3.547 \(\int \cot ^{\frac{11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\)
Optimal. Leaf size=297 \[ -\frac{2 a^2 (3 B+4 i A) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{2 a^2 (46 A-45 i B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{8 a^2 (60 B+59 i A) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{315 d}-\frac{8 a^2 (197 A-195 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{315 d}+\frac{(4+4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d} \]
[Out]
((4 + 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot
[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (8*a^2*(197*A - (195*I)*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(
315*d) + (8*a^2*((59*I)*A + 60*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(315*d) + (2*a^2*(46*A - (45*
I)*B)*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a^2*((4*I)*A + 3*B)*Cot[c + d*x]^(7/2)*Sqrt[
a + I*a*Tan[c + d*x]])/(21*d) - (2*a*A*Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2))/(9*d)
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Rubi [A] time = 1.1288, antiderivative size = 297, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used =
{4241, 3593, 3598, 12, 3544, 205} \[ -\frac{2 a^2 (3 B+4 i A) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{2 a^2 (46 A-45 i B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{8 a^2 (60 B+59 i A) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{315 d}-\frac{8 a^2 (197 A-195 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{315 d}+\frac{(4+4 i) a^{5/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d} \]
Antiderivative was successfully verified.
[In]
Int[Cot[c + d*x]^(11/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
[Out]
((4 + 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot
[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (8*a^2*(197*A - (195*I)*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(
315*d) + (8*a^2*((59*I)*A + 60*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(315*d) + (2*a^2*(46*A - (45*
I)*B)*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a^2*((4*I)*A + 3*B)*Cot[c + d*x]^(7/2)*Sqrt[
a + I*a*Tan[c + d*x]])/(21*d) - (2*a*A*Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2))/(9*d)
Rule 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{11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac{11}{2}}(c+d x)} \, dx\\ &=-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac{1}{9} \left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a+i a \tan (c+d x))^{3/2} \left (\frac{3}{2} a (4 i A+3 B)-\frac{3}{2} a (2 A-3 i B) \tan (c+d x)\right )}{\tan ^{\frac{9}{2}}(c+d x)} \, dx\\ &=-\frac{2 a^2 (4 i A+3 B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac{1}{63} \left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)} \left (-\frac{3}{4} a^2 (46 A-45 i B)-\frac{3}{4} a^2 (38 i A+39 B) \tan (c+d x)\right )}{\tan ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (46 A-45 i B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a^2 (4 i A+3 B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac{\left (8 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)} \left (-\frac{3}{2} a^3 (59 i A+60 B)+\frac{3}{2} a^3 (46 A-45 i B) \tan (c+d x)\right )}{\tan ^{\frac{5}{2}}(c+d x)} \, dx}{315 a}\\ &=\frac{8 a^2 (59 i A+60 B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{315 d}+\frac{2 a^2 (46 A-45 i B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a^2 (4 i A+3 B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac{\left (16 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)} \left (\frac{3}{4} a^4 (197 A-195 i