3.541 \(\int \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\)
Optimal. Leaf size=245 \[ \frac{(2-2 i) a^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{2 a (7 B+8 i A) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}+\frac{4 a (19 A-21 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{4 a (63 B+67 i A) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d} \]
[Out]
((2 - 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot
[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (4*a*((67*I)*A + 63*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(105*
d) + (4*a*(19*A - (21*I)*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a*((8*I)*A + 7*B)*Cot[
c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) - (2*a*A*Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]])/(7*d
)
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Rubi [A] time = 0.89362, antiderivative size = 245, 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} \[ \frac{(2-2 i) a^{3/2} (A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{2 a (7 B+8 i A) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}+\frac{4 a (19 A-21 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{4 a (63 B+67 i A) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d} \]
Antiderivative was successfully verified.
[In]
Int[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]
[Out]
((2 - 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot
[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (4*a*((67*I)*A + 63*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(105*
d) + (4*a*(19*A - (21*I)*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a*((8*I)*A + 7*B)*Cot[
c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) - (2*a*A*Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]])/(7*d
)
Rule 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))^{3/2} (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))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{9}{2}}(c+d x)} \, dx\\ &=-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}+\frac{1}{7} \left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)} \left (\frac{1}{2} a (8 i A+7 B)-\frac{1}{2} a (6 A-7 i B) \tan (c+d x)\right )}{\tan ^{\frac{7}{2}}(c+d x)} \, dx\\ &=-\frac{2 a (8 i A+7 B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}+\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)} \left (-\frac{1}{2} a^2 (19 A-21 i B)-a^2 (8 i A+7 B) \tan (c+d x)\right )}{\tan ^{\frac{5}{2}}(c+d x)} \, dx}{35 a}\\ &=\frac{4 a (19 A-21 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a (8 i A+7 B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}+\frac{\left (8 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)} \left (-\frac{1}{4} a^3 (67 i A+63 B)+\frac{1}{2} a^3 (19 A-21 i B) \tan (c+d x)\right )}{\tan ^{\frac{3}{2}}(c+d x)} \, dx}{105 a^2}\\ &=\frac{4 a (67 i A+63 B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{4 a (19 A-21 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a (8 i A+7 B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}+\frac{\left (16 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{105 a^4 (A-i B) \sqrt{a+i a \tan (c+d x)}}{8 \sqrt{\tan (c+d x)}} \, dx}{105 a^3}\\ &=\frac{4 a (67 i A+63 B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{4 a (19 A-21 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a (8 i A+7 B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}+\left (2 a (A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx\\ &=\frac{4 a (67 i A+63 B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{4 a (19 A-21 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a (8 i A+7 B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{\left (4 i a^3 (A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \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{(2+2 i) a^{3/2} (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 (67 i A+63 B) \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{105 d}+\frac{4 a (19 A-21 i B) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{2 a (8 i A+7 B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}-\frac{2 a A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}\\ \end{align*}
