3.548 \(\int \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\)
Optimal. Leaf size=251 \[ -\frac{2 a^2 (7 B+10 i A) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}+\frac{2 a^2 (80 A-77 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{4 a^2 (133 B+130 i A) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{(4-4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d} \]
[Out]
((4 - 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*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)
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Rubi [A] time = 0.94605, antiderivative size = 251, normalized size of antiderivative = 1.,
number of steps used = 8, 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} \[ -\frac{2 a^2 (7 B+10 i A) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}+\frac{2 a^2 (80 A-77 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{4 a^2 (133 B+130 i A) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{(4-4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d} \]
Antiderivative was successfully verified.
[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 4241
Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ
[u, x]
Rule 3593
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(a^2*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^
(n + 1))/(d*f*(b*c + a*d)*(n + 1)), x] - Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m -
1) + b*d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ
[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]
Rule 3598
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*d - B*c)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(f
*(n + 1)*(c^2 + d^2)), x] - Dist[1/(a*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n
+ 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c*m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 3544
Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
(-2*a*b)/f, Subst[Int[1/(a*c - b*d - 2*a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rubi steps
\begin{align*} \int \cot ^{\frac{9}{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{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 ) \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{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 [A] time = 9.69823, size = 332, normalized size = 1.32 \[ \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left (-4 i \sqrt{2} (A-i B) e^{-3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left (1+e^{2 i (c+d x)}\right )}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )-\frac{(\cos (2 c)-i \sin (2 c)) \sqrt{\cot (c+d x)} \csc ^3(c+d x) \sqrt{\sec (c+d x)} ((-35 A+77 i B) \cos (c+d x)+(95 A-77 i B) \cos (3 (c+d x))+2 \sin (c+d x) ((287 B+305 i A) \cos (2 (c+d x))-215 i A-245 B))}{210 (\cos (d x)+i \sin (d x))^2}\right )}{d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
[Out]
((((-4*I)*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)))]*Sqr
t[(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]^3*Sqrt[Sec[c + d*x]]*(Cos[2*c] - I*Sin[2*c])*((-
35*A + (77*I)*B)*Cos[c + d*x] + (95*A - (77*I)*B)*Cos[3*(c + d*x)] + 2*((-215*I)*A - 245*B + ((305*I)*A + 287*
B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(210*(Cos[d*x] + I*Sin[d*x])^2))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c
+ d*x]))/(d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))
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Maple [B] time = 0.507, size = 3126, normalized size = 12.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x)
[Out]
