\(\int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\) [541]

Optimal result

Integrand size = 38, antiderivative size = 245 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {(2-2 i) a^{3/2} (A-i B) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {4 a (67 i A+63 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d} \]

[Out]

(2-2*I)*a^(3/2)*(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+4/105*a*(19*A-21*I*B)*cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^(1/2)/d-2/35*a*(8*I*A+7*B)*cot(d*x+c)
^(5/2)*(a+I*a*tan(d*x+c))^(1/2)/d-2/7*a*A*cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(1/2)/d+4/105*a*(67*I*A+63*B)*co
t(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4326, 3674, 3679, 12, 3625, 211} \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {(2-2 i) a^{3/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 a (7 B+8 i A) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (63 B+67 i A) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d} \]

[In]

Int[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

((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 + (4*a*((67*I)*A + 63*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(105*
d) + (4*a*(19*A - (21*I)*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a*((8*I)*A + 7*B)*Cot[
c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) - (2*a*A*Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]])/(7*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 211

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

Rule 3625

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 3674

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] && E
qQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]

Rule 3679

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 4326

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*} \text {integral}& = \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx \\ & = -\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}+\frac {1}{7} \left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)} \left (\frac {1}{2} a (8 i A+7 B)-\frac {1}{2} a (6 A-7 i B) \tan (c+d x)\right )}{\tan ^{\frac {7}{2}}(c+d x)} \, dx \\ & = -\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}+\frac {\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}{2} a^2 (19 A-21 i B)-a^2 (8 i A+7 B) \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx}{35 a} \\ & = \frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 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 {1}{4} a^3 (67 i A+63 B)+\frac {1}{2} a^3 (19 A-21 i B) \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{105 a^2} \\ & = \frac {4 a (67 i A+63 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}+\frac {\left (16 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {105 a^4 (A-i B) \sqrt {a+i a \tan (c+d x)}}{8 \sqrt {\tan (c+d x)}} \, dx}{105 a^3} \\ & = \frac {4 a (67 i A+63 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}+\left (2 a (A-i 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 {4 a (67 i A+63 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}-\frac {\left (4 i a^3 (A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {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 {(2+2 i) a^{3/2} (i A+B) \text {arctanh}\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 {4 a (67 i A+63 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a (19 A-21 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a (8 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.45 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.47 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {2 \cot ^{\frac {5}{2}}(c+d x) \left (-15 a^2 A (i+\cot (c+d x))^2 \tan (c+d x)-3 (3 i A+7 B) (a+i a \tan (c+d x))^2+(29 A-21 i B) \tan (c+d x) (a+i a \tan (c+d x))^2-\frac {105 a (A-i B) \tan ^2(c+d x) \left (-\sqrt [4]{-1} a \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)} (-i+\tan (c+d x))+\sqrt {a} \text {arcsinh}\left (\frac {\sqrt {i a \tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {i a \tan (c+d x)} (-i+\tan (c+d x))+\sqrt {1+i \tan (c+d x)} \left (a (-i+\tan (c+d x))+i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )\right )}{\sqrt {1+i \tan (c+d x)}}\right )}{105 d \sqrt {a+i a \tan (c+d x)}} \]

[In]

Integrate[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

(2*Cot[c + d*x]^(5/2)*(-15*a^2*A*(I + Cot[c + d*x])^2*Tan[c + d*x] - 3*((3*I)*A + 7*B)*(a + I*a*Tan[c + d*x])^
2 + (29*A - (21*I)*B)*Tan[c + d*x]*(a + I*a*Tan[c + d*x])^2 - (105*a*(A - I*B)*Tan[c + d*x]^2*(-((-1)^(1/4)*a*
ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]]*(-I + Tan[c + d*x])) + Sqrt[a]*ArcSinh[Sqrt[I*a*Tan[
c + d*x]]/Sqrt[a]]*Sqrt[I*a*Tan[c + d*x]]*(-I + Tan[c + d*x]) + Sqrt[1 + I*Tan[c + d*x]]*(a*(-I + Tan[c + d*x]
) + I*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[I*a*Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[I*a*Tan[c + d*x]]*Sqrt
[a + I*a*Tan[c + d*x]])))/Sqrt[1 + I*Tan[c + d*x]]))/(105*d*Sqrt[a + I*a*Tan[c + d*x]])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 803 vs. \(2 (200 ) = 400\).