B)+\frac{3}{2} a^4 (59 i A+60 B) \tan (c+d x)\right )}{\tan ^{\frac{3}{2}}(c+d x)} \, dx}{945 a^2}\\ &=-\frac{8 a^2 (197 A-195 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{315 d}+\frac{8 a^2 (59 i A+60 B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{315 d}+\frac{2 a^2 (46 A-45 i B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a^2 (4 i A+3 B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac{\left (32 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{945 a^5 (i A+B) \sqrt{a+i a \tan (c+d x)}}{8 \sqrt{\tan (c+d x)}} \, dx}{945 a^3}\\ &=-\frac{8 a^2 (197 A-195 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{315 d}+\frac{8 a^2 (59 i A+60 B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{315 d}+\frac{2 a^2 (46 A-45 i B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a^2 (4 i A+3 B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\left (4 a^2 (i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx\\ &=-\frac{8 a^2 (197 A-195 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{315 d}+\frac{8 a^2 (59 i A+60 B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{315 d}+\frac{2 a^2 (46 A-45 i B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a^2 (4 i A+3 B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}-\frac{\left (8 i a^4 (i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-i a-2 a^2 x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}\\ &=\frac{(4-4 i) a^{5/2} (i A+B) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{d}-\frac{8 a^2 (197 A-195 i B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{315 d}+\frac{8 a^2 (59 i A+60 B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{315 d}+\frac{2 a^2 (46 A-45 i B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a^2 (4 i A+3 B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}\\ \end{align*}
Mathematica [A] time = 11.5458, size = 354, normalized size = 1.19 \[ \frac{(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left (4 \sqrt{2} (A-i B) e^{-3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left (1+e^{2 i (c+d x)}\right )}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )+\frac{(\cos (2 c)-i \sin (2 c)) \sqrt{\cot (c+d x)} \csc ^4(c+d x) \sqrt{\sec (c+d x)} (12 (251 A-260 i B) \cos (2 (c+d x))+(-961 A+915 i B) \cos (4 (c+d x))+282 i A \sin (2 (c+d x))-331 i A \sin (4 (c+d x))-2331 A+390 B \sin (2 (c+d x))-285 B \sin (4 (c+d x))+2205 i B)}{1260 (\cos (d x)+i \sin (d x))^2}\right )}{d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[c + d*x]^(11/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
[Out]
(((4*Sqrt[2]*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*
(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]
)/E^((3*I)*(c + d*x)) + (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^4*Sqrt[Sec[c + d*x]]*(Cos[2*c] - I*Sin[2*c])*(-2331*A
+ (2205*I)*B + 12*(251*A - (260*I)*B)*Cos[2*(c + d*x)] + (-961*A + (915*I)*B)*Cos[4*(c + d*x)] + (282*I)*A*Si
n[2*(c + d*x)] + 390*B*Sin[2*(c + d*x)] - (331*I)*A*Sin[4*(c + d*x)] - 285*B*Sin[4*(c + d*x)]))/(1260*(Cos[d*x
] + I*Sin[d*x])^2))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] +
B*Sin[c + d*x]))
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Maple [B] time = 0.566, size = 3412, normalized size = 11.5 \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)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x)
[Out]
-1/315/d*a^2*2^(1/2)*(240*B*2^(1/2)*cos(d*x+c)*sin(d*x+c)+630*I*A*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)*cos(d*x+c)^4*sin(d*x+c)-285*B*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)+1260*
