\(\int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [548]

Optimal result

Integrand size = 38, antiderivative size = 251 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {(4-4 i) a^{5/2} (A-i B) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {4 a^2 (130 i A+133 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {2 a^2 (80 A-77 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (10 i A+7 B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d} \]

[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+2/105*a^2*(80*A-77*I*B)*cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^(1/2)/d-2/35*a^2*(10*I*A+7*B)*cot(d
*x+c)^(5/2)*(a+I*a*tan(d*x+c))^(1/2)/d+4/105*a^2*(130*I*A+133*B)*cot(d*x+c)^(1/2)*(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))^(3/2)/d

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 251, 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} \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {(4-4 i) a^{5/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 a^2 (7 B+10 i A) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}+\frac {2 a^2 (80 A-77 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 a^2 (133 B+130 i A) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d} \]

[In]

Int[Cot[c + d*x]^(9/2)*(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 + (4*a^2*((130*I)*A + 133*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(
105*d) + (2*a^2*(80*A - (77*I)*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a^2*((10*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)*(a + I*a*Tan[c + d*x])^
(3/2))/(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))^{5/2} (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) (a+i a \tan (c+d x))^{3/2}}{7 d}+\frac {1}{7} \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 (10 i A+7 B)-\frac {1}{2} a (4 A-7 i B) \tan (c+d x)\right )}{\tan ^{\frac {7}{2}}(c+d x)} \, dx \\ & = -\frac {2 a^2 (10 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) (a+i a \tan (c+d x))^{3/2}}{7 d}+\frac {1}{35} \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 (80 A-77 i B)-\frac {3}{4} a^2 (20 i A+21 B) \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 a^2 (80 A-77 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (10 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) (a+i a \tan (c+d x))^{3/2}}{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 (130 i A+133 B)+\frac {1}{4} a^3 (80 A-77 i B) \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{105 a} \\ & = \frac {4 a^2 (130 i A+133 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {2 a^2 (80 A-77 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (10 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) (a+i a \tan (c+d x))^{3/2}}{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)}}{4 \sqrt {\tan (c+d x)}} \, dx}{105 a^2} \\ & = \frac {4 a^2 (130 i A+133 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {2 a^2 (80 A-77 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (10 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) (a+i a \tan (c+d x))^{3/2}}{7 d}+\left (4 a^2 (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^2 (130 i A+133 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {2 a^2 (80 A-77 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (10 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) (a+i a \tan (c+d x))^{3/2}}{7 d}-\frac {\left (8 i a^4 (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 {(4+4 i) a^{5/2} (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^2 (130 i A+133 B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {2 a^2 (80 A-77 i B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (10 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) (a+i a \tan (c+d x))^{3/2}}{7 d} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(524\) vs. \(2(251)=502\).

Time = 9.42 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.09 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-\frac {2 A (a+i a \tan (c+d x))^{5/2}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (-\frac {a (5 i A+7 B) (a+i a \tan (c+d x))^{5/2}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {7}{2} a (A-i B) \left (\frac {4 i \sqrt {2} a^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {4 i a^{5/2} \text {arcsinh}\left (\frac {\sqrt {i a \tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+i \tan (c+d x)} \sqrt {i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {5 (-1)^{3/4} a^2 \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {1+i \tan (c+d x)}}-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {14 i a^2 \sqrt {a+i a \tan (c+d x)}}{3 d \sqrt {\tan (c+d x)}}-\frac {i a^{3/2} \text {arcsinh}\left (\frac {\sqrt {i a \tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d \sqrt {1+i \tan (c+d x)} \sqrt {\tan (c+d x)}}\right )\right )}{7 a}\right ) \]

[In]

Integrate[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-2*A*(a + I*a*Tan[c + d*x])^(5/2))/(7*d*Tan[c + d*x]^(7/2)) + (2*(-1/5
*(a*((5*I)*A + 7*B)*(a + I*a*Tan[c + d*x])^(5/2))/(d*Tan[c + d*x]^(5/2)) - (7*a*(A - I*B)*(((4*I)*Sqrt[2]*a^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]])/(d*Sqrt[Tan[c + d
*x]]) - ((4*I)*a^(5/2)*ArcSinh[Sqrt[I*a*Tan[c + d*x]]/Sqrt[a]]*Sqrt[1 + I*Tan[c + d*x]]*Sqrt[I*a*Tan[c + d*x]]
)/(d*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (5*(-1)^(3/4)*a^2*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]]
*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[1 + I*Tan[c + d*x]]) - (2*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*Tan[c + d*
x]^(3/2)) - (((14*I)/3)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) - (I*a^(3/2)*ArcSinh[Sqrt[I*a*T
an[c + d*x]]/Sqrt[a]]*Sqrt[I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[1 + I*Tan[c + d*x]]*Sqrt[Tan[
c + d*x]])))/2))/(7*a))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 805 vs. \(2 (206 ) = 412\).