-1/105/d*a^2*2^(1/2)*(-840*B*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*((cos(d*x+c)-1)/
sin(d*x+c))^(1/2)-420*B*cos(d*x+c)^2*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*
2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-s
in(d*x+c)+1))-840*B*cos(d*x+c)^2*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^
(1/2)+1)-343*B*2^(1/2)*cos(d*x+c)*sin(d*x+c)+400*I*A*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)-305*I*A*2^(1/2)*cos(d*x+c
)^2*sin(d*x+c)-840*I*A*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*((cos(d*x+c)-1)/sin(d*
x+c))^(1/2)-840*I*A*cos(d*x+c)^2*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^
(1/2)-1)-420*I*A*cos(d*x+c)^2*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)
*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+
c)+1))+840*I*B*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*((cos(d*x+c)-1)/sin(d*x+c))^(1
/2)+840*I*B*cos(d*x+c)^2*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)
+420*I*B*cos(d*x+c)^2*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x
+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))-3
40*I*A*2^(1/2)*cos(d*x+c)*sin(d*x+c)+420*I*A*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*
((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+420*I*A*cos(d*x+c)^4*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1
)/sin(d*x+c))^(1/2)*2^(1/2)-1)+210*I*A*cos(d*x+c)^4*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*ln(-(((cos(d*x+c)-1)/sin
(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+
c)-cos(d*x+c)-sin(d*x+c)+1))-420*I*B*cos(d*x+c)^4*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*((cos(d*
x+c)-1)/sin(d*x+c))^(1/2)-420*I*B*cos(d*x+c)^4*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*
x+c))^(1/2)*2^(1/2)-1)-210*I*B*cos(d*x+c)^4*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))
^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d
*x+c)+sin(d*x+c)-1))+364*B*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)-95*A*cos(d*x+c)^3*2^(1/2)+80*A*cos(d*x+c)*2^(1/2)-2
66*I*B*2^(1/2)-840*A*cos(d*x+c)^2*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*((cos(d*x+c)-1)/sin(d*x+
c))^(1/2)-840*A*cos(d*x+c)^2*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2
)+1)-420*A*cos(d*x+c)^2*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d
*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))
-645*A*2^(1/2)*cos(d*x+c)^2+400*A*cos(d*x+c)^4*2^(1/2)+260*A*2^(1/2)+420*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^
(1/2)*2^(1/2)-1)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+420*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)
-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+210*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2
)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)
+sin(d*x+c)-1))+420*B*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+21
0*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin
(d*x+c)-1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))+420*B*((cos(d*x+c)-
1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+266*B*2^(1/2)*sin(d*x+c)-287*B*cos(d*
x+c)^2*sin(d*x+c)*2^(1/2)+420*A*cos(d*x+c)^4*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+
c))^(1/2)*2^(1/2)-1)+210*A*cos(d*x+c)^4*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/
2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c
)+sin(d*x+c)-1))+210*B*cos(d*x+c)^4*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d
*x+c)-1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))*((cos(d*x+c)-1)/sin(d
*x+c))^(1/2)+420*B*cos(d*x+c)^4*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(