Time = 0.58 (sec) , antiderivative size = 804, normalized size of antiderivative = 3.28

method	result	size
derivativedivides	\(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {9}{2}} \tan \left (d x +c \right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a \left (504 B \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right )^{3} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+536 i A \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right )^{3} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+105 i \sqrt {i a}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) \sqrt {2}\, a \tan \left (d x +c \right )^{4}+152 A \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right )^{2} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-420 i B \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) \sqrt {-i a}\, a \tan \left (d x +c \right )^{4}+420 A \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) \sqrt {-i a}\, a \tan \left (d x +c \right )^{4}+210 i \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) \sqrt {-i a}\, a \tan \left (d x +c \right )^{4}-105 \sqrt {i a}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) \sqrt {2}\, a \tan \left (d x +c \right )^{4}-96 i A \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+210 \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) \sqrt {-i a}\, a \tan \left (d x +c \right )^{4}-168 i B \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right )^{2} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-84 B \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-60 A \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}\right )}{210 d \sqrt {i a}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}}\)	\(804\)
default	\(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {9}{2}} \tan \left (d x +c \right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a \left (504 B \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right )^{3} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+536 i A \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right )^{3} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+105 i \sqrt {i a}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) \sqrt {2}\, a \tan \left (d x +c \right )^{4}+152 A \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right )^{2} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-420 i B \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) \sqrt {-i a}\, a \tan \left (d x +c \right )^{4}+420 A \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) \sqrt {-i a}\, a \tan \left (d x +c \right )^{4}+210 i \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) \sqrt {-i a}\, a \tan \left (d x +c \right )^{4}-105 \sqrt {i a}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) \sqrt {2}\, a \tan \left (d x +c \right )^{4}-96 i A \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+210 \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) \sqrt {-i a}\, a \tan \left (d x +c \right )^{4}-168 i B \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right )^{2} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-84 B \sqrt {i a}\, \sqrt {-i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-60 A \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}\right )}{210 d \sqrt {i a}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}}\)	\(804\)

[In]

int(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/210/d*(1/tan(d*x+c))^(9/2)*tan(d*x+c)*(a*(1+I*tan(d*x+c)))^(1/2)*a*(504*B*(I*a)^(1/2)*(-I*a)^(1/2)*tan(d*x+c
)^3*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+536*I*A*(I*a)^(1/2)*(-I*a)^(1/2)*tan(d*x+c)^3*(a*tan(d*x+c)*(1+I*tan
(d*x+c)))^(1/2)+105*I*(I*a)^(1/2)*ln(-(-2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+I*a-3*a*t
an(d*x+c))/(tan(d*x+c)+I))*2^(1/2)*a*tan(d*x+c)^4+152*A*(I*a)^(1/2)*(-I*a)^(1/2)*tan(d*x+c)^2*(a*tan(d*x+c)*(1
+I*tan(d*x+c)))^(1/2)-420*I*B*ln(1/2*(2*I*a*tan(d*x+c)+2*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(I*a)^(1/2)+a)/
(I*a)^(1/2))*(-I*a)^(1/2)*a*tan(d*x+c)^4+420*A*ln(1/2*(2*I*a*tan(d*x+c)+2*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2
)*(I*a)^(1/2)+a)/(I*a)^(1/2))*(-I*a)^(1/2)*a*tan(d*x+c)^4+210*I*ln(1/2*(2*I*a*tan(d*x+c)+2*(a*tan(d*x+c)*(1+I*
tan(d*x+c)))^(1/2)*(I*a)^(1/2)+a)/(I*a)^(1/2))*(-I*a)^(1/2)*a*tan(d*x+c)^4-105*(I*a)^(1/2)*ln(-(-2*2^(1/2)*(-I
*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+I*a-3*a*tan(d*x+c))/(tan(d*x+c)+I))*2^(1/2)*a*tan(d*x+c)^4-96*
I*A*(I*a)^(1/2)*(-I*a)^(1/2)*tan(d*x+c)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+210*ln(1/2*(2*I*a*tan(d*x+c)+2*(
a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(I*a)^(1/2)+a)/(I*a)^(1/2))*(-I*a)^(1/2)*a*tan(d*x+c)^4-168*I*B*(I*a)^(1/
2)*(-I*a)^(1/2)*tan(d*x+c)^2*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-84*B*(I*a)^(1/2)*(-I*a)^(1/2)*tan(d*x+c)*(a
*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-60*A*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(I*a)^(1/2)*(-I*a)^(1/2))/(I*a)
^(1/2)/(-I*a)^(1/2)/(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (187) = 374\).