Time = 1.67 (sec) , antiderivative size = 806, normalized size of antiderivative = 3.21

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^{2} \left (532 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 )}-90 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 )}+520 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 )}+160 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}-154 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 )}-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}+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}+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}+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}-42 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 )}-30 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 )}{105 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 )}}\)	\(806\)
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^{2} \left (532 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 )}-90 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 )}+520 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 )}+160 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}-154 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 )}-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}+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}+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}+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}-42 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 )}-30 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 )}{105 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 )}}\)	\(806\)

[In]

int(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/105/d*(1/tan(d*x+c))^(9/2)*tan(d*x+c)*(a*(1+I*tan(d*x+c)))^(1/2)*a^2*(532*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)-90*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)+520*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)+160*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*t
an(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)^4-154
*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)-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+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*tan(d*x+c))/(t
an(d*x+c)+I))*2^(1/2)*a*tan(d*x+c)^4+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+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)^4-42*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)-30*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. 577 vs. \(2 (193) = 386\).

Time = 0.27 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.30 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {2 \, {\left (105 \, \sqrt {2} \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{5}}{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^{3} e^{\left (i \, d x + i \, c\right )} + \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{5}}{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^{2}}\right ) - 105 \, \sqrt {2} \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{5}}{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^{3} e^{\left (i \, d x + i \, c\right )} - \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{5}}{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^{2}}\right ) - 2 \, \sqrt {2} {\left (2 \, {\left (-100 i \, A - 91 \, B\right )} a^{2} e^{\left (7 i \, d x + 7 i \, c\right )} + 7 \, {\left (55 i \, A + 61 \, B\right )} a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} + 350 \, {\left (-i \, A - B\right )} a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + 105 \, {\left (i \, A + 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}}\right )}}{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))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

2/105*(105*sqrt(2)*sqrt(-(I*A^2 + 2*A*B - 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(4*((A - I*B)*a^3*e^(I*d*x + I*c) + sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^5/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)) - 105*sqrt(2)*sqrt(-(I*A^2 + 2*A*B - 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(4*((A - I*B)*a^3*e^(I*d*x + I*c) - sqrt
(-(I*A^2 + 2*A*B - I*B^2)*a^5/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)) - 2*sqrt(2)*(2*(-100*I*A -
91*B)*a^2*e^(7*I*d*x + 7*I*c) + 7*(55*I*A + 61*B)*a^2*e^(5*I*d*x + 5*I*c) + 350*(-I*A - B)*a^2*e^(3*I*d*x + 3*
I*c) + 105*(I*A + 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)))/(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. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)**(9/2)*(a+I*a*tan(d*x+c))**(5/2)*(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. 4087 vs. \(2 (193) = 386\).