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)*cos(d*x+c)^4*sin(d*x
+c)-2281*A*cos(d*x+c)^3*2^(1/2)+1024*A*cos(d*x+c)*2^(1/2)+1714*A*2^(1/2)*cos(d*x+c)^2+1260*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)*cos(d*x+c)^4*sin(d*x+c)+1260*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)*cos(d*x+c)^4*sin(d*x+c)+1260*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)*cos(d*x+c)^4*sin(d*x+
c)+630*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)^4
*sin(d*x+c)-1260*I*A*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)*co
s(d*x+c)^2*sin(d*x+c)-2520*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)*cos(d*x+c)^2*sin(d*x+c)-961*A*cos(d*x+c)^4*2^(1/2)-788*A*2^(1/2)-1260*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/
2)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-630*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*sin(
d*x+c)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((cos(d*x+c)-1)/sin
(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1))+780*I*B*2^(1/2)-2520*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)*cos(d*x+c)^2*sin(d*x+c)-2520*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)*cos(d*x+c)^2*sin(d*x+c)-2520*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)*cos(d*x+c)^2*sin(d*x+c)-1260*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)^2*sin(d*x+
c)+780*B*2^(1/2)*sin(d*x+c)+1292*A*cos(d*x+c)^5*2^(1/2)-1935*B*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)-1200*I*B*cos(d*
x+c)^5*2^(1/2)+915*I*B*cos(d*x+c)^4*2^(1/2)+2220*I*B*cos(d*x+c)^3*2^(1/2)-1695*I*B*cos(d*x+c)^2*2^(1/2)+788*I*
A*sin(d*x+c)*2^(1/2)-1020*I*B*cos(d*x+c)*2^(1/2)+1260*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)*sin(d*x+c)+1260*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)*sin(d*x+c)+630*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))*sin(d*x+c)-1260*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)*cos(d*x+c)^4*sin(d*x+c)-1260*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/si
n(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^4*sin(d*x+c)-630*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))*cos(d*x+c)^4*sin(d*x+c)+630*B*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)*cos(d*x+c)^4*sin(d*x+c)+1260*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arct
an(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^4*sin(d*x+c)+1260*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)*cos(d*x+c)^4*sin(d*x+c)+1292*I*A*cos(d*x+c)^4*sin(d*x+c
)*2^(1/2)-331*I*A*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)-1950*I*A*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)+630*I*A*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)*sin(d*x+c)+1260*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)*sin(d*x+c)+1260*I*A*((cos(d*x+c)-1)/si
n(d*x+c))^(1/2)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)*sin(d*x+c)+236*I*A*cos(d*x+c)*sin(d*x+c)*2
^(1/2)+1260*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)
+1260*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+630*B
*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*sin(d*x+c)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*
x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))-1260*A*((co
s(d*x+c)-1)/sin(d*x+c))^(1/2)*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+2520*A*((cos(d*x+
c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)+2520*A*((c
os(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+126
0*A*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*ln(-(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*s
in(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))-2520*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)