Time = 0.26 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.32 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {105 \, \sqrt {2} \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {4 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} + \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - 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 (-i \, A - B\right )} a}\right ) - 105 \, \sqrt {2} \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {4 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} - \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - 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 (-i \, A - B\right )} a}\right ) - 2 \, \sqrt {2} {\left ({\left (-211 i \, A - 189 \, B\right )} a e^{\left (7 i \, d x + 7 i \, c\right )} + 7 \, {\left (53 i \, A + 57 \, B\right )} a e^{\left (5 i \, d x + 5 i \, c\right )} + 35 \, {\left (-11 i \, A - 9 \, B\right )} a e^{\left (3 i \, d x + 3 i \, c\right )} + 105 \, {\left (i \, A + B\right )} a 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}}}{105 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]

[In]

integrate(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/105*(105*sqrt(2)*sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^3/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(4*((A - I*B)*a^2*e^(I*d*x + I*c) + sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^3/d^2)*(d*e
^(2*I*d*x + 2*I*c) - 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)/((-I*A - B)*a)) - 105*sqrt(2)*sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^3/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(4*((A - I*B)*a^2*e^(I*d*x + I*c) - sqrt(-
(I*A^2 + 2*A*B - I*B^2)*a^3/d^2)*(d*e^(2*I*d*x + 2*I*c) - 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)/((-I*A - B)*a)) - 2*sqrt(2)*((-211*I*A - 189*B)
*a*e^(7*I*d*x + 7*I*c) + 7*(53*I*A + 57*B)*a*e^(5*I*d*x + 5*I*c) + 35*(-11*I*A - 9*B)*a*e^(3*I*d*x + 3*I*c) +
105*(I*A + B)*a*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)))/(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)

Sympy [F(-1)]

Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)**(9/2)*(a+I*a*tan(d*x+c))**(3/2)*(A+B*tan(d*x+c)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3787 vs. \(2 (187) = 374\).

