 6350400)*A + (6350400*I + 6350400)*B)*a^2)*cos(3*d*x + 3*c) + (((2
439360*I - 2439360)*A + (1512000*I + 1512000)*B)*a^2*cos(d*x + c) + (-(2439360*I + 2439360)*A + (1512000*I - 1
512000)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + ((-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*cos(
2*d*x + 2*c)^2 + (-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I +
12700800)*A - (12700800*I - 12700800)*B)*a^2*cos(2*d*x + 2*c) + (-(6350400*I + 6350400)*A + (6350400*I - 63504
00)*B)*a^2)*sin(3*d*x + 3*c))*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a) + ((((6350400*
I - 6350400)*A + (6350400*I + 6350400)*B)*a^2*cos(2*d*x + 2*c)^4 + ((6350400*I - 6350400)*A + (6350400*I + 635
0400)*B)*a^2*sin(2*d*x + 2*c)^4 + (-(25401600*I - 25401600)*A - (25401600*I + 25401600)*B)*a^2*cos(2*d*x + 2*c
)^3 + ((38102400*I - 38102400)*A + (38102400*I + 38102400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(25401600*I - 2540160
0)*A - (25401600*I + 25401600)*B)*a^2*cos(2*d*x + 2*c) + ((6350400*I - 6350400)*A + (6350400*I + 6350400)*B)*a
^2 + (((12700800*I - 12700800)*A + (12700800*I + 12700800)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(25401600*I - 2540160
0)*A - (25401600*I + 25401600)*B)*a^2*cos(2*d*x + 2*c) + ((12700800*I - 12700800)*A + (12700800*I + 12700800)*
B)*a^2)*sin(2*d*x + 2*c)^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*arctan2(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)) + (((3175200*I + 3175200)*A - (3175200*I - 3175200)*B)*a^2*cos(2*d*x + 2*c)^4 + ((3175200*I + 3175200)*A
- (3175200*I - 3175200)*B)*a^2*sin(2*d*x + 2*c)^4 + (-(12700800*I + 12700800)*A + (12700800*I - 12700800)*B)*
a^2*cos(2*d*x + 2*c)^3 + ((19051200*I + 19051200)*A - (19051200*I - 19051200)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(1
2700800*I + 12700800)*A + (12700800*I - 12700800)*B)*a^2*cos(2*d*x + 2*c) + ((3175200*I + 3175200)*A - (317520
0*I - 3175200)*B)*a^2 + (((6350400*I + 6350400)*A - (6350400*I - 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(12700
800*I + 12700800)*A + (12700800*I - 12700800)*B)*a^2*cos(2*d*x + 2*c) + ((6350400*I + 6350400)*A - (6350400*I
- 6350400)*B)*a^2)*sin(2*d*x + 2*c)^2)*log(4*cos(d*x + c)^2 + 4*sin(d*x + c)^2 + 4*sqrt(cos(2*d*x + 2*c)^2 + s
in(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)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + sin(d*x + c)*s
in(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) + (((-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*cos(9*d*x + 9*c) +
((7408800*I + 7408800)*A - (13759200*I - 13759200)*B)*a^2*cos(7*d*x + 7*c) + (-(8414280*I + 8414280)*A + (1317
7080*I - 13177080)*B)*a^2*cos(5*d*x + 5*c) + ((631260*I + 631260)*A - (6717060*I - 6717060)*B)*a^2*cos(3*d*x +
3*c) + ((1079820*I + 1079820)*A + (948780*I - 948780)*B)*a^2*cos(d*x + c) + (-(6350400*I - 6350400)*A - (6350
400*I + 6350400)*B)*a^2*sin(9*d*x + 9*c) + ((7408800*I - 7408800)*A + (13759200*I + 13759200)*B)*a^2*sin(7*d*x