Mathematica [A] time = 7.45957, size = 320, normalized size = 1.31 \[ \frac{(a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \left (-2 i \sqrt{2} (A-i B) e^{-2 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left (1+e^{2 i (c+d x)}\right )}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )-\frac{1}{210} \sqrt{\cot (c+d x)} \csc ^3(c+d x) \sqrt{\sec (c+d x)} (\cos (c+d x)-i \sin (c+d x)) (7 (A+6 i B) \cos (c+d x)+(53 A-42 i B) \cos (3 (c+d x))+2 \sin (c+d x) ((147 B+158 i A) \cos (2 (c+d x))-110 i A-105 B))\right )}{d \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]
[Out]
((((-2*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^((2*I)*(c + d*x)) - (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^3*Sqrt[Sec[c + d*x]]*(Cos[c + d*x] - I*Sin[c + d
*x])*(7*(A + (6*I)*B)*Cos[c + d*x] + (53*A - (42*I)*B)*Cos[3*(c + d*x)] + 2*((-110*I)*A - 105*B + ((158*I)*A +
147*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/210)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*
x]^(5/2)*(A*Cos[c + d*x] + B*Sin[c + d*x]))
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Maple [B] time = 0.505, size = 3124, normalized size = 12.8 \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))^(3/2)*(A+B*tan(d*x+c)),x)
[Out]
-1/105/d*a*2^(1/2)*(-420*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)/si
n(d*x+c))^(1/2)-210*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))-420*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)-168*B*2^(1/2)*cos(d*x+c)*sin(d*x+c)+189*B*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)-53*A*cos(d*x+c)^3*2^(1/2)+38*
A*cos(d*x+c)*2^(1/2)-420*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)/si
n(d*x+c))^(1/2)-420*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)-210*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))-330*A*2^(1/2)*cos(d*x+c)^2+211*A*cos(d*x+c)^4*2^(1/2)+134*A*2^(1/2)+210*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)+210*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)+105*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))+210*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)+105*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))+210*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)-126*I*B*2^(1/2)+126*B*2^(1/2)*s
in(d*x+c)-147*B*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)+210*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)+105*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))+105*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)+210*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*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)-189*I*B*2^(1/2)*cos(d*x+c)^4+42*I*B*2^(1/2)*cos(d*x+c)^3+315*I*B*2^(1/2)*cos(d*x+c)^2
+134*I*A*2^(1/2)*sin(d*x+c)+105*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)+210*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)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-42*I*B*2^(1/2)*cos(
d*x+c)-210*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)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)-210*I*B*cos(d*x+c)^4*((co
s(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)-105*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))+211*I*A*2^(1/2)*cos(d*x+c)^3*s
