3.762 \(\int \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx\)
Optimal. Leaf size=222 \[ -\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{6 i a^2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}+\frac{32 a^2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{104 i a^2 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{(4-4 i) a^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d} \]
[Out]
((4 - 4*I)*a^(5/2)*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 + (((104*I)/21)*a^2*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d + (32*a^2*Cot[c +
d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(21*d) - (((6*I)/7)*a^2*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/
d - (2*a^2*Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]])/(7*d)
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Rubi [A] time = 0.645841, antiderivative size = 222, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used =
{4241, 3553, 3598, 12, 3544, 205} \[ -\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{6 i a^2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}+\frac{32 a^2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{104 i a^2 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{(4-4 i) a^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
[In]
Int[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(5/2),x]
[Out]
((4 - 4*I)*a^(5/2)*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 + (((104*I)/21)*a^2*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d + (32*a^2*Cot[c +
d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(21*d) - (((6*I)/7)*a^2*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/
d - (2*a^2*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 3553
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(a^2*(b*c - a*d)*(a + b*Tan[e + f*x])^(m - 2)*(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 - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*(b*c*(m
- 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x], x] /; Fr
eeQ[{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] && GtQ[m, 1] && LtQ[
n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
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} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a+i a \tan (c+d x))^{5/2}}{\tan ^{\frac{9}{2}}(c+d x)} \, dx\\ &=-\frac{2 a^2 \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{15 i a^2}{2}+\frac{13}{2} a^2 \tan (c+d x)\right )}{\tan ^{\frac{7}{2}}(c+d x)} \, dx\\ &=-\frac{6 i a^2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{2 a^2 \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 (20 a^3+15 i a^3 \tan (c+d x)\right )}{\tan ^{\frac{5}{2}}(c+d x)} \, dx}{35 a}\\ &=\frac{32 a^2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{6 i a^2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{2 a^2 \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{65 i a^4}{2}-20 a^4 \tan (c+d x)\right )}{\tan ^{\frac{3}{2}}(c+d x)} \, dx}{105 a^2}\\ &=\frac{104 i a^2 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{32 a^2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{6 i a^2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{2 a^2 \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^5 \sqrt{a+i a \tan (c+d x)}}{4 \sqrt{\tan (c+d x)}} \, dx}{105 a^3}\\ &=\frac{104 i a^2 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{32 a^2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{6 i a^2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}+\left (4 a^2 \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{104 i a^2 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{32 a^2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{6 i a^2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{\left (8 i a^4 \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} \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{104 i a^2 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{32 a^2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}-\frac{6 i a^2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}\\ \end{align*}