Time = 1.69 (sec) , antiderivative size = 4087, normalized size of antiderivative = 16.28 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/105*(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*(3*(140*(-(I - 1)*A - (I + 1)*B
)*a^2*cos(7*d*x + 7*c) + 140*((I - 1)*A + (2*I + 2)*B)*a^2*cos(5*d*x + 5*c) + 21*(-(4*I - 4)*A - (9*I + 9)*B)*
a^2*cos(3*d*x + 3*c) + ((4*I - 4)*A + (49*I + 49)*B)*a^2*cos(d*x + c) + 140*((I + 1)*A - (I - 1)*B)*a^2*sin(7*
d*x + 7*c) + 140*(-(I + 1)*A + (2*I - 2)*B)*a^2*sin(5*d*x + 5*c) + 21*((4*I + 4)*A - (9*I - 9)*B)*a^2*sin(3*d*
x + 3*c) + (-(4*I + 4)*A + (49*I - 49)*B)*a^2*sin(d*x + c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)
- 1)) + 4*(((65*I - 65)*A + (56*I + 56)*B)*a^2*cos(d*x + c) + (-(65*I + 65)*A + (56*I - 56)*B)*a^2*sin(d*x +
c) + (((65*I - 65)*A + (56*I + 56)*B)*a^2*cos(d*x + c) + (-(65*I + 65)*A + (56*I - 56)*B)*a^2*sin(d*x + c))*co
s(2*d*x + 2*c)^2 + (((65*I - 65)*A + (56*I + 56)*B)*a^2*cos(d*x + c) + (-(65*I + 65)*A + (56*I - 56)*B)*a^2*si
n(d*x + c))*sin(2*d*x + 2*c)^2 + 105*((-(I - 1)*A - (I + 1)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(I - 1)*A - (I + 1)*
B)*a^2*sin(2*d*x + 2*c)^2 + 2*((I - 1)*A + (I + 1)*B)*a^2*cos(2*d*x + 2*c) + (-(I - 1)*A - (I + 1)*B)*a^2)*cos
(3*d*x + 3*c) + 2*((-(65*I - 65)*A - (56*I + 56)*B)*a^2*cos(d*x + c) + ((65*I + 65)*A - (56*I - 56)*B)*a^2*sin
(d*x + c))*cos(2*d*x + 2*c) + 105*(((I + 1)*A - (I - 1)*B)*a^2*cos(2*d*x + 2*c)^2 + ((I + 1)*A - (I - 1)*B)*a^
2*sin(2*d*x + 2*c)^2 + 2*(-(I + 1)*A + (I - 1)*B)*a^2*cos(2*d*x + 2*c) + ((I + 1)*A - (I - 1)*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)) + 3*(140*(-(I + 1)*A + (I - 1)*B)*a^2*cos(7
*d*x + 7*c) + 140*((I + 1)*A - (2*I - 2)*B)*a^2*cos(5*d*x + 5*c) + 21*(-(4*I + 4)*A + (9*I - 9)*B)*a^2*cos(3*d
*x + 3*c) + ((4*I + 4)*A - (49*I - 49)*B)*a^2*cos(d*x + c) + 140*(-(I - 1)*A - (I + 1)*B)*a^2*sin(7*d*x + 7*c)
+ 140*((I - 1)*A + (2*I + 2)*B)*a^2*sin(5*d*x + 5*c) + 21*(-(4*I - 4)*A - (9*I + 9)*B)*a^2*sin(3*d*x + 3*c) +
((4*I - 4)*A + (49*I + 49)*B)*a^2*sin(d*x + c))*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 4*
(((65*I + 65)*A - (56*I - 56)*B)*a^2*cos(d*x + c) + ((65*I - 65)*A + (56*I + 56)*B)*a^2*sin(d*x + c) + (((65*I
+ 65)*A - (56*I - 56)*B)*a^2*cos(d*x + c) + ((65*I - 65)*A + (56*I + 56)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c
)^2 + (((65*I + 65)*A - (56*I - 56)*B)*a^2*cos(d*x + c) + ((65*I - 65)*A + (56*I + 56)*B)*a^2*sin(d*x + c))*si
n(2*d*x + 2*c)^2 + 105*((-(I + 1)*A + (I - 1)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(I + 1)*A + (I - 1)*B)*a^2*sin(2*d
*x + 2*c)^2 + 2*((I + 1)*A - (I - 1)*B)*a^2*cos(2*d*x + 2*c) + (-(I + 1)*A + (I - 1)*B)*a^2)*cos(3*d*x + 3*c)
+ 2*((-(65*I + 65)*A + (56*I - 56)*B)*a^2*cos(d*x + c) + (-(65*I - 65)*A - (56*I + 56)*B)*a^2*sin(d*x + c))*co
s(2*d*x + 2*c) + 105*((-(I - 1)*A - (I + 1)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(I - 1)*A - (I + 1)*B)*a^2*sin(2*d*x
+ 2*c)^2 + 2*((I - 1)*A + (I + 1)*B)*a^2*cos(2*d*x + 2*c) + (-(I - 1)*A - (I + 1)*B)*a^2)*sin(3*d*x + 3*c))*s
in(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a) + 210*(2*((-(I + 1)*A + (I - 1)*B)*a^2*cos(2*
d*x + 2*c)^4 + (-(I + 1)*A + (I - 1)*B)*a^2*sin(2*d*x + 2*c)^4 + 4*((I + 1)*A - (I - 1)*B)*a^2*cos(2*d*x + 2*c