+ 7*c) + (-(8414280*I - 8414280)*A - (13177080*I + 13177080)*B)*a^2*sin(5*d*x + 5*c) + ((631260*I - 631260)*A
+ (6717060*I + 6717060)*B)*a^2*sin(3*d*x + 3*c) + ((1079820*I - 1079820)*A - (948780*I + 948780)*B)*a^2*sin(d
*x + c))*cos(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + ((-(7159320*I + 7159320)*A + (6811560*I -
6811560)*B)*a^2*cos(d*x + c) + (-(7159320*I - 7159320)*A - (6811560*I + 6811560)*B)*a^2*sin(d*x + c) + ((-(715
9320*I + 7159320)*A + (6811560*I - 6811560)*B)*a^2*cos(d*x + c) + (-(7159320*I - 7159320)*A - (6811560*I + 681
1560)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(7159320*I + 7159320)*A + (6811560*I - 6811560)*B)*a^2*cos(
d*x + c) + (-(7159320*I - 7159320)*A - (6811560*I + 6811560)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + ((-(635
0400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(6350400*I + 6350400)*A + (6350400*I
- 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I + 12700800)*A - (12700800*I - 12700800)*B)*a^2*cos(2*d*x
+ 2*c) + (-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2)*cos(5*d*x + 5*c) + (((14817600*I + 14817600
)*A - (14817600*I - 14817600)*B)*a^2*cos(2*d*x + 2*c)^2 + ((14817600*I + 14817600)*A - (14817600*I - 14817600)
*B)*a^2*sin(2*d*x + 2*c)^2 + (-(29635200*I + 29635200)*A + (29635200*I - 29635200)*B)*a^2*cos(2*d*x + 2*c) + (
(14817600*I + 14817600)*A - (14817600*I - 14817600)*B)*a^2)*cos(3*d*x + 3*c) + (((14318640*I + 14318640)*A - (
13623120*I - 13623120)*B)*a^2*cos(d*x + c) + ((14318640*I - 14318640)*A + (13623120*I + 13623120)*B)*a^2*sin(d
*x + c))*cos(2*d*x + 2*c) + ((-(6350400*I - 6350400)*A - (6350400*I + 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(
6350400*I - 6350400)*A - (6350400*I + 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I - 12700800)*A + (12700
800*I + 12700800)*B)*a^2*cos(2*d*x + 2*c) + (-(6350400*I - 6350400)*A - (6350400*I + 6350400)*B)*a^2)*sin(5*d*
x + 5*c) + (((14817600*I - 14817600)*A + (14817600*I + 14817600)*B)*a^2*cos(2*d*x + 2*c)^2 + ((14817600*I - 14
817600)*A + (14817600*I + 14817600)*B)*a^2*sin(2*d*x + 2*c)^2 + (-(29635200*I - 29635200)*A - (29635200*I + 29
635200)*B)*a^2*cos(2*d*x + 2*c) + ((14817600*I - 14817600)*A + (14817600*I + 14817600)*B)*a^2)*sin(3*d*x + 3*c
))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + ((((12378240*I + 12378240)*A - (13305600*I - 133
05600)*B)*a^2*cos(d*x + c) + ((12378240*I - 12378240)*A + (13305600*I + 13305600)*B)*a^2*sin(d*x + c))*cos(2*d
*x + 2*c)^4 + (((12378240*I + 12378240)*A - (13305600*I - 13305600)*B)*a^2*cos(d*x + c) + ((12378240*I - 12378
240)*A + (13305600*I + 13305600)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^4 + ((-(49512960*I + 49512960)*A + (532
22400*I - 53222400)*B)*a^2*cos(d*x + c) + (-(49512960*I - 49512960)*A - (53222400*I + 53222400)*B)*a^2*sin(d*x
+ c))*cos(2*d*x + 2*c)^3 + ((12378240*I + 12378240)*A - (13305600*I - 13305600)*B)*a^2*cos(d*x + c) + ((12378
240*I - 12378240)*A + (13305600*I + 13305600)*B)*a^2*sin(d*x + c) + (((74269440*I + 74269440)*A - (79833600*I