in(d*x+c)-158*I*A*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)-210*I*A*cos(d*x+c)^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))*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)-420*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)*arctan(((c
os(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)*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)*arctan(((cos(d*x
+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)-1)+210*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))-172*I*A*2^(1/2)*cos(d*x+c)*sin(d*x+c)+105*I*A*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)*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)+210*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)*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)*arctan(((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+1)-105*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))+210*A*cos(d*x+c)^4*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*arctan(((co
s(d*x+c)-1)/sin(d*x+c))^(1/2)*2^(1/2)+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 = 14.4057, size = 5021, normalized size = 20.49 \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))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")
[Out]
1/176400*(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*((((352800*I - 352800)*A + (3
52800*I + 352800)*B)*a*cos(7*d*x + 7*c) + (-(176400*I - 176400)*A - (529200*I + 529200)*B)*a*cos(5*d*x + 5*c)
+ ((167580*I - 167580)*A + (255780*I + 255780)*B)*a*cos(3*d*x + 3*c) + ((59220*I - 59220)*A - (79380*I + 79380
)*B)*a*cos(d*x + c) + (-(352800*I + 352800)*A + (352800*I - 352800)*B)*a*sin(7*d*x + 7*c) + ((176400*I + 17640
0)*A - (529200*I - 529200)*B)*a*sin(5*d*x + 5*c) + (-(167580*I + 167580)*A + (255780*I - 255780)*B)*a*sin(3*d*
x + 3*c) + (-(59220*I + 59220)*A - (79380*I - 79380)*B)*a*sin(d*x + c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(
2*d*x + 2*c) - 1)) + (((-(236880*I - 236880)*A - (199920*I + 199920)*B)*a*cos(d*x + c) + ((236880*I + 236880)*
A - (199920*I - 199920)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (-(236880*I - 236880)*A - (199920*I + 199920)*
B)*a*cos(d*x + c) + ((-(236880*I - 236880)*A - (199920*I + 199920)*B)*a*cos(d*x + c) + ((236880*I + 236880)*A
- (199920*I - 199920)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2 + ((236880*I + 236880)*A - (199920*I - 199920)*B)*
a*sin(d*x + c) + (((352800*I - 352800)*A + (352800*I + 352800)*B)*a*cos(2*d*x + 2*c)^2 + ((352800*I - 352800)*
A + (352800*I + 352800)*B)*a*sin(2*d*x + 2*c)^2 + (-(705600*I - 705600)*A - (705600*I + 705600)*B)*a*cos(2*d*x
+ 2*c) + ((352800*I - 352800)*A + (352800*I + 352800)*B)*a)*cos(3*d*x + 3*c) + (((473760*I - 473760)*A + (399
840*I + 399840)*B)*a*cos(d*x + c) + (-(473760*I + 473760)*A + (399840*I - 399840)*B)*a*sin(d*x + c))*cos(2*d*x
+ 2*c) + ((-(352800*I + 352800)*A + (352800*I - 352800)*B)*a*cos(2*d*x + 2*c)^2 + (-(352800*I + 352800)*A + (
352800*I - 352800)*B)*a*sin(2*d*x + 2*c)^2 + ((705600*I + 705600)*A - (705600*I - 705600)*B)*a*cos(2*d*x + 2*c
) + (-(352800*I + 352800)*A + (352800*I - 352800)*B)*a)*sin(3*d*x + 3*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), co
s(2*d*x + 2*c) - 1)) + (((352800*I + 352800)*A - (352800*I - 352800)*B)*a*cos(7*d*x + 7*c) + (-(176400*I + 176
400)*A + (529200*I - 529200)*B)*a*cos(5*d*x + 5*c) + ((167580*I + 167580)*A - (255780*I - 255780)*B)*a*cos(3*d
*x + 3*c) + ((59220*I + 59220)*A + (79380*I - 79380)*B)*a*cos(d*x + c) + ((352800*I - 352800)*A + (352800*I +
352800)*B)*a*sin(7*d*x + 7*c) + (-(176400*I - 176400)*A - (529200*I + 529200)*B)*a*sin(5*d*x + 5*c) + ((167580
*I - 167580)*A + (255780*I + 255780)*B)*a*sin(3*d*x + 3*c) + ((59220*I - 59220)*A - (79380*I + 79380)*B)*a*sin
(d*x + c))*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((-(236880*I + 236880)*A + (199920*I -