*2^(1/2)+1)-2520*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*arctan(((cos(d*x+c)-1)/sin(d*x+c)
)^(1/2)*2^(1/2)-1)-1260*B*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*ln(-(((cos(d*x+c)-1)/sin(d
*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-cos(d*x+c)-sin(d*x+c)+1)/(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)
+cos(d*x+c)+sin(d*x+c)-1))+1200*B*cos(d*x+c)^4*sin(d*x+c)*2^(1/2))*(cos(d*x+c)/sin(d*x+c))^(11/2)*(a*(I*sin(d*
x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*sin(d*x+c)/(I*sin(d*x+c)+cos(d*x+c)-1)/cos(d*x+c)^5
________________________________________________________________________________________
Maxima [B] time = 26.9287, size = 6020, normalized size = 20.27 \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)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")
[Out]
1/1587600*(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*(((-(6350400*I + 6350400)*A
+ (6350400*I - 6350400)*B)*a^2*cos(7*d*x + 7*c) + ((21168000*I + 21168000)*A - (21168000*I - 21168000)*B)*a^2*
cos(5*d*x + 5*c) + (-(25824960*I + 25824960)*A + (25824960*I - 25824960)*B)*a^2*cos(3*d*x + 3*c) + ((11429460*
I + 11429460)*A - (11139660*I - 11139660)*B)*a^2*cos(d*x + c) + (-(6350400*I - 6350400)*A - (6350400*I + 63504
00)*B)*a^2*sin(7*d*x + 7*c) + ((21168000*I - 21168000)*A + (21168000*I + 21168000)*B)*a^2*sin(5*d*x + 5*c) + (
-(25824960*I - 25824960)*A - (25824960*I + 25824960)*B)*a^2*sin(3*d*x + 3*c) + ((11429460*I - 11429460)*A + (1
1139660*I + 11139660)*B)*a^2*sin(d*x + c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((12196
80*I + 1219680)*A - (756000*I - 756000)*B)*a^2*cos(d*x + c) + ((1219680*I - 1219680)*A + (756000*I + 756000)*B
)*a^2*sin(d*x + c) + (((1219680*I + 1219680)*A - (756000*I - 756000)*B)*a^2*cos(d*x + c) + ((1219680*I - 12196
80)*A + (756000*I + 756000)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (((1219680*I + 1219680)*A - (756000*I -
756000)*B)*a^2*cos(d*x + c) + ((1219680*I - 1219680)*A + (756000*I + 756000)*B)*a^2*sin(d*x + c))*sin(2*d*x +
2*c)^2 + ((-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(6350400*I + 6350400
)*A + (6350400*I - 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I + 12700800)*A - (12700800*I - 12700800)*B
)*a^2*cos(2*d*x + 2*c) + (-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2)*cos(3*d*x + 3*c) + ((-(2439
360*I + 2439360)*A + (1512000*I - 1512000)*B)*a^2*cos(d*x + c) + (-(2439360*I - 2439360)*A - (1512000*I + 1512
000)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + ((-(6350400*I - 6350400)*A - (6350400*I + 6350400)*B)*a^2*cos(2*d
*x + 2*c)^2 + (-(6350400*I - 6350400)*A - (6350400*I + 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I - 127
00800)*A + (12700800*I + 12700800)*B)*a^2*cos(2*d*x + 2*c) + (-(6350400*I - 6350400)*A - (6350400*I + 6350400)
*B)*a^2)*sin(3*d*x + 3*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((6350400*I - 6350400)*
A + (6350400*I + 6350400)*B)*a^2*cos(7*d*x + 7*c) + (-(21168000*I - 21168000)*A - (21168000*I + 21168000)*B)*a
^2*cos(5*d*x + 5*c) + ((25824960*I - 25824960)*A + (25824960*I + 25824960)*B)*a^2*cos(3*d*x + 3*c) + (-(114294
60*I - 11429460)*A - (11139660*I + 11139660)*B)*a^2*cos(d*x + c) + (-(6350400*I + 6350400)*A + (6350400*I - 63
50400)*B)*a^2*sin(7*d*x + 7*c) + ((21168000*I + 21168000)*A - (21168000*I - 21168000)*B)*a^2*sin(5*d*x + 5*c)
+ (-(25824960*I + 25824960)*A + (25824960*I - 25824960)*B)*a^2*sin(3*d*x + 3*c) + ((11429460*I + 11429460)*A -
(11139660*I - 11139660)*B)*a^2*sin(d*x + c))*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + ((-(1