Mathematica [A] time = 2.39077, size = 167, normalized size = 0.75 \[ \frac{4 i a^2 e^{-i (c+d x)} \sqrt{\cot (c+d x)} \left (-21 e^{i (c+d x)}+70 e^{3 i (c+d x)}-77 e^{5 i (c+d x)}+40 e^{7 i (c+d x)}-21 \left (-1+e^{2 i (c+d x)}\right )^{7/2} \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )\right ) \sqrt{a+i a \tan (c+d x)}}{21 d \left (-1+e^{2 i (c+d x)}\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(5/2),x]
[Out]
(((4*I)/21)*a^2*(-21*E^(I*(c + d*x)) + 70*E^((3*I)*(c + d*x)) - 77*E^((5*I)*(c + d*x)) + 40*E^((7*I)*(c + d*x)
) - 21*(-1 + E^((2*I)*(c + d*x)))^(7/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c +
d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^3)
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Maple [B] time = 0.429, size = 1581, normalized size = 7.1 \begin{align*} \text{result 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),x)
[Out]
-1/21/d*2^(1/2)*a^2*(52*2^(1/2)+84*((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+42*((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)*si
n(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+84*I*((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)+52
*I*2^(1/2)*sin(d*x+c)+42*I*((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)*si
n(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))+84*I*((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*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*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)-84*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))+84*((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)^4-129*2^(1/2)*cos(d*x+c)^2+16*2^(1/2)*cos(d*x+
c)+84*I*((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+84*
I*((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+42*I*((co
s(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+80*I*2^(1/2)*c
os(d*x+c)^3*sin(d*x+c)-61*I*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)-168*I*((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)*cos(d*x+c)^2-168*I*((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)^2-84*I*((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)^2-68*I*sin(d*x+c)*cos(d*x+c)*2^(1/2)-19*2^(1/2)*cos(d*x+c)^3+84*((c
os(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)+84*((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*((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))+80*cos(d*x+c)^4*2^(1/2))*(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
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Maxima [B] time = 4.27711, size = 4211, normalized size = 18.97 \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),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^2*cos(7*
d*x + 7*c) - (44100*I - 44100)*a^2*cos(5*d*x + 5*c) + (26460*I - 26460)*a^2*cos(3*d*x + 3*c) - (1260*I - 1260)
*a^2*cos(d*x + c) - (44100*I + 44100)*a^2*sin(7*d*x + 7*c) + (44100*I + 44100)*a^2*sin(5*d*x + 5*c) - (26460*I
+ 26460)*a^2*sin(3*d*x + 3*c) + (1260*I + 1260)*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^2*cos(d*x + c) + (27300*I + 27300)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2
- (27300*I - 27300)*a^2*cos(d*x + c) + (-(27300*I - 27300)*a^2*cos(d*x + c) + (27300*I + 27300)*a^2*sin(d*x +
c))*sin(2*d*x + 2*c)^2 + (27300*I + 27300)*a^2*sin(d*x + c) + ((44100*I - 44100)*a^2*cos(2*d*x + 2*c)^2 + (44
100*I - 44100)*a^2*sin(2*d*x + 2*c)^2 - (88200*I - 88200)*a^2*cos(2*d*x + 2*c) + (44100*I - 44100)*a^2)*cos(3*
d*x + 3*c) + ((54600*I - 54600)*a^2*cos(d*x + c) - (54600*I + 54600)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + (-(4
4100*I + 44100)*a^2*cos(2*d*x + 2*c)^2 - (44100*I + 44100)*a^2*sin(2*d*x + 2*c)^2 + (88200*I + 88200)*a^2*cos(
2*d*x + 2*c) - (44100*I + 44100)*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^2*cos(7*d*x + 7*c) - (44100*I + 44100)*a^2*cos(5*d*x + 5*c) + (26460*I + 26460)*a^2*
cos(3*d*x + 3*c) - (1260*I + 1260)*a^2*cos(d*x + c) + (44100*I - 44100)*a^2*sin(7*d*x + 7*c) - (44100*I - 4410