Time = 3.82 (sec) , antiderivative size = 3787, normalized size of antiderivative = 15.46 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/420*(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*(3*(280*(-(I - 1)*A - (I + 1)*B
)*a*cos(7*d*x + 7*c) + 140*((I - 1)*A + (3*I + 3)*B)*a*cos(5*d*x + 5*c) + 7*(-(19*I - 19)*A - (29*I + 29)*B)*a
*cos(3*d*x + 3*c) + (-(47*I - 47)*A + (63*I + 63)*B)*a*cos(d*x + c) + 280*((I + 1)*A - (I - 1)*B)*a*sin(7*d*x
+ 7*c) + 140*(-(I + 1)*A + (3*I - 3)*B)*a*sin(5*d*x + 5*c) + 7*((19*I + 19)*A - (29*I - 29)*B)*a*sin(3*d*x + 3
*c) + ((47*I + 47)*A + (63*I - 63)*B)*a*sin(d*x + c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))
+ 4*((((141*I - 141)*A + (119*I + 119)*B)*a*cos(d*x + c) + (-(141*I + 141)*A + (119*I - 119)*B)*a*sin(d*x + c
))*cos(2*d*x + 2*c)^2 + ((141*I - 141)*A + (119*I + 119)*B)*a*cos(d*x + c) + (((141*I - 141)*A + (119*I + 119)
*B)*a*cos(d*x + c) + (-(141*I + 141)*A + (119*I - 119)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2 + (-(141*I + 141)
*A + (119*I - 119)*B)*a*sin(d*x + c) + 210*((-(I - 1)*A - (I + 1)*B)*a*cos(2*d*x + 2*c)^2 + (-(I - 1)*A - (I +
1)*B)*a*sin(2*d*x + 2*c)^2 + 2*((I - 1)*A + (I + 1)*B)*a*cos(2*d*x + 2*c) + (-(I - 1)*A - (I + 1)*B)*a)*cos(3
*d*x + 3*c) + 2*((-(141*I - 141)*A - (119*I + 119)*B)*a*cos(d*x + c) + ((141*I + 141)*A - (119*I - 119)*B)*a*s
in(d*x + c))*cos(2*d*x + 2*c) + 210*(((I + 1)*A - (I - 1)*B)*a*cos(2*d*x + 2*c)^2 + ((I + 1)*A - (I - 1)*B)*a*
sin(2*d*x + 2*c)^2 + 2*(-(I + 1)*A + (I - 1)*B)*a*cos(2*d*x + 2*c) + ((I + 1)*A - (I - 1)*B)*a)*sin(3*d*x + 3*
c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 3*(280*(-(I + 1)*A + (I - 1)*B)*a*cos(7*d*x + 7
*c) + 140*((I + 1)*A - (3*I - 3)*B)*a*cos(5*d*x + 5*c) + 7*(-(19*I + 19)*A + (29*I - 29)*B)*a*cos(3*d*x + 3*c)
+ (-(47*I + 47)*A - (63*I - 63)*B)*a*cos(d*x + c) + 280*(-(I - 1)*A - (I + 1)*B)*a*sin(7*d*x + 7*c) + 140*((I
- 1)*A + (3*I + 3)*B)*a*sin(5*d*x + 5*c) + 7*(-(19*I - 19)*A - (29*I + 29)*B)*a*sin(3*d*x + 3*c) + (-(47*I -
47)*A + (63*I + 63)*B)*a*sin(d*x + c))*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 4*((((141*I
+ 141)*A - (119*I - 119)*B)*a*cos(d*x + c) + ((141*I - 141)*A + (119*I + 119)*B)*a*sin(d*x + c))*cos(2*d*x + 2
*c)^2 + ((141*I + 141)*A - (119*I - 119)*B)*a*cos(d*x + c) + (((141*I + 141)*A - (119*I - 119)*B)*a*cos(d*x +
c) + ((141*I - 141)*A + (119*I + 119)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2 + ((141*I - 141)*A + (119*I + 119)
*B)*a*sin(d*x + c) + 210*((-(I + 1)*A + (I - 1)*B)*a*cos(2*d*x + 2*c)^2 + (-(I + 1)*A + (I - 1)*B)*a*sin(2*d*x
+ 2*c)^2 + 2*((I + 1)*A - (I - 1)*B)*a*cos(2*d*x + 2*c) + (-(I + 1)*A + (I - 1)*B)*a)*cos(3*d*x + 3*c) + 2*((
-(141*I + 141)*A + (119*I - 119)*B)*a*cos(d*x + c) + (-(141*I - 141)*A - (119*I + 119)*B)*a*sin(d*x + c))*cos(
2*d*x + 2*c) + 210*((-(I - 1)*A - (I + 1)*B)*a*cos(2*d*x + 2*c)^2 + (-(I - 1)*A - (I + 1)*B)*a*sin(2*d*x + 2*c
)^2 + 2*((I - 1)*A + (I + 1)*B)*a*cos(2*d*x + 2*c) + (-(I - 1)*A - (I + 1)*B)*a)*sin(3*d*x + 3*c))*sin(3/2*arc
tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a) + 420*(2*((-(I + 1)*A + (I - 1)*B)*a*cos(2*d*x + 2*c)^4
+ (-(I + 1)*A + (I - 1)*B)*a*sin(2*d*x + 2*c)^4 + 4*((I + 1)*A - (I - 1)*B)*a*cos(2*d*x + 2*c)^3 + 6*(-(I + 1