)^3 + 6*(-(I + 1)*A + (I - 1)*B)*a^2*cos(2*d*x + 2*c)^2 + 4*((I + 1)*A - (I - 1)*B)*a^2*cos(2*d*x + 2*c) + (-(
I + 1)*A + (I - 1)*B)*a^2 + 2*((-(I + 1)*A + (I - 1)*B)*a^2*cos(2*d*x + 2*c)^2 + 2*((I + 1)*A - (I - 1)*B)*a^2
*cos(2*d*x + 2*c) + (-(I + 1)*A + (I - 1)*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)) + (((I - 1)*A + (I + 1)*B)*a^2*cos(2*d*x + 2*c)^4 + ((I - 1)*A
+ (I + 1)*B)*a^2*sin(2*d*x + 2*c)^4 + 4*(-(I - 1)*A - (I + 1)*B)*a^2*cos(2*d*x + 2*c)^3 + 6*((I - 1)*A + (I +
1)*B)*a^2*cos(2*d*x + 2*c)^2 + 4*(-(I - 1)*A - (I + 1)*B)*a^2*cos(2*d*x + 2*c) + ((I - 1)*A + (I + 1)*B)*a^2
+ 2*(((I - 1)*A + (I + 1)*B)*a^2*cos(2*d*x + 2*c)^2 + 2*(-(I - 1)*A - (I + 1)*B)*a^2*cos(2*d*x + 2*c) + ((I -
1)*A + (I + 1)*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
+ 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*c
os(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*co
s(2*d*x + 2*c) + 1)^(1/4)*sqrt(a) + (((-(608*I - 608)*A - (581*I + 581)*B)*a^2*cos(d*x + c) + ((608*I + 608)*A
- (581*I - 581)*B)*a^2*sin(d*x + c) + ((-(608*I - 608)*A - (581*I + 581)*B)*a^2*cos(d*x + c) + ((608*I + 608)
*A - (581*I - 581)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(608*I - 608)*A - (581*I + 581)*B)*a^2*cos(d*x
+ c) + ((608*I + 608)*A - (581*I - 581)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + 420*((-(I - 1)*A - (I + 1)*
B)*a^2*cos(2*d*x + 2*c)^2 + (-(I - 1)*A - (I + 1)*B)*a^2*sin(2*d*x + 2*c)^2 + 2*((I - 1)*A + (I + 1)*B)*a^2*co
s(2*d*x + 2*c) + (-(I - 1)*A - (I + 1)*B)*a^2)*cos(5*d*x + 5*c) + 980*(((I - 1)*A + (I + 1)*B)*a^2*cos(2*d*x +
2*c)^2 + ((I - 1)*A + (I + 1)*B)*a^2*sin(2*d*x + 2*c)^2 + 2*(-(I - 1)*A - (I + 1)*B)*a^2*cos(2*d*x + 2*c) + (
(I - 1)*A + (I + 1)*B)*a^2)*cos(3*d*x + 3*c) + 2*(((608*I - 608)*A + (581*I + 581)*B)*a^2*cos(d*x + c) + (-(60
8*I + 608)*A + (581*I - 581)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + 420*(((I + 1)*A - (I - 1)*B)*a^2*cos(2*d*
x + 2*c)^2 + ((I + 1)*A - (I - 1)*B)*a^2*sin(2*d*x + 2*c)^2 + 2*(-(I + 1)*A + (I - 1)*B)*a^2*cos(2*d*x + 2*c)
+ ((I + 1)*A - (I - 1)*B)*a^2)*sin(5*d*x + 5*c) + 980*((-(I + 1)*A + (I - 1)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(I
+ 1)*A + (I - 1)*B)*a^2*sin(2*d*x + 2*c)^2 + 2*((I + 1)*A - (I - 1)*B)*a^2*cos(2*d*x + 2*c) + (-(I + 1)*A + (I
- 1)*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)) + 4*((((115*I - 115)*
A + (133*I + 133)*B)*a^2*cos(d*x + c) + (-(115*I + 115)*A + (133*I - 133)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c
)^4 + (((115*I - 115)*A + (133*I + 133)*B)*a^2*cos(d*x + c) + (-(115*I + 115)*A + (133*I - 133)*B)*a^2*sin(d*x
+ c))*sin(2*d*x + 2*c)^4 + 4*((-(115*I - 115)*A - (133*I + 133)*B)*a^2*cos(d*x + c) + ((115*I + 115)*A - (133
*I - 133)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^3 + ((115*I - 115)*A + (133*I + 133)*B)*a^2*cos(d*x + c) + (-(
115*I + 115)*A + (133*I - 133)*B)*a^2*sin(d*x + c) + 6*(((115*I - 115)*A + (133*I + 133)*B)*a^2*cos(d*x + c) +
(-(115*I + 115)*A + (133*I - 133)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + 2*(((115*I - 115)*A + (133*I + 13
3)*B)*a^2*cos(d*x + c) + (-(115*I + 115)*A + (133*I - 133)*B)*a^2*sin(d*x + c) + (((115*I - 115)*A + (133*I +
133)*B)*a^2*cos(d*x + c) + (-(115*I + 115)*A + (133*I - 133)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + 2*((-(1