0)*a^2*sin(5*d*x + 5*c) + (26460*I - 26460)*a^2*sin(3*d*x + 3*c) - (1260*I - 1260)*a^2*sin(d*x + c))*sin(7/2*a
rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + ((-(27300*I + 27300)*a^2*cos(d*x + c) - (27300*I - 27300)*a^
2*sin(d*x + c))*cos(2*d*x + 2*c)^2 - (27300*I + 27300)*a^2*cos(d*x + c) + (-(27300*I + 27300)*a^2*cos(d*x + c)
- (27300*I - 27300)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 - (27300*I - 27300)*a^2*sin(d*x + c) + ((44100*I + 4
4100)*a^2*cos(2*d*x + 2*c)^2 + (44100*I + 44100)*a^2*sin(2*d*x + 2*c)^2 - (88200*I + 88200)*a^2*cos(2*d*x + 2*
c) + (44100*I + 44100)*a^2)*cos(3*d*x + 3*c) + ((54600*I + 54600)*a^2*cos(d*x + c) + (54600*I - 54600)*a^2*sin
(d*x + c))*cos(2*d*x + 2*c) + ((44100*I - 44100)*a^2*cos(2*d*x + 2*c)^2 + (44100*I - 44100)*a^2*sin(2*d*x + 2*
c)^2 - (88200*I - 88200)*a^2*cos(2*d*x + 2*c) + (44100*I - 44100)*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) + (((44100*I + 44100)*a^2*cos(2*d*x + 2*c)^4 + (44100*I + 44100)*
a^2*sin(2*d*x + 2*c)^4 - (176400*I + 176400)*a^2*cos(2*d*x + 2*c)^3 + (264600*I + 264600)*a^2*cos(2*d*x + 2*c)
^2 - (176400*I + 176400)*a^2*cos(2*d*x + 2*c) + ((88200*I + 88200)*a^2*cos(2*d*x + 2*c)^2 - (176400*I + 176400
)*a^2*cos(2*d*x + 2*c) + (88200*I + 88200)*a^2)*sin(2*d*x + 2*c)^2 + (44100*I + 44100)*a^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^2*cos(2*d*x + 2*
c)^4 - (22050*I - 22050)*a^2*sin(2*d*x + 2*c)^4 + (88200*I - 88200)*a^2*cos(2*d*x + 2*c)^3 - (132300*I - 13230
0)*a^2*cos(2*d*x + 2*c)^2 + (88200*I - 88200)*a^2*cos(2*d*x + 2*c) + (-(44100*I - 44100)*a^2*cos(2*d*x + 2*c)^
2 + (88200*I - 88200)*a^2*cos(2*d*x + 2*c) - (44100*I - 44100)*a^2)*sin(2*d*x + 2*c)^2 - (22050*I - 22050)*a^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) + ((((6
3840*I - 63840)*a^2*cos(d*x + c) - (63840*I + 63840)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (63840*I - 63840)*
a^2*cos(d*x + c) + ((63840*I - 63840)*a^2*cos(d*x + c) - (63840*I + 63840)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^
2 - (63840*I + 63840)*a^2*sin(d*x + c) + ((44100*I - 44100)*a^2*cos(2*d*x + 2*c)^2 + (44100*I - 44100)*a^2*sin
(2*d*x + 2*c)^2 - (88200*I - 88200)*a^2*cos(2*d*x + 2*c) + (44100*I - 44100)*a^2)*cos(5*d*x + 5*c) + (-(102900
*I - 102900)*a^2*cos(2*d*x + 2*c)^2 - (102900*I - 102900)*a^2*sin(2*d*x + 2*c)^2 + (205800*I - 205800)*a^2*cos
(2*d*x + 2*c) - (102900*I - 102900)*a^2)*cos(3*d*x + 3*c) + (-(127680*I - 127680)*a^2*cos(d*x + c) + (127680*I
+ 127680)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + (-(44100*I + 44100)*a^2*cos(2*d*x + 2*c)^2 - (44100*I + 44100)
*a^2*sin(2*d*x + 2*c)^2 + (88200*I + 88200)*a^2*cos(2*d*x + 2*c) - (44100*I + 44100)*a^2)*sin(5*d*x + 5*c) + (
(102900*I + 102900)*a^2*cos(2*d*x + 2*c)^2 + (102900*I + 102900)*a^2*sin(2*d*x + 2*c)^2 - (205800*I + 205800)*
a^2*cos(2*d*x + 2*c) + (102900*I + 102900)*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^2*cos(d*x + c) + (48300*I + 48300)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^4
+ (-(48300*I - 48300)*a^2*cos(d*x + c) + (48300*I + 48300)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^4 + ((193200*I -
193200)*a^2*cos(d*x + c) - (193200*I + 193200)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^3 + (-(289800*I - 289800)*a
^2*cos(d*x + c) + (289800*I + 289800)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 - (48300*I - 48300)*a^2*cos(d*x + c
) + ((-(96600*I - 96600)*a^2*cos(d*x + c) + (96600*I + 96600)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 - (96600*I
- 96600)*a^2*cos(d*x + c) + (96600*I + 96600)*a^2*sin(d*x + c) + ((193200*I - 193200)*a^2*cos(d*x + c) - (1932
00*I + 193200)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + (48300*I + 48300)*a^2*sin(d*x + c) + (
(193200*I - 193200)*a^2*cos(d*x + c) - (193200*I + 193200)*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^2*cos(d*x + c) + (63840*I - 63840)*a^2*sin(d
*x + c))*cos(2*d*x + 2*c)^2 + (63840*I + 63840)*a^2*cos(d*x + c) + ((63840*I + 63840)*a^2*cos(d*x + c) + (6384