1/2)+1)+420*B*cos(d*x+c)^4*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-
1)+420*A*cos(d*x+c)^4*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-36
4*I*B*2^(1/2)*cos(d*x+c)^4+77*I*B*2^(1/2)*cos(d*x+c)^3+630*I*B*2^(1/2)*cos(d*x+c)^2+260*I*A*2^(1/2)*sin(d*x+c)
+420*I*A*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)+420*I*A*((cos(d
*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+210*I*A*((cos(d*x+c)-1)/sin(d*x
+c))^(1/2)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((cos(d*x+c)-1)
/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))-77*I*B*2^(1/2)*cos(d*x+c)-420*I*B*arctan(((cos
(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-420*I*B*((cos(d*x+c)-1)/sin(d*x+c))^
(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-210*I*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*ln(-(((cos
(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^
(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)))*(cos(d*x+c)/sin(d*x+c))^(9/2)*(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)^4
________________________________________________________________________________________
Maxima [B] time = 9.94747, size = 5426, normalized size = 21.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")
[Out]
1/11025*(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*((((44100*I - 44100)*A + (4410
0*I + 44100)*B)*a^2*cos(7*d*x + 7*c) + (-(44100*I - 44100)*A - (88200*I + 88200)*B)*a^2*cos(5*d*x + 5*c) + ((2
6460*I - 26460)*A + (59535*I + 59535)*B)*a^2*cos(3*d*x + 3*c) + (-(1260*I - 1260)*A - (15435*I + 15435)*B)*a^2
*cos(d*x + c) + (-(44100*I + 44100)*A + (44100*I - 44100)*B)*a^2*sin(7*d*x + 7*c) + ((44100*I + 44100)*A - (88
200*I - 88200)*B)*a^2*sin(5*d*x + 5*c) + (-(26460*I + 26460)*A + (59535*I - 59535)*B)*a^2*sin(3*d*x + 3*c) + (
(1260*I + 1260)*A - (15435*I - 15435)*B)*a^2*sin(d*x + c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)
- 1)) + ((-(27300*I - 27300)*A - (23520*I + 23520)*B)*a^2*cos(d*x + c) + ((27300*I + 27300)*A - (23520*I - 235
20)*B)*a^2*sin(d*x + c) + ((-(27300*I - 27300)*A - (23520*I + 23520)*B)*a^2*cos(d*x + c) + ((27300*I + 27300)*
A - (23520*I - 23520)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(27300*I - 27300)*A - (23520*I + 23520)*B)*
a^2*cos(d*x + c) + ((27300*I + 27300)*A - (23520*I - 23520)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + (((44100
*I - 44100)*A + (44100*I + 44100)*B)*a^2*cos(2*d*x + 2*c)^2 + ((44100*I - 44100)*A + (44100*I + 44100)*B)*a^2*
sin(2*d*x + 2*c)^2 + (-(88200*I - 88200)*A - (88200*I + 88200)*B)*a^2*cos(2*d*x + 2*c) + ((44100*I - 44100)*A
+ (44100*I + 44100)*B)*a^2)*cos(3*d*x + 3*c) + (((54600*I - 54600)*A + (47040*I + 47040)*B)*a^2*cos(d*x + c) +
(-(54600*I + 54600)*A + (47040*I - 47040)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + ((-(44100*I + 44100)*A + (4
4100*I - 44100)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(44100*I + 44100)*A + (44100*I - 44100)*B)*a^2*sin(2*d*x + 2*c)^
2 + ((88200*I + 88200)*A - (88200*I - 88200)*B)*a^2*cos(2*d*x + 2*c) + (-(44100*I + 44100)*A + (44100*I - 4410
0)*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)) + (((44100*I + 44100)*A
- (44100*I - 44100)*B)*a^2*cos(7*d*x + 7*c) + (-(44100*I + 44100)*A + (88200*I - 88200)*B)*a^2*cos(5*d*x + 5*c
) + ((26460*I + 26460)*A - (59535*I - 59535)*B)*a^2*cos(3*d*x + 3*c) + (-(1260*I + 1260)*A + (15435*I - 15435)
*B)*a^2*cos(d*x + c) + ((44100*I - 44100)*A + (44100*I + 44100)*B)*a^2*sin(7*d*x + 7*c) + (-(44100*I - 44100)*
A - (88200*I + 88200)*B)*a^2*sin(5*d*x + 5*c) + ((26460*I - 26460)*A + (59535*I + 59535)*B)*a^2*sin(3*d*x + 3*