- 79833600)*B)*a^2*cos(d*x + c) + ((74269440*I - 74269440)*A + (79833600*I + 79833600)*B)*a^2*sin(d*x + c))*co
s(2*d*x + 2*c)^2 + (((24756480*I + 24756480)*A - (26611200*I - 26611200)*B)*a^2*cos(d*x + c) + ((24756480*I -
24756480)*A + (26611200*I + 26611200)*B)*a^2*sin(d*x + c) + (((24756480*I + 24756480)*A - (26611200*I - 266112
00)*B)*a^2*cos(d*x + c) + ((24756480*I - 24756480)*A + (26611200*I + 26611200)*B)*a^2*sin(d*x + c))*cos(2*d*x
+ 2*c)^2 + ((-(49512960*I + 49512960)*A + (53222400*I - 53222400)*B)*a^2*cos(d*x + c) + (-(49512960*I - 495129
60)*A - (53222400*I + 53222400)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + ((-(49512960*I + 4
9512960)*A + (53222400*I - 53222400)*B)*a^2*cos(d*x + c) + (-(49512960*I - 49512960)*A - (53222400*I + 5322240
0)*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)) + (((635040
0*I - 6350400)*A + (6350400*I + 6350400)*B)*a^2*cos(9*d*x + 9*c) + (-(7408800*I - 7408800)*A - (13759200*I + 1
3759200)*B)*a^2*cos(7*d*x + 7*c) + ((8414280*I - 8414280)*A + (13177080*I + 13177080)*B)*a^2*cos(5*d*x + 5*c)
+ (-(631260*I - 631260)*A - (6717060*I + 6717060)*B)*a^2*cos(3*d*x + 3*c) + (-(1079820*I - 1079820)*A + (94878
0*I + 948780)*B)*a^2*cos(d*x + c) + (-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*sin(9*d*x + 9*c)
+ ((7408800*I + 7408800)*A - (13759200*I - 13759200)*B)*a^2*sin(7*d*x + 7*c) + (-(8414280*I + 8414280)*A + (13
177080*I - 13177080)*B)*a^2*sin(5*d*x + 5*c) + ((631260*I + 631260)*A - (6717060*I - 6717060)*B)*a^2*sin(3*d*x
+ 3*c) + ((1079820*I + 1079820)*A + (948780*I - 948780)*B)*a^2*sin(d*x + c))*sin(9/2*arctan2(sin(2*d*x + 2*c)
, cos(2*d*x + 2*c) - 1)) + (((7159320*I - 7159320)*A + (6811560*I + 6811560)*B)*a^2*cos(d*x + c) + (-(7159320*
I + 7159320)*A + (6811560*I - 6811560)*B)*a^2*sin(d*x + c) + (((7159320*I - 7159320)*A + (6811560*I + 6811560)
*B)*a^2*cos(d*x + c) + (-(7159320*I + 7159320)*A + (6811560*I - 6811560)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)
^2 + (((7159320*I - 7159320)*A + (6811560*I + 6811560)*B)*a^2*cos(d*x + c) + (-(7159320*I + 7159320)*A + (6811
560*I - 6811560)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + (((6350400*I - 6350400)*A + (6350400*I + 6350400)*B
)*a^2*cos(2*d*x + 2*c)^2 + ((6350400*I - 6350400)*A + (6350400*I + 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + (-(127
00800*I - 12700800)*A - (12700800*I + 12700800)*B)*a^2*cos(2*d*x + 2*c) + ((6350400*I - 6350400)*A + (6350400*
I + 6350400)*B)*a^2)*cos(5*d*x + 5*c) + ((-(14817600*I - 14817600)*A - (14817600*I + 14817600)*B)*a^2*cos(2*d*
x + 2*c)^2 + (-(14817600*I - 14817600)*A - (14817600*I + 14817600)*B)*a^2*sin(2*d*x + 2*c)^2 + ((29635200*I -
29635200)*A + (29635200*I + 29635200)*B)*a^2*cos(2*d*x + 2*c) + (-(14817600*I - 14817600)*A - (14817600*I + 14
817600)*B)*a^2)*cos(3*d*x + 3*c) + ((-(14318640*I - 14318640)*A - (13623120*I + 13623120)*B)*a^2*cos(d*x + c)
+ ((14318640*I + 14318640)*A - (13623120*I - 13623120)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + ((-(6350400*I +