)*A + (I - 1)*B)*a*cos(2*d*x + 2*c)^2 + 4*((I + 1)*A - (I - 1)*B)*a*cos(2*d*x + 2*c) + 2*((-(I + 1)*A + (I - 1
)*B)*a*cos(2*d*x + 2*c)^2 + 2*((I + 1)*A - (I - 1)*B)*a*cos(2*d*x + 2*c) + (-(I + 1)*A + (I - 1)*B)*a)*sin(2*d
*x + 2*c)^2 + (-(I + 1)*A + (I - 1)*B)*a)*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)) + (((I - 1)*A + (I + 1)*B)*a*cos(2*d*x + 2*c)^4 + ((I - 1)*A + (I + 1)*B)*a*sin(2*d*x + 2*
c)^4 + 4*(-(I - 1)*A - (I + 1)*B)*a*cos(2*d*x + 2*c)^3 + 6*((I - 1)*A + (I + 1)*B)*a*cos(2*d*x + 2*c)^2 + 4*(-
(I - 1)*A - (I + 1)*B)*a*cos(2*d*x + 2*c) + 2*(((I - 1)*A + (I + 1)*B)*a*cos(2*d*x + 2*c)^2 + 2*(-(I - 1)*A -
(I + 1)*B)*a*cos(2*d*x + 2*c) + ((I - 1)*A + (I + 1)*B)*a)*sin(2*d*x + 2*c)^2 + ((I - 1)*A + (I + 1)*B)*a)*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)*cos(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) + ((((-(1249
*I - 1249)*A - (1183*I + 1183)*B)*a*cos(d*x + c) + ((1249*I + 1249)*A - (1183*I - 1183)*B)*a*sin(d*x + c))*cos
(2*d*x + 2*c)^2 + (-(1249*I - 1249)*A - (1183*I + 1183)*B)*a*cos(d*x + c) + ((-(1249*I - 1249)*A - (1183*I + 1
183)*B)*a*cos(d*x + c) + ((1249*I + 1249)*A - (1183*I - 1183)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2 + ((1249*I
+ 1249)*A - (1183*I - 1183)*B)*a*sin(d*x + c) + 840*((-(I - 1)*A - (I + 1)*B)*a*cos(2*d*x + 2*c)^2 + (-(I - 1
)*A - (I + 1)*B)*a*sin(2*d*x + 2*c)^2 + 2*((I - 1)*A + (I + 1)*B)*a*cos(2*d*x + 2*c) + (-(I - 1)*A - (I + 1)*B
)*a)*cos(5*d*x + 5*c) + 1960*(((I - 1)*A + (I + 1)*B)*a*cos(2*d*x + 2*c)^2 + ((I - 1)*A + (I + 1)*B)*a*sin(2*d
*x + 2*c)^2 + 2*(-(I - 1)*A - (I + 1)*B)*a*cos(2*d*x + 2*c) + ((I - 1)*A + (I + 1)*B)*a)*cos(3*d*x + 3*c) + 2*
(((1249*I - 1249)*A + (1183*I + 1183)*B)*a*cos(d*x + c) + (-(1249*I + 1249)*A + (1183*I - 1183)*B)*a*sin(d*x +
c))*cos(2*d*x + 2*c) + 840*(((I + 1)*A - (I - 1)*B)*a*cos(2*d*x + 2*c)^2 + ((I + 1)*A - (I - 1)*B)*a*sin(2*d*
x + 2*c)^2 + 2*(-(I + 1)*A + (I - 1)*B)*a*cos(2*d*x + 2*c) + ((I + 1)*A - (I - 1)*B)*a)*sin(5*d*x + 5*c) + 196
0*((-(I + 1)*A + (I - 1)*B)*a*cos(2*d*x + 2*c)^2 + (-(I + 1)*A + (I - 1)*B)*a*sin(2*d*x + 2*c)^2 + 2*((I + 1)*
A - (I - 1)*B)*a*cos(2*d*x + 2*c) + (-(I + 1)*A + (I - 1)*B)*a)*sin(3*d*x + 3*c))*cos(5/2*arctan2(sin(2*d*x +
2*c), cos(2*d*x + 2*c) - 1)) + 16*((((52*I - 52)*A + (63*I + 63)*B)*a*cos(d*x + c) + (-(52*I + 52)*A + (63*I -
63)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^4 + (((52*I - 52)*A + (63*I + 63)*B)*a*cos(d*x + c) + (-(52*I + 52)*A
+ (63*I - 63)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^4 + 4*((-(52*I - 52)*A - (63*I + 63)*B)*a*cos(d*x + c) + ((
52*I + 52)*A - (63*I - 63)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^3 + 6*(((52*I - 52)*A + (63*I + 63)*B)*a*cos(d*
x + c) + (-(52*I + 52)*A + (63*I - 63)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((52*I - 52)*A + (63*I + 63)*B)
*a*cos(d*x + c) + 2*((((52*I - 52)*A + (63*I + 63)*B)*a*cos(d*x + c) + (-(52*I + 52)*A + (63*I - 63)*B)*a*sin(
d*x + c))*cos(2*d*x + 2*c)^2 + ((52*I - 52)*A + (63*I + 63)*B)*a*cos(d*x + c) + (-(52*I + 52)*A + (63*I - 63)*
B)*a*sin(d*x + c) + 2*((-(52*I - 52)*A - (63*I + 63)*B)*a*cos(d*x + c) + ((52*I + 52)*A - (63*I - 63)*B)*a*sin
(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + (-(52*I + 52)*A + (63*I - 63)*B)*a*sin(d*x + c) + 4*((-(52*I