c) + (-(1260*I - 1260)*A - (15435*I + 15435)*B)*a^2*sin(d*x + c))*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c) - 1)) + ((-(27300*I + 27300)*A + (23520*I - 23520)*B)*a^2*cos(d*x + c) + (-(27300*I - 27300)*A - (23520
*I + 23520)*B)*a^2*sin(d*x + c) + ((-(27300*I + 27300)*A + (23520*I - 23520)*B)*a^2*cos(d*x + c) + (-(27300*I
- 27300)*A - (23520*I + 23520)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(27300*I + 27300)*A + (23520*I - 2
3520)*B)*a^2*cos(d*x + c) + (-(27300*I - 27300)*A - (23520*I + 23520)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2
+ (((44100*I + 44100)*A - (44100*I - 44100)*B)*a^2*cos(2*d*x + 2*c)^2 + ((44100*I + 44100)*A - (44100*I - 4410
0)*B)*a^2*sin(2*d*x + 2*c)^2 + (-(88200*I + 88200)*A + (88200*I - 88200)*B)*a^2*cos(2*d*x + 2*c) + ((44100*I +
44100)*A - (44100*I - 44100)*B)*a^2)*cos(3*d*x + 3*c) + (((54600*I + 54600)*A - (47040*I - 47040)*B)*a^2*cos(
d*x + c) + ((54600*I - 54600)*A + (47040*I + 47040)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + (((44100*I - 44100
)*A + (44100*I + 44100)*B)*a^2*cos(2*d*x + 2*c)^2 + ((44100*I - 44100)*A + (44100*I + 44100)*B)*a^2*sin(2*d*x
+ 2*c)^2 + (-(88200*I - 88200)*A - (88200*I + 88200)*B)*a^2*cos(2*d*x + 2*c) + ((44100*I - 44100)*A + (44100*I
+ 44100)*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) + ((((44
100*I + 44100)*A - (44100*I - 44100)*B)*a^2*cos(2*d*x + 2*c)^4 + ((44100*I + 44100)*A - (44100*I - 44100)*B)*a
^2*sin(2*d*x + 2*c)^4 + (-(176400*I + 176400)*A + (176400*I - 176400)*B)*a^2*cos(2*d*x + 2*c)^3 + ((264600*I +
264600)*A - (264600*I - 264600)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(176400*I + 176400)*A + (176400*I - 176400)*B)*
a^2*cos(2*d*x + 2*c) + ((44100*I + 44100)*A - (44100*I - 44100)*B)*a^2 + (((88200*I + 88200)*A - (88200*I - 88
200)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(176400*I + 176400)*A + (176400*I - 176400)*B)*a^2*cos(2*d*x + 2*c) + ((882
00*I + 88200)*A - (88200*I - 88200)*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)) + ((-(22050*I - 22050)*A - (22050*I + 22050)*B)*a^2*cos(2*d*x + 2*c)
^4 + (-(22050*I - 22050)*A - (22050*I + 22050)*B)*a^2*sin(2*d*x + 2*c)^4 + ((88200*I - 88200)*A + (88200*I + 8
8200)*B)*a^2*cos(2*d*x + 2*c)^3 + (-(132300*I - 132300)*A - (132300*I + 132300)*B)*a^2*cos(2*d*x + 2*c)^2 + ((
88200*I - 88200)*A + (88200*I + 88200)*B)*a^2*cos(2*d*x + 2*c) + (-(22050*I - 22050)*A - (22050*I + 22050)*B)*
a^2 + ((-(44100*I - 44100)*A - (44100*I + 44100)*B)*a^2*cos(2*d*x + 2*c)^2 + ((88200*I - 88200)*A + (88200*I +
88200)*B)*a^2*cos(2*d*x + 2*c) + (-(44100*I - 44100)*A - (44100*I + 44100)*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*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) + ((((63840*I -
63840)*A + (61005*I + 61005)*B)*a^2*cos(d*x + c) + (-(63840*I + 63840)*A + (61005*I - 61005)*B)*a^2*sin(d*x +
c) + (((63840*I - 63840)*A + (61005*I + 61005)*B)*a^2*cos(d*x + c) + (-(63840*I + 63840)*A + (61005*I - 61005
)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (((63840*I - 63840)*A + (61005*I + 61005)*B)*a^2*cos(d*x + c) + (-
(63840*I + 63840)*A + (61005*I - 61005)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + (((44100*I - 44100)*A + (441
00*I + 44100)*B)*a^2*cos(2*d*x + 2*c)^2 + ((44100*I - 44100)*A + (44100*I + 44100)*B)*a^2*sin(2*d*x + 2*c)^2 +
(-(88200*I - 88200)*A - (88200*I + 88200)*B)*a^2*cos(2*d*x + 2*c) + ((44100*I - 44100)*A + (44100*I + 44100)*
B)*a^2)*cos(5*d*x + 5*c) + ((-(102900*I - 102900)*A - (102900*I + 102900)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(10290
0*I - 102900)*A - (102900*I + 102900)*B)*a^2*sin(2*d*x + 2*c)^2 + ((205800*I - 205800)*A + (205800*I + 205800)