219680*I - 1219680)*A - (756000*I + 756000)*B)*a^2*cos(d*x + c) + ((1219680*I + 1219680)*A - (756000*I - 75600
0)*B)*a^2*sin(d*x + c) + ((-(1219680*I - 1219680)*A - (756000*I + 756000)*B)*a^2*cos(d*x + c) + ((1219680*I +
1219680)*A - (756000*I - 756000)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(1219680*I - 1219680)*A - (75600
0*I + 756000)*B)*a^2*cos(d*x + c) + ((1219680*I + 1219680)*A - (756000*I - 756000)*B)*a^2*sin(d*x + c))*sin(2*
d*x + 2*c)^2 + (((6350400*I - 6350400)*A + (6350400*I + 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + ((6350400*I - 635
0400)*A + (6350400*I + 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + (-(12700800*I - 12700800)*A - (12700800*I + 127008
00)*B)*a^2*cos(2*d*x + 2*c) + ((6350400*I - 6350400)*A + (6350400*I + 6350400)*B)*a^2)*cos(3*d*x + 3*c) + (((2
439360*I - 2439360)*A + (1512000*I + 1512000)*B)*a^2*cos(d*x + c) + (-(2439360*I + 2439360)*A + (1512000*I - 1
512000)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + ((-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*cos(
2*d*x + 2*c)^2 + (-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I +
12700800)*A - (12700800*I - 12700800)*B)*a^2*cos(2*d*x + 2*c) + (-(6350400*I + 6350400)*A + (6350400*I - 63504
00)*B)*a^2)*sin(3*d*x + 3*c))*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a) + ((((6350400*
I - 6350400)*A + (6350400*I + 6350400)*B)*a^2*cos(2*d*x + 2*c)^4 + ((6350400*I - 6350400)*A + (6350400*I + 635
0400)*B)*a^2*sin(2*d*x + 2*c)^4 + (-(25401600*I - 25401600)*A - (25401600*I + 25401600)*B)*a^2*cos(2*d*x + 2*c
)^3 + ((38102400*I - 38102400)*A + (38102400*I + 38102400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(25401600*I - 2540160
0)*A - (25401600*I + 25401600)*B)*a^2*cos(2*d*x + 2*c) + ((6350400*I - 6350400)*A + (6350400*I + 6350400)*B)*a
^2 + (((12700800*I - 12700800)*A + (12700800*I + 12700800)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(25401600*I - 2540160
0)*A - (25401600*I + 25401600)*B)*a^2*cos(2*d*x + 2*c) + ((12700800*I - 12700800)*A + (12700800*I + 12700800)*
B)*a^2)*sin(2*d*x + 2*c)^2)*arctan2(2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)
*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 2*sin(d*x + c), 2*(cos(2*d*x + 2*c)^2 + sin(2*d*x
+ 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 2*cos(d*x
+ c)) + (((3175200*I + 3175200)*A - (3175200*I - 3175200)*B)*a^2*cos(2*d*x + 2*c)^4 + ((3175200*I + 3175200)*A
- (3175200*I - 3175200)*B)*a^2*sin(2*d*x + 2*c)^4 + (-(12700800*I + 12700800)*A + (12700800*I - 12700800)*B)*
a^2*cos(2*d*x + 2*c)^3 + ((19051200*I + 19051200)*A - (19051200*I - 19051200)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(1
2700800*I + 12700800)*A + (12700800*I - 12700800)*B)*a^2*cos(2*d*x + 2*c) + ((3175200*I + 3175200)*A - (317520
0*I - 3175200)*B)*a^2 + (((6350400*I + 6350400)*A - (6350400*I - 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(12700
800*I + 12700800)*A + (12700800*I - 12700800)*B)*a^2*cos(2*d*x + 2*c) + ((6350400*I + 6350400)*A - (6350400*I
- 6350400)*B)*a^2)*sin(2*d*x + 2*c)^2)*log(4*cos(d*x + c)^2 + 4*sin(d*x + c)^2 + 4*sqrt(cos(2*d*x + 2*c)^2 + s
in(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2 + sin(
1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2) + 8*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2
*d*x + 2*c) + 1)^(1/4)*(cos(d*x + c)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + sin(d*x + c)*s
in(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))))*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*
d*x + 2*c) + 1)^(1/4)*sqrt(a) + (((-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*cos(9*d*x + 9*c) +