0*I - 63840)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + (63840*I - 63840)*a^2*sin(d*x + c) + ((44100*I + 44100)*a^
2*cos(2*d*x + 2*c)^2 + (44100*I + 44100)*a^2*sin(2*d*x + 2*c)^2 - (88200*I + 88200)*a^2*cos(2*d*x + 2*c) + (44
100*I + 44100)*a^2)*cos(5*d*x + 5*c) + (-(102900*I + 102900)*a^2*cos(2*d*x + 2*c)^2 - (102900*I + 102900)*a^2*
sin(2*d*x + 2*c)^2 + (205800*I + 205800)*a^2*cos(2*d*x + 2*c) - (102900*I + 102900)*a^2)*cos(3*d*x + 3*c) + (-
(127680*I + 127680)*a^2*cos(d*x + c) - (127680*I - 127680)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + ((44100*I - 44
100)*a^2*cos(2*d*x + 2*c)^2 + (44100*I - 44100)*a^2*sin(2*d*x + 2*c)^2 - (88200*I - 88200)*a^2*cos(2*d*x + 2*c
) + (44100*I - 44100)*a^2)*sin(5*d*x + 5*c) + (-(102900*I - 102900)*a^2*cos(2*d*x + 2*c)^2 - (102900*I - 10290
0)*a^2*sin(2*d*x + 2*c)^2 + (205800*I - 205800)*a^2*cos(2*d*x + 2*c) - (102900*I - 102900)*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^2*cos(d*x + c) - (48300*
I - 48300)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^4 + (-(48300*I + 48300)*a^2*cos(d*x + c) - (48300*I - 48300)*a^2
*sin(d*x + c))*sin(2*d*x + 2*c)^4 + ((193200*I + 193200)*a^2*cos(d*x + c) + (193200*I - 193200)*a^2*sin(d*x +
c))*cos(2*d*x + 2*c)^3 + (-(289800*I + 289800)*a^2*cos(d*x + c) - (289800*I - 289800)*a^2*sin(d*x + c))*cos(2*
d*x + 2*c)^2 - (48300*I + 48300)*a^2*cos(d*x + c) + ((-(96600*I + 96600)*a^2*cos(d*x + c) - (96600*I - 96600)*
a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 - (96600*I + 96600)*a^2*cos(d*x + c) - (96600*I - 96600)*a^2*sin(d*x + c)
+ ((193200*I + 193200)*a^2*cos(d*x + c) + (193200*I - 193200)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x +
2*c)^2 - (48300*I - 48300)*a^2*sin(d*x + c) + ((193200*I + 193200)*a^2*cos(d*x + c) + (193200*I - 193200)*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*co
s(2*d*x + 2*c) + 1)^(1/4)*d)
________________________________________________________________________________________
Fricas [B] time = 1.42297, size = 1339, normalized size = 6.03 \begin{align*} \frac{\sqrt{2}{\left (320 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 616 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 560 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 168 i \, a^{2}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} e^{\left (i \, d x + i \, c\right )} + 21 \, \sqrt{-\frac{32 i \, a^{5}}{d^{2}}}{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac{{\left (4 \, \sqrt{2}{\left (a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} e^{\left (i \, d x + i \, c\right )} + i \, \sqrt{-\frac{32 i \, a^{5}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{2}}\right ) - 21 \, \sqrt{-\frac{32 i \, a^{5}}{d^{2}}}{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac{{\left (4 \, \sqrt{2}{\left (a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} e^{\left (i \, d x + i \, c\right )} - i \, \sqrt{-\frac{32 i \, a^{5}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{2}}\right )}{42 \,{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \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),x, algorithm="fricas")
[Out]
1/42*(sqrt(2)*(320*I*a^2*e^(6*I*d*x + 6*I*c) - 616*I*a^2*e^(4*I*d*x + 4*I*c) + 560*I*a^2*e^(2*I*d*x + 2*I*c) -
168*I*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) + 21*sqrt(-32*I*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(1/4*(4*sqrt(2)*(a^2*e^(2*I*d*x + 2*I*c) - 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^5/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2
*I*d*x - 2*I*c)/a^2) - 21*sqrt(-32*I*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(1/4*(4*sqrt(2)*(a^2*e^(2*I*d*x + 2*I*c) - 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^5/d^2)*d*e^(2*I*d*x + 2*I*
c))*e^(-2*I*d*x - 2*I*c)/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),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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),x, algorithm="giac")
[Out]
integrate((I*a*tan(d*x + c) + a)^(5/2)*cot(d*x + c)^(9/2), x)