*B)*a^2*cos(2*d*x + 2*c) + (-(102900*I - 102900)*A - (102900*I + 102900)*B)*a^2)*cos(3*d*x + 3*c) + ((-(127680
*I - 127680)*A - (122010*I + 122010)*B)*a^2*cos(d*x + c) + ((127680*I + 127680)*A - (122010*I - 122010)*B)*a^2
*sin(d*x + c))*cos(2*d*x + 2*c) + ((-(44100*I + 44100)*A + (44100*I - 44100)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(44
100*I + 44100)*A + (44100*I - 44100)*B)*a^2*sin(2*d*x + 2*c)^2 + ((88200*I + 88200)*A - (88200*I - 88200)*B)*a
^2*cos(2*d*x + 2*c) + (-(44100*I + 44100)*A + (44100*I - 44100)*B)*a^2)*sin(5*d*x + 5*c) + (((102900*I + 10290
0)*A - (102900*I - 102900)*B)*a^2*cos(2*d*x + 2*c)^2 + ((102900*I + 102900)*A - (102900*I - 102900)*B)*a^2*sin
(2*d*x + 2*c)^2 + (-(205800*I + 205800)*A + (205800*I - 205800)*B)*a^2*cos(2*d*x + 2*c) + ((102900*I + 102900)
*A - (102900*I - 102900)*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)) +
(((-(48300*I - 48300)*A - (55860*I + 55860)*B)*a^2*cos(d*x + c) + ((48300*I + 48300)*A - (55860*I - 55860)*B)*
a^2*sin(d*x + c))*cos(2*d*x + 2*c)^4 + ((-(48300*I - 48300)*A - (55860*I + 55860)*B)*a^2*cos(d*x + c) + ((4830
0*I + 48300)*A - (55860*I - 55860)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^4 + (((193200*I - 193200)*A + (223440
*I + 223440)*B)*a^2*cos(d*x + c) + (-(193200*I + 193200)*A + (223440*I - 223440)*B)*a^2*sin(d*x + c))*cos(2*d*
x + 2*c)^3 + (-(48300*I - 48300)*A - (55860*I + 55860)*B)*a^2*cos(d*x + c) + ((48300*I + 48300)*A - (55860*I -
55860)*B)*a^2*sin(d*x + c) + ((-(289800*I - 289800)*A - (335160*I + 335160)*B)*a^2*cos(d*x + c) + ((289800*I
+ 289800)*A - (335160*I - 335160)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(96600*I - 96600)*A - (111720*I
+ 111720)*B)*a^2*cos(d*x + c) + ((96600*I + 96600)*A - (111720*I - 111720)*B)*a^2*sin(d*x + c) + ((-(96600*I
- 96600)*A - (111720*I + 111720)*B)*a^2*cos(d*x + c) + ((96600*I + 96600)*A - (111720*I - 111720)*B)*a^2*sin(d
*x + c))*cos(2*d*x + 2*c)^2 + (((193200*I - 193200)*A + (223440*I + 223440)*B)*a^2*cos(d*x + c) + (-(193200*I
+ 193200)*A + (223440*I - 223440)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + (((193200*I - 19
3200)*A + (223440*I + 223440)*B)*a^2*cos(d*x + c) + (-(193200*I + 193200)*A + (223440*I - 223440)*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)) + (((63840*I + 63840)*A -
(61005*I - 61005)*B)*a^2*cos(d*x + c) + ((63840*I - 63840)*A + (61005*I + 61005)*B)*a^2*sin(d*x + c) + (((6384
0*I + 63840)*A - (61005*I - 61005)*B)*a^2*cos(d*x + c) + ((63840*I - 63840)*A + (61005*I + 61005)*B)*a^2*sin(d
*x + c))*cos(2*d*x + 2*c)^2 + (((63840*I + 63840)*A - (61005*I - 61005)*B)*a^2*cos(d*x + c) + ((63840*I - 6384
0)*A + (61005*I + 61005)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + (((44100*I + 44100)*A - (44100*I - 44100)*B
)*a^2*cos(2*d*x + 2*c)^2 + ((44100*I + 44100)*A - (44100*I - 44100)*B)*a^2*sin(2*d*x + 2*c)^2 + (-(88200*I + 8
8200)*A + (88200*I - 88200)*B)*a^2*cos(2*d*x + 2*c) + ((44100*I + 44100)*A - (44100*I - 44100)*B)*a^2)*cos(5*d
*x + 5*c) + ((-(102900*I + 102900)*A + (102900*I - 102900)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(102900*I + 102900)*A
+ (102900*I - 102900)*B)*a^2*sin(2*d*x + 2*c)^2 + ((205800*I + 205800)*A - (205800*I - 205800)*B)*a^2*cos(2*d
*x + 2*c) + (-(102900*I + 102900)*A + (102900*I - 102900)*B)*a^2)*cos(3*d*x + 3*c) + ((-(127680*I + 127680)*A
+ (122010*I - 122010)*B)*a^2*cos(d*x + c) + (-(127680*I - 127680)*A - (122010*I + 122010)*B)*a^2*sin(d*x + c))
*cos(2*d*x + 2*c) + (((44100*I - 44100)*A + (44100*I + 44100)*B)*a^2*cos(2*d*x + 2*c)^2 + ((44100*I - 44100)*A