((7408800*I + 7408800)*A - (13759200*I - 13759200)*B)*a^2*cos(7*d*x + 7*c) + (-(8414280*I + 8414280)*A + (1317
7080*I - 13177080)*B)*a^2*cos(5*d*x + 5*c) + ((631260*I + 631260)*A - (6717060*I - 6717060)*B)*a^2*cos(3*d*x +
3*c) + ((1079820*I + 1079820)*A + (948780*I - 948780)*B)*a^2*cos(d*x + c) + (-(6350400*I - 6350400)*A - (6350
400*I + 6350400)*B)*a^2*sin(9*d*x + 9*c) + ((7408800*I - 7408800)*A + (13759200*I + 13759200)*B)*a^2*sin(7*d*x
+ 7*c) + (-(8414280*I - 8414280)*A - (13177080*I + 13177080)*B)*a^2*sin(5*d*x + 5*c) + ((631260*I - 631260)*A
+ (6717060*I + 6717060)*B)*a^2*sin(3*d*x + 3*c) + ((1079820*I - 1079820)*A - (948780*I + 948780)*B)*a^2*sin(d
*x + c))*cos(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + ((-(7159320*I + 7159320)*A + (6811560*I -
6811560)*B)*a^2*cos(d*x + c) + (-(7159320*I - 7159320)*A - (6811560*I + 6811560)*B)*a^2*sin(d*x + c) + ((-(715
9320*I + 7159320)*A + (6811560*I - 6811560)*B)*a^2*cos(d*x + c) + (-(7159320*I - 7159320)*A - (6811560*I + 681
1560)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(7159320*I + 7159320)*A + (6811560*I - 6811560)*B)*a^2*cos(
d*x + c) + (-(7159320*I - 7159320)*A - (6811560*I + 6811560)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + ((-(635
0400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(6350400*I + 6350400)*A + (6350400*I
- 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I + 12700800)*A - (12700800*I - 12700800)*B)*a^2*cos(2*d*x
+ 2*c) + (-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2)*cos(5*d*x + 5*c) + (((14817600*I + 14817600
)*A - (14817600*I - 14817600)*B)*a^2*cos(2*d*x + 2*c)^2 + ((14817600*I + 14817600)*A - (14817600*I - 14817600)
*B)*a^2*sin(2*d*x + 2*c)^2 + (-(29635200*I + 29635200)*A + (29635200*I - 29635200)*B)*a^2*cos(2*d*x + 2*c) + (
(14817600*I + 14817600)*A - (14817600*I - 14817600)*B)*a^2)*cos(3*d*x + 3*c) + (((14318640*I + 14318640)*A - (
13623120*I - 13623120)*B)*a^2*cos(d*x + c) + ((14318640*I - 14318640)*A + (13623120*I + 13623120)*B)*a^2*sin(d
*x + c))*cos(2*d*x + 2*c) + ((-(6350400*I - 6350400)*A - (6350400*I + 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(
6350400*I - 6350400)*A - (6350400*I + 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I - 12700800)*A + (12700
800*I + 12700800)*B)*a^2*cos(2*d*x + 2*c) + (-(6350400*I - 6350400)*A - (6350400*I + 6350400)*B)*a^2)*sin(5*d*
x + 5*c) + (((14817600*I - 14817600)*A + (14817600*I + 14817600)*B)*a^2*cos(2*d*x + 2*c)^2 + ((14817600*I - 14
817600)*A + (14817600*I + 14817600)*B)*a^2*sin(2*d*x + 2*c)^2 + (-(29635200*I - 29635200)*A - (29635200*I + 29
635200)*B)*a^2*cos(2*d*x + 2*c) + ((14817600*I - 14817600)*A + (14817600*I + 14817600)*B)*a^2)*sin(3*d*x + 3*c
))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + ((((12378240*I + 12378240)*A - (13305600*I - 133
05600)*B)*a^2*cos(d*x + c) + ((12378240*I - 12378240)*A + (13305600*I + 13305600)*B)*a^2*sin(d*x + c))*cos(2*d
*x + 2*c)^4 + (((12378240*I + 12378240)*A - (13305600*I - 13305600)*B)*a^2*cos(d*x + c) + ((12378240*I - 12378
240)*A + (13305600*I + 13305600)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^4 + ((-(49512960*I + 49512960)*A + (532
22400*I - 53222400)*B)*a^2*cos(d*x + c) + (-(49512960*I - 49512960)*A - (53222400*I + 53222400)*B)*a^2*sin(d*x
+ c))*cos(2*d*x + 2*c)^3 + ((12378240*I + 12378240)*A - (13305600*I - 13305600)*B)*a^2*cos(d*x + c) + ((12378
240*I - 12378240)*A + (13305600*I + 13305600)*B)*a^2*sin(d*x + c) + (((74269440*I + 74269440)*A - (79833600*I
- 79833600)*B)*a^2*cos(d*x + c) + ((74269440*I - 74269440)*A + (79833600*I + 79833600)*B)*a^2*sin(d*x + c))*co