199920)*B)*a*cos(d*x + c) + (-(236880*I - 236880)*A - (199920*I + 199920)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^
2 + (-(236880*I + 236880)*A + (199920*I - 199920)*B)*a*cos(d*x + c) + ((-(236880*I + 236880)*A + (199920*I - 1
99920)*B)*a*cos(d*x + c) + (-(236880*I - 236880)*A - (199920*I + 199920)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2
+ (-(236880*I - 236880)*A - (199920*I + 199920)*B)*a*sin(d*x + c) + (((352800*I + 352800)*A - (352800*I - 352
800)*B)*a*cos(2*d*x + 2*c)^2 + ((352800*I + 352800)*A - (352800*I - 352800)*B)*a*sin(2*d*x + 2*c)^2 + (-(70560
0*I + 705600)*A + (705600*I - 705600)*B)*a*cos(2*d*x + 2*c) + ((352800*I + 352800)*A - (352800*I - 352800)*B)*
a)*cos(3*d*x + 3*c) + (((473760*I + 473760)*A - (399840*I - 399840)*B)*a*cos(d*x + c) + ((473760*I - 473760)*A
+ (399840*I + 399840)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c) + (((352800*I - 352800)*A + (352800*I + 352800)*B)*
a*cos(2*d*x + 2*c)^2 + ((352800*I - 352800)*A + (352800*I + 352800)*B)*a*sin(2*d*x + 2*c)^2 + (-(705600*I - 70
5600)*A - (705600*I + 705600)*B)*a*cos(2*d*x + 2*c) + ((352800*I - 352800)*A + (352800*I + 352800)*B)*a)*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) + ((((352800*I + 352800)*A - (3
52800*I - 352800)*B)*a*cos(2*d*x + 2*c)^4 + ((352800*I + 352800)*A - (352800*I - 352800)*B)*a*sin(2*d*x + 2*c)
^4 + (-(1411200*I + 1411200)*A + (1411200*I - 1411200)*B)*a*cos(2*d*x + 2*c)^3 + ((2116800*I + 2116800)*A - (2
116800*I - 2116800)*B)*a*cos(2*d*x + 2*c)^2 + (-(1411200*I + 1411200)*A + (1411200*I - 1411200)*B)*a*cos(2*d*x
+ 2*c) + (((705600*I + 705600)*A - (705600*I - 705600)*B)*a*cos(2*d*x + 2*c)^2 + (-(1411200*I + 1411200)*A +
(1411200*I - 1411200)*B)*a*cos(2*d*x + 2*c) + ((705600*I + 705600)*A - (705600*I - 705600)*B)*a)*sin(2*d*x + 2
*c)^2 + ((352800*I + 352800)*A - (352800*I - 352800)*B)*a)*arctan2(2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2
- 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 2*sin(d*x + c), 2*(
cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(
2*d*x + 2*c) - 1)) + 2*cos(d*x + c)) + ((-(176400*I - 176400)*A - (176400*I + 176400)*B)*a*cos(2*d*x + 2*c)^4
+ (-(176400*I - 176400)*A - (176400*I + 176400)*B)*a*sin(2*d*x + 2*c)^4 + ((705600*I - 705600)*A + (705600*I +
705600)*B)*a*cos(2*d*x + 2*c)^3 + (-(1058400*I - 1058400)*A - (1058400*I + 1058400)*B)*a*cos(2*d*x + 2*c)^2 +
((705600*I - 705600)*A + (705600*I + 705600)*B)*a*cos(2*d*x + 2*c) + ((-(352800*I - 352800)*A - (352800*I + 3
52800)*B)*a*cos(2*d*x + 2*c)^2 + ((705600*I - 705600)*A + (705600*I + 705600)*B)*a*cos(2*d*x + 2*c) + (-(35280
0*I - 352800)*A - (352800*I + 352800)*B)*a)*sin(2*d*x + 2*c)^2 + (-(176400*I - 176400)*A - (176400*I + 176400)
*B)*a)*log(4*cos(d*x + c)^2 + 4*sin(d*x + c)^2 + 4*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x
+ 2*c) + 1)*(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), co
s(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) +
(((((524580*I - 524580)*A + (496860*I + 496860)*B)*a*cos(d*x + c) + (-(524580*I + 524580)*A + (496860*I - 4968
60)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((524580*I - 524580)*A + (496860*I + 496860)*B)*a*cos(d*x + c) + (
((524580*I - 524580)*A + (496860*I + 496860)*B)*a*cos(d*x + c) + (-(524580*I + 524580)*A + (496860*I - 496860)
*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2 + (-(524580*I + 524580)*A + (496860*I - 496860)*B)*a*sin(d*x + c) + (((
352800*I - 352800)*A + (352800*I + 352800)*B)*a*cos(2*d*x + 2*c)^2 + ((352800*I - 352800)*A + (352800*I + 3528
00)*B)*a*sin(2*d*x + 2*c)^2 + (-(705600*I - 705600)*A - (705600*I + 705600)*B)*a*cos(2*d*x + 2*c) + ((352800*I
- 352800)*A + (352800*I + 352800)*B)*a)*cos(5*d*x + 5*c) + ((-(823200*I - 823200)*A - (823200*I + 823200)*B)*