15*I - 115)*A - (133*I + 133)*B)*a^2*cos(d*x + c) + ((115*I + 115)*A - (133*I - 133)*B)*a^2*sin(d*x + c))*cos(
2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*((-(115*I - 115)*A - (133*I + 133)*B)*a^2*cos(d*x + c) + ((115*I + 115)*A
- (133*I - 133)*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
)) + ((-(608*I + 608)*A + (581*I - 581)*B)*a^2*cos(d*x + c) + (-(608*I - 608)*A - (581*I + 581)*B)*a^2*sin(d*x
+ c) + ((-(608*I + 608)*A + (581*I - 581)*B)*a^2*cos(d*x + c) + (-(608*I - 608)*A - (581*I + 581)*B)*a^2*sin(
d*x + c))*cos(2*d*x + 2*c)^2 + ((-(608*I + 608)*A + (581*I - 581)*B)*a^2*cos(d*x + c) + (-(608*I - 608)*A - (5
81*I + 581)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + 420*((-(I + 1)*A + (I - 1)*B)*a^2*cos(2*d*x + 2*c)^2 + (
-(I + 1)*A + (I - 1)*B)*a^2*sin(2*d*x + 2*c)^2 + 2*((I + 1)*A - (I - 1)*B)*a^2*cos(2*d*x + 2*c) + (-(I + 1)*A
+ (I - 1)*B)*a^2)*cos(5*d*x + 5*c) + 980*(((I + 1)*A - (I - 1)*B)*a^2*cos(2*d*x + 2*c)^2 + ((I + 1)*A - (I - 1
)*B)*a^2*sin(2*d*x + 2*c)^2 + 2*(-(I + 1)*A + (I - 1)*B)*a^2*cos(2*d*x + 2*c) + ((I + 1)*A - (I - 1)*B)*a^2)*c
os(3*d*x + 3*c) + 2*(((608*I + 608)*A - (581*I - 581)*B)*a^2*cos(d*x + c) + ((608*I - 608)*A + (581*I + 581)*B
)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + 420*((-(I - 1)*A - (I + 1)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(I - 1)*A - (I
+ 1)*B)*a^2*sin(2*d*x + 2*c)^2 + 2*((I - 1)*A + (I + 1)*B)*a^2*cos(2*d*x + 2*c) + (-(I - 1)*A - (I + 1)*B)*a^
2)*sin(5*d*x + 5*c) + 980*(((I - 1)*A + (I + 1)*B)*a^2*cos(2*d*x + 2*c)^2 + ((I - 1)*A + (I + 1)*B)*a^2*sin(2*
d*x + 2*c)^2 + 2*(-(I - 1)*A - (I + 1)*B)*a^2*cos(2*d*x + 2*c) + ((I - 1)*A + (I + 1)*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)) + 4*((((115*I + 115)*A - (133*I - 133)*B)*a^2*cos(d
*x + c) + ((115*I - 115)*A + (133*I + 133)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^4 + (((115*I + 115)*A - (133*
I - 133)*B)*a^2*cos(d*x + c) + ((115*I - 115)*A + (133*I + 133)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^4 + 4*((
-(115*I + 115)*A + (133*I - 133)*B)*a^2*cos(d*x + c) + (-(115*I - 115)*A - (133*I + 133)*B)*a^2*sin(d*x + c))*
cos(2*d*x + 2*c)^3 + ((115*I + 115)*A - (133*I - 133)*B)*a^2*cos(d*x + c) + ((115*I - 115)*A + (133*I + 133)*B
)*a^2*sin(d*x + c) + 6*(((115*I + 115)*A - (133*I - 133)*B)*a^2*cos(d*x + c) + ((115*I - 115)*A + (133*I + 133
)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + 2*(((115*I + 115)*A - (133*I - 133)*B)*a^2*cos(d*x + c) + ((115*I
- 115)*A + (133*I + 133)*B)*a^2*sin(d*x + c) + (((115*I + 115)*A - (133*I - 133)*B)*a^2*cos(d*x + c) + ((115*I
- 115)*A + (133*I + 133)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + 2*((-(115*I + 115)*A + (133*I - 133)*B)*a^
2*cos(d*x + c) + (-(115*I - 115)*A - (133*I + 133)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 +
4*((-(115*I + 115)*A + (133*I - 133)*B)*a^2*cos(d*x + c) + (-(115*I - 115)*A - (133*I + 133)*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)

Giac [F]

\[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{\frac {9}{2}} \,d x } \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]

[In]

int(cot(c + d*x)^(9/2)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2),x)

[Out]

int(cot(c + d*x)^(9/2)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2), x)