6350400)*A + (6350400*I - 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(6350400*I + 6350400)*A + (6350400*I - 63504
00)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I + 12700800)*A - (12700800*I - 12700800)*B)*a^2*cos(2*d*x + 2*c) +
(-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2)*sin(5*d*x + 5*c) + (((14817600*I + 14817600)*A - (1
4817600*I - 14817600)*B)*a^2*cos(2*d*x + 2*c)^2 + ((14817600*I + 14817600)*A - (14817600*I - 14817600)*B)*a^2*
sin(2*d*x + 2*c)^2 + (-(29635200*I + 29635200)*A + (29635200*I - 29635200)*B)*a^2*cos(2*d*x + 2*c) + ((1481760
0*I + 14817600)*A - (14817600*I - 14817600)*B)*a^2)*sin(3*d*x + 3*c))*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*
d*x + 2*c) - 1)) + (((-(12378240*I - 12378240)*A - (13305600*I + 13305600)*B)*a^2*cos(d*x + c) + ((12378240*I
+ 12378240)*A - (13305600*I - 13305600)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^4 + ((-(12378240*I - 12378240)*A
- (13305600*I + 13305600)*B)*a^2*cos(d*x + c) + ((12378240*I + 12378240)*A - (13305600*I - 13305600)*B)*a^2*s
in(d*x + c))*sin(2*d*x + 2*c)^4 + (((49512960*I - 49512960)*A + (53222400*I + 53222400)*B)*a^2*cos(d*x + c) +
(-(49512960*I + 49512960)*A + (53222400*I - 53222400)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^3 + (-(12378240*I
- 12378240)*A - (13305600*I + 13305600)*B)*a^2*cos(d*x + c) + ((12378240*I + 12378240)*A - (13305600*I - 13305
600)*B)*a^2*sin(d*x + c) + ((-(74269440*I - 74269440)*A - (79833600*I + 79833600)*B)*a^2*cos(d*x + c) + ((7426
9440*I + 74269440)*A - (79833600*I - 79833600)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(24756480*I - 2475
6480)*A - (26611200*I + 26611200)*B)*a^2*cos(d*x + c) + ((24756480*I + 24756480)*A - (26611200*I - 26611200)*B
)*a^2*sin(d*x + c) + ((-(24756480*I - 24756480)*A - (26611200*I + 26611200)*B)*a^2*cos(d*x + c) + ((24756480*I
+ 24756480)*A - (26611200*I - 26611200)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (((49512960*I - 49512960)*A
+ (53222400*I + 53222400)*B)*a^2*cos(d*x + c) + (-(49512960*I + 49512960)*A + (53222400*I - 53222400)*B)*a^2*
sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + (((49512960*I - 49512960)*A + (53222400*I + 53222400)*B)*
a^2*cos(d*x + c) + (-(49512960*I + 49512960)*A + (53222400*I - 53222400)*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)^4 + sin(2*d*x + 2*c)^4
- 4*cos(2*d*x + 2*c)^3 + 2*(cos(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c)^2 + 6*cos(2*d*x + 2
*c)^2 - 4*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)*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{11/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^(11/2)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2),x)

[Out]

int(cot(c + d*x)^(11/2)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**(11/2)*(a+I*a*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)

[Out]

Timed out

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