- 52)*A - (63*I + 63)*B)*a*cos(d*x + c) + ((52*I + 52)*A - (63*I - 63)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c))*c
os(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((-(1249*I + 1249)*A + (1183*I - 1183)*B)*a*cos(d*x
+ c) + (-(1249*I - 1249)*A - (1183*I + 1183)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (-(1249*I + 1249)*A + (1
183*I - 1183)*B)*a*cos(d*x + c) + ((-(1249*I + 1249)*A + (1183*I - 1183)*B)*a*cos(d*x + c) + (-(1249*I - 1249)
*A - (1183*I + 1183)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2 + (-(1249*I - 1249)*A - (1183*I + 1183)*B)*a*sin(d*
x + c) + 840*((-(I + 1)*A + (I - 1)*B)*a*cos(2*d*x + 2*c)^2 + (-(I + 1)*A + (I - 1)*B)*a*sin(2*d*x + 2*c)^2 +
2*((I + 1)*A - (I - 1)*B)*a*cos(2*d*x + 2*c) + (-(I + 1)*A + (I - 1)*B)*a)*cos(5*d*x + 5*c) + 1960*(((I + 1)*A
- (I - 1)*B)*a*cos(2*d*x + 2*c)^2 + ((I + 1)*A - (I - 1)*B)*a*sin(2*d*x + 2*c)^2 + 2*(-(I + 1)*A + (I - 1)*B)
*a*cos(2*d*x + 2*c) + ((I + 1)*A - (I - 1)*B)*a)*cos(3*d*x + 3*c) + 2*(((1249*I + 1249)*A - (1183*I - 1183)*B)
*a*cos(d*x + c) + ((1249*I - 1249)*A + (1183*I + 1183)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c) + 840*((-(I - 1)*A
- (I + 1)*B)*a*cos(2*d*x + 2*c)^2 + (-(I - 1)*A - (I + 1)*B)*a*sin(2*d*x + 2*c)^2 + 2*((I - 1)*A + (I + 1)*B)*
a*cos(2*d*x + 2*c) + (-(I - 1)*A - (I + 1)*B)*a)*sin(5*d*x + 5*c) + 1960*(((I - 1)*A + (I + 1)*B)*a*cos(2*d*x
+ 2*c)^2 + ((I - 1)*A + (I + 1)*B)*a*sin(2*d*x + 2*c)^2 + 2*(-(I - 1)*A - (I + 1)*B)*a*cos(2*d*x + 2*c) + ((I
- 1)*A + (I + 1)*B)*a)*sin(3*d*x + 3*c))*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 16*((((52*
I + 52)*A - (63*I - 63)*B)*a*cos(d*x + c) + ((52*I - 52)*A + (63*I + 63)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^4
+ (((52*I + 52)*A - (63*I - 63)*B)*a*cos(d*x + c) + ((52*I - 52)*A + (63*I + 63)*B)*a*sin(d*x + c))*sin(2*d*x
+ 2*c)^4 + 4*((-(52*I + 52)*A + (63*I - 63)*B)*a*cos(d*x + c) + (-(52*I - 52)*A - (63*I + 63)*B)*a*sin(d*x +
c))*cos(2*d*x + 2*c)^3 + 6*(((52*I + 52)*A - (63*I - 63)*B)*a*cos(d*x + c) + ((52*I - 52)*A + (63*I + 63)*B)*a
*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((52*I + 52)*A - (63*I - 63)*B)*a*cos(d*x + c) + 2*((((52*I + 52)*A - (63*
I - 63)*B)*a*cos(d*x + c) + ((52*I - 52)*A + (63*I + 63)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((52*I + 52)*
A - (63*I - 63)*B)*a*cos(d*x + c) + ((52*I - 52)*A + (63*I + 63)*B)*a*sin(d*x + c) + 2*((-(52*I + 52)*A + (63*
I - 63)*B)*a*cos(d*x + c) + (-(52*I - 52)*A - (63*I + 63)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c
)^2 + ((52*I - 52)*A + (63*I + 63)*B)*a*sin(d*x + c) + 4*((-(52*I + 52)*A + (63*I - 63)*B)*a*cos(d*x + c) + (-
(52*I - 52)*A - (63*I + 63)*B)*a*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)

Giac [F]

\[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{\frac {9}{2}} \,d x } \]

[In]

integrate(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^(3/2)*cot(d*x + c)^(9/2), x)

Mupad [F(-1)]

Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \]

[In]

int(cot(c + d*x)^(9/2)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(3/2),x)

[Out]

int(cot(c + d*x)^(9/2)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(3/2), x)