s(2*d*x + 2*c)^2 + (((24756480*I + 24756480)*A - (26611200*I - 26611200)*B)*a^2*cos(d*x + c) + ((24756480*I -
24756480)*A + (26611200*I + 26611200)*B)*a^2*sin(d*x + c) + (((24756480*I + 24756480)*A - (26611200*I - 266112
00)*B)*a^2*cos(d*x + c) + ((24756480*I - 24756480)*A + (26611200*I + 26611200)*B)*a^2*sin(d*x + c))*cos(2*d*x
+ 2*c)^2 + ((-(49512960*I + 49512960)*A + (53222400*I - 53222400)*B)*a^2*cos(d*x + c) + (-(49512960*I - 495129
60)*A - (53222400*I + 53222400)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + ((-(49512960*I + 4
9512960)*A + (53222400*I - 53222400)*B)*a^2*cos(d*x + c) + (-(49512960*I - 49512960)*A - (53222400*I + 5322240
0)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((635040
0*I - 6350400)*A + (6350400*I + 6350400)*B)*a^2*cos(9*d*x + 9*c) + (-(7408800*I - 7408800)*A - (13759200*I + 1
3759200)*B)*a^2*cos(7*d*x + 7*c) + ((8414280*I - 8414280)*A + (13177080*I + 13177080)*B)*a^2*cos(5*d*x + 5*c)
+ (-(631260*I - 631260)*A - (6717060*I + 6717060)*B)*a^2*cos(3*d*x + 3*c) + (-(1079820*I - 1079820)*A + (94878
0*I + 948780)*B)*a^2*cos(d*x + c) + (-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2*sin(9*d*x + 9*c)
+ ((7408800*I + 7408800)*A - (13759200*I - 13759200)*B)*a^2*sin(7*d*x + 7*c) + (-(8414280*I + 8414280)*A + (13
177080*I - 13177080)*B)*a^2*sin(5*d*x + 5*c) + ((631260*I + 631260)*A - (6717060*I - 6717060)*B)*a^2*sin(3*d*x
+ 3*c) + ((1079820*I + 1079820)*A + (948780*I - 948780)*B)*a^2*sin(d*x + c))*sin(9/2*arctan2(sin(2*d*x + 2*c)
, cos(2*d*x + 2*c) - 1)) + (((7159320*I - 7159320)*A + (6811560*I + 6811560)*B)*a^2*cos(d*x + c) + (-(7159320*
I + 7159320)*A + (6811560*I - 6811560)*B)*a^2*sin(d*x + c) + (((7159320*I - 7159320)*A + (6811560*I + 6811560)
*B)*a^2*cos(d*x + c) + (-(7159320*I + 7159320)*A + (6811560*I - 6811560)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)
^2 + (((7159320*I - 7159320)*A + (6811560*I + 6811560)*B)*a^2*cos(d*x + c) + (-(7159320*I + 7159320)*A + (6811
560*I - 6811560)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + (((6350400*I - 6350400)*A + (6350400*I + 6350400)*B
)*a^2*cos(2*d*x + 2*c)^2 + ((6350400*I - 6350400)*A + (6350400*I + 6350400)*B)*a^2*sin(2*d*x + 2*c)^2 + (-(127
00800*I - 12700800)*A - (12700800*I + 12700800)*B)*a^2*cos(2*d*x + 2*c) + ((6350400*I - 6350400)*A + (6350400*
I + 6350400)*B)*a^2)*cos(5*d*x + 5*c) + ((-(14817600*I - 14817600)*A - (14817600*I + 14817600)*B)*a^2*cos(2*d*
x + 2*c)^2 + (-(14817600*I - 14817600)*A - (14817600*I + 14817600)*B)*a^2*sin(2*d*x + 2*c)^2 + ((29635200*I -
29635200)*A + (29635200*I + 29635200)*B)*a^2*cos(2*d*x + 2*c) + (-(14817600*I - 14817600)*A - (14817600*I + 14
817600)*B)*a^2)*cos(3*d*x + 3*c) + ((-(14318640*I - 14318640)*A - (13623120*I + 13623120)*B)*a^2*cos(d*x + c)
+ ((14318640*I + 14318640)*A - (13623120*I - 13623120)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + ((-(6350400*I +
6350400)*A + (6350400*I - 6350400)*B)*a^2*cos(2*d*x + 2*c)^2 + (-(6350400*I + 6350400)*A + (6350400*I - 63504
00)*B)*a^2*sin(2*d*x + 2*c)^2 + ((12700800*I + 12700800)*A - (12700800*I - 12700800)*B)*a^2*cos(2*d*x + 2*c) +
(-(6350400*I + 6350400)*A + (6350400*I - 6350400)*B)*a^2)*sin(5*d*x + 5*c) + (((14817600*I + 14817600)*A - (1
4817600*I - 14817600)*B)*a^2*cos(2*d*x + 2*c)^2 + ((14817600*I + 14817600)*A - (14817600*I - 14817600)*B)*a^2*
sin(2*d*x + 2*c)^2 + (-(29635200*I + 29635200)*A + (29635200*I - 29635200)*B)*a^2*cos(2*d*x + 2*c) + ((1481760
0*I + 14817600)*A - (14817600*I - 14817600)*B)*a^2)*sin(3*d*x + 3*c))*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*
d*x + 2*c) - 1)) + (((-(12378240*I - 12378240)*A - (13305600*I + 13305600)*B)*a^2*cos(d*x + c) + ((12378240*I