+ (44100*I + 44100)*B)*a^2*sin(2*d*x + 2*c)^2 + (-(88200*I - 88200)*A - (88200*I + 88200)*B)*a^2*cos(2*d*x +
2*c) + ((44100*I - 44100)*A + (44100*I + 44100)*B)*a^2)*sin(5*d*x + 5*c) + ((-(102900*I - 102900)*A - (102900*
I + 102900)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(102900*I - 102900)*A - (102900*I + 102900)*B)*a^2*sin(2*d*x + 2*c)^
2 + ((205800*I - 205800)*A + (205800*I + 205800)*B)*a^2*cos(2*d*x + 2*c) + (-(102900*I - 102900)*A - (102900*I
+ 102900)*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)) + (((-(48300*I +
48300)*A + (55860*I - 55860)*B)*a^2*cos(d*x + c) + (-(48300*I - 48300)*A - (55860*I + 55860)*B)*a^2*sin(d*x +
c))*cos(2*d*x + 2*c)^4 + ((-(48300*I + 48300)*A + (55860*I - 55860)*B)*a^2*cos(d*x + c) + (-(48300*I - 48300)
*A - (55860*I + 55860)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^4 + (((193200*I + 193200)*A - (223440*I - 223440)
*B)*a^2*cos(d*x + c) + ((193200*I - 193200)*A + (223440*I + 223440)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^3 +
(-(48300*I + 48300)*A + (55860*I - 55860)*B)*a^2*cos(d*x + c) + (-(48300*I - 48300)*A - (55860*I + 55860)*B)*a
^2*sin(d*x + c) + ((-(289800*I + 289800)*A + (335160*I - 335160)*B)*a^2*cos(d*x + c) + (-(289800*I - 289800)*A
- (335160*I + 335160)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(96600*I + 96600)*A + (111720*I - 111720)*
B)*a^2*cos(d*x + c) + (-(96600*I - 96600)*A - (111720*I + 111720)*B)*a^2*sin(d*x + c) + ((-(96600*I + 96600)*A
+ (111720*I - 111720)*B)*a^2*cos(d*x + c) + (-(96600*I - 96600)*A - (111720*I + 111720)*B)*a^2*sin(d*x + c))*
cos(2*d*x + 2*c)^2 + (((193200*I + 193200)*A - (223440*I - 223440)*B)*a^2*cos(d*x + c) + ((193200*I - 193200)*
A + (223440*I + 223440)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + (((193200*I + 193200)*A -
(223440*I - 223440)*B)*a^2*cos(d*x + c) + ((193200*I - 193200)*A + (223440*I + 223440)*B)*a^2*sin(d*x + c))*co
s(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)
________________________________________________________________________________________
Fricas [B] time = 1.56895, size = 1659, normalized size = 6.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")
[Out]
1/210*(sqrt(2)*((1600*I*A + 1456*B)*a^2*e^(6*I*d*x + 6*I*c) + (-3080*I*A - 3416*B)*a^2*e^(4*I*d*x + 4*I*c) + (
2800*I*A + 2800*B)*a^2*e^(2*I*d*x + 2*I*c) + (-840*I*A - 840*B)*a^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I
*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*e^(I*d*x + I*c) - 105*sqrt((-32*I*A^2 - 64*A*B + 32*I*B^2
)*a^5/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log((sqrt(2)*((4*I*
A + 4*B)*a^2*e^(2*I*d*x + 2*I*c) + (-4*I*A - 4*B)*a^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x +
2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*e^(I*d*x + I*c) + sqrt((-32*I*A^2 - 64*A*B + 32*I*B^2)*a^5/d^2)*d*e^(2*
I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/((4*I*A + 4*B)*a^2)) + 105*sqrt((-32*I*A^2 - 64*A*B + 32*I*B^2)*a^5/d^2)*
(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log((sqrt(2)*((4*I*A + 4*B)*a^
2*e^(2*I*d*x + 2*I*c) + (-4*I*A - 4*B)*a^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)
/(e^(2*I*d*x + 2*I*c) - 1))*e^(I*d*x + I*c) - sqrt((-32*I*A^2 - 64*A*B + 32*I*B^2)*a^5/d^2)*d*e^(2*I*d*x + 2*I
*c))*e^(-2*I*d*x - 2*I*c)/((4*I*A + 4*B)*a^2)))/(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*
d*x + 2*I*c) - d)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)**(9/2)*(a+I*a*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \tan \left (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cot \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)