a*cos(2*d*x + 2*c)^2 + (-(823200*I - 823200)*A - (823200*I + 823200)*B)*a*sin(2*d*x + 2*c)^2 + ((1646400*I - 1
646400)*A + (1646400*I + 1646400)*B)*a*cos(2*d*x + 2*c) + (-(823200*I - 823200)*A - (823200*I + 823200)*B)*a)*
cos(3*d*x + 3*c) + ((-(1049160*I - 1049160)*A - (993720*I + 993720)*B)*a*cos(d*x + c) + ((1049160*I + 1049160)
*A - (993720*I - 993720)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c) + ((-(352800*I + 352800)*A + (352800*I - 352800)*
B)*a*cos(2*d*x + 2*c)^2 + (-(352800*I + 352800)*A + (352800*I - 352800)*B)*a*sin(2*d*x + 2*c)^2 + ((705600*I +
705600)*A - (705600*I - 705600)*B)*a*cos(2*d*x + 2*c) + (-(352800*I + 352800)*A + (352800*I - 352800)*B)*a)*s
in(5*d*x + 5*c) + (((823200*I + 823200)*A - (823200*I - 823200)*B)*a*cos(2*d*x + 2*c)^2 + ((823200*I + 823200)
*A - (823200*I - 823200)*B)*a*sin(2*d*x + 2*c)^2 + (-(1646400*I + 1646400)*A + (1646400*I - 1646400)*B)*a*cos(
2*d*x + 2*c) + ((823200*I + 823200)*A - (823200*I - 823200)*B)*a)*sin(3*d*x + 3*c))*cos(5/2*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c) - 1)) + (((-(349440*I - 349440)*A - (423360*I + 423360)*B)*a*cos(d*x + c) + ((349440*
I + 349440)*A - (423360*I - 423360)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^4 + ((-(349440*I - 349440)*A - (423360
*I + 423360)*B)*a*cos(d*x + c) + ((349440*I + 349440)*A - (423360*I - 423360)*B)*a*sin(d*x + c))*sin(2*d*x + 2
*c)^4 + (((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*cos(d*x + c) + (-(1397760*I + 1397760)*A + (169
3440*I - 1693440)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^3 + ((-(2096640*I - 2096640)*A - (2540160*I + 2540160)*B
)*a*cos(d*x + c) + ((2096640*I + 2096640)*A - (2540160*I - 2540160)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (-
(349440*I - 349440)*A - (423360*I + 423360)*B)*a*cos(d*x + c) + (((-(698880*I - 698880)*A - (846720*I + 846720
)*B)*a*cos(d*x + c) + ((698880*I + 698880)*A - (846720*I - 846720)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (-(
698880*I - 698880)*A - (846720*I + 846720)*B)*a*cos(d*x + c) + ((698880*I + 698880)*A - (846720*I - 846720)*B)
*a*sin(d*x + c) + (((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*cos(d*x + c) + (-(1397760*I + 1397760
)*A + (1693440*I - 1693440)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + ((349440*I + 349440)*A -
(423360*I - 423360)*B)*a*sin(d*x + c) + (((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*cos(d*x + c) +
(-(1397760*I + 1397760)*A + (1693440*I - 1693440)*B)*a*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)) + ((((524580*I + 524580)*A - (496860*I - 496860)*B)*a*cos(d*x + c) + ((5245
80*I - 524580)*A + (496860*I + 496860)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^2 + ((524580*I + 524580)*A - (49686
0*I - 496860)*B)*a*cos(d*x + c) + (((524580*I + 524580)*A - (496860*I - 496860)*B)*a*cos(d*x + c) + ((524580*I
- 524580)*A + (496860*I + 496860)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^2 + ((524580*I - 524580)*A + (496860*I
+ 496860)*B)*a*sin(d*x + c) + (((352800*I + 352800)*A - (352800*I - 352800)*B)*a*cos(2*d*x + 2*c)^2 + ((352800
*I + 352800)*A - (352800*I - 352800)*B)*a*sin(2*d*x + 2*c)^2 + (-(705600*I + 705600)*A + (705600*I - 705600)*B
)*a*cos(2*d*x + 2*c) + ((352800*I + 352800)*A - (352800*I - 352800)*B)*a)*cos(5*d*x + 5*c) + ((-(823200*I + 82
3200)*A + (823200*I - 823200)*B)*a*cos(2*d*x + 2*c)^2 + (-(823200*I + 823200)*A + (823200*I - 823200)*B)*a*sin
(2*d*x + 2*c)^2 + ((1646400*I + 1646400)*A - (1646400*I - 1646400)*B)*a*cos(2*d*x + 2*c) + (-(823200*I + 82320
0)*A + (823200*I - 823200)*B)*a)*cos(3*d*x + 3*c) + ((-(1049160*I + 1049160)*A + (993720*I - 993720)*B)*a*cos(
d*x + c) + (-(1049160*I - 1049160)*A - (993720*I + 993720)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c) + (((352800*I -