+ 12378240)*A - (13305600*I - 13305600)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^4 + ((-(12378240*I - 12378240)*A
- (13305600*I + 13305600)*B)*a^2*cos(d*x + c) + ((12378240*I + 12378240)*A - (13305600*I - 13305600)*B)*a^2*s
in(d*x + c))*sin(2*d*x + 2*c)^4 + (((49512960*I - 49512960)*A + (53222400*I + 53222400)*B)*a^2*cos(d*x + c) +
(-(49512960*I + 49512960)*A + (53222400*I - 53222400)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^3 + (-(12378240*I
- 12378240)*A - (13305600*I + 13305600)*B)*a^2*cos(d*x + c) + ((12378240*I + 12378240)*A - (13305600*I - 13305
600)*B)*a^2*sin(d*x + c) + ((-(74269440*I - 74269440)*A - (79833600*I + 79833600)*B)*a^2*cos(d*x + c) + ((7426
9440*I + 74269440)*A - (79833600*I - 79833600)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((-(24756480*I - 2475
6480)*A - (26611200*I + 26611200)*B)*a^2*cos(d*x + c) + ((24756480*I + 24756480)*A - (26611200*I - 26611200)*B
)*a^2*sin(d*x + c) + ((-(24756480*I - 24756480)*A - (26611200*I + 26611200)*B)*a^2*cos(d*x + c) + ((24756480*I
+ 24756480)*A - (26611200*I - 26611200)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (((49512960*I - 49512960)*A
+ (53222400*I + 53222400)*B)*a^2*cos(d*x + c) + (-(49512960*I + 49512960)*A + (53222400*I - 53222400)*B)*a^2*
sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + (((49512960*I - 49512960)*A + (53222400*I + 53222400)*B)*
a^2*cos(d*x + c) + (-(49512960*I + 49512960)*A + (53222400*I - 53222400)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)
)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a))/((cos(2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4
- 4*cos(2*d*x + 2*c)^3 + 2*(cos(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c)^2 + 6*cos(2*d*x + 2
*c)^2 - 4*cos(2*d*x + 2*c) + 1)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*d)
________________________________________________________________________________________
Fricas [B] time = 1.54211, size = 1813, normalized size = 6.1 \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)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")
[Out]
-1/630*(8*sqrt(2)*(2*(323*A - 300*I*B)*a^2*e^(8*I*d*x + 8*I*c) - 27*(61*A - 65*I*B)*a^2*e^(6*I*d*x + 6*I*c) +
63*(37*A - 35*I*B)*a^2*e^(4*I*d*x + 4*I*c) - 1365*(A - I*B)*a^2*e^(2*I*d*x + 2*I*c) + 315*(A - I*B)*a^2)*sqrt(
a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*e^(I*d*x + I*c) - 315
*sqrt((32*I*A^2 + 64*A*B - 32*I*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*
x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((4*I*A + 4*B)*a^2*e^(2*I*d*x + 2*I*c) + (-4*I*A - 4*B)
*a^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*e^(I*d*x +
I*c) + I*sqrt((32*I*A^2 + 64*A*B - 32*I*B^2)*a^5/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/((4*I*A + 4
*B)*a^2)) + 315*sqrt((32*I*A^2 + 64*A*B - 32*I*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c)
+ 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log((sqrt(2)*((4*I*A + 4*B)*a^2*e^(2*I*d*x + 2*I*c) +
(-4*I*A - 4*B)*a^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) -
1))*e^(I*d*x + I*c) - I*sqrt((32*I*A^2 + 64*A*B - 32*I*B^2)*a^5/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I
*c)/((4*I*A + 4*B)*a^2)))/(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(
2*I*d*x + 2*I*c) + d)
________________________________________________________________________________________
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)**(11/2)*(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{11}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="giac")
[Out]
integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^(5/2)*cot(d*x + c)^(11/2), x)