352800)*A + (352800*I + 352800)*B)*a*cos(2*d*x + 2*c)^2 + ((352800*I - 352800)*A + (352800*I + 352800)*B)*a*s
in(2*d*x + 2*c)^2 + (-(705600*I - 705600)*A - (705600*I + 705600)*B)*a*cos(2*d*x + 2*c) + ((352800*I - 352800)
*A + (352800*I + 352800)*B)*a)*sin(5*d*x + 5*c) + ((-(823200*I - 823200)*A - (823200*I + 823200)*B)*a*cos(2*d*
x + 2*c)^2 + (-(823200*I - 823200)*A - (823200*I + 823200)*B)*a*sin(2*d*x + 2*c)^2 + ((1646400*I - 1646400)*A
+ (1646400*I + 1646400)*B)*a*cos(2*d*x + 2*c) + (-(823200*I - 823200)*A - (823200*I + 823200)*B)*a)*sin(3*d*x
+ 3*c))*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + (((-(349440*I + 349440)*A + (423360*I - 423
360)*B)*a*cos(d*x + c) + (-(349440*I - 349440)*A - (423360*I + 423360)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^4 +
((-(349440*I + 349440)*A + (423360*I - 423360)*B)*a*cos(d*x + c) + (-(349440*I - 349440)*A - (423360*I + 4233
60)*B)*a*sin(d*x + c))*sin(2*d*x + 2*c)^4 + (((1397760*I + 1397760)*A - (1693440*I - 1693440)*B)*a*cos(d*x + c
) + ((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c)^3 + ((-(2096640*I + 2
096640)*A + (2540160*I - 2540160)*B)*a*cos(d*x + c) + (-(2096640*I - 2096640)*A - (2540160*I + 2540160)*B)*a*s
in(d*x + c))*cos(2*d*x + 2*c)^2 + (-(349440*I + 349440)*A + (423360*I - 423360)*B)*a*cos(d*x + c) + (((-(69888
0*I + 698880)*A + (846720*I - 846720)*B)*a*cos(d*x + c) + (-(698880*I - 698880)*A - (846720*I + 846720)*B)*a*s
in(d*x + c))*cos(2*d*x + 2*c)^2 + (-(698880*I + 698880)*A + (846720*I - 846720)*B)*a*cos(d*x + c) + (-(698880*
I - 698880)*A - (846720*I + 846720)*B)*a*sin(d*x + c) + (((1397760*I + 1397760)*A - (1693440*I - 1693440)*B)*a
*cos(d*x + c) + ((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*
x + 2*c)^2 + (-(349440*I - 349440)*A - (423360*I + 423360)*B)*a*sin(d*x + c) + (((1397760*I + 1397760)*A - (16
93440*I - 1693440)*B)*a*cos(d*x + c) + ((1397760*I - 1397760)*A + (1693440*I + 1693440)*B)*a*sin(d*x + c))*cos
(2*d*x + 2*c))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a))/((cos(2*d*x + 2*c)^4 + sin(2
*d*x + 2*c)^4 - 4*cos(2*d*x + 2*c)^3 + 2*(cos(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c)^2 + 6*
cos(2*d*x + 2*c)^2 - 4*cos(2*d*x + 2*c) + 1)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1
)^(1/4)*d)
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Fricas [B] time = 1.50792, size = 1619, 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))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")
[Out]
1/210*(sqrt(2)*((844*I*A + 756*B)*a*e^(6*I*d*x + 6*I*c) + (-1484*I*A - 1596*B)*a*e^(4*I*d*x + 4*I*c) + (1540*I
*A + 1260*B)*a*e^(2*I*d*x + 2*I*c) + (-420*I*A - 420*B)*a)*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((-8*I*A^2 - 16*A*B + 8*I*B^2)*a^3/d^2)*(
d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log((sqrt(2)*((2*I*A + 2*B)*a*e
^(2*I*d*x + 2*I*c) + (-2*I*A - 2*B)*a)*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((-8*I*A^2 - 16*A*B + 8*I*B^2)*a^3/d^2)*d*e^(2*I*d*x + 2*I*c))*e^
(-2*I*d*x - 2*I*c)/((2*I*A + 2*B)*a)) + 105*sqrt((-8*I*A^2 - 16*A*B + 8*I*B^2)*a^3/d^2)*(d*e^(6*I*d*x + 6*I*c)
- 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log((sqrt(2)*((2*I*A + 2*B)*a*e^(2*I*d*x + 2*I*c) +
(-2*I*A - 2*B)*a)*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((-8*I*A^2 - 16*A*B + 8*I*B^2)*a^3/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/((
2*I*A + 2*B)*a)))/(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))**(3/2)*(A+B*tan(d*x+c)),x)
[Out]
Timed out
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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{3}{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))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="giac")
[Out]
integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^(3/2)*cot(d*x + c)^(9/2), x)