3.8.62 \(\int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx\) [762]
Optimal. Leaf size=222 \[ \frac {(4-4 i) a^{5/2} \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {104 i a^2 \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{21 d}+\frac {32 a^2 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {6 i a^2 \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d} \]
[Out]
(4-4*I)*a^(5/2)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(
1/2)/d+32/21*a^2*cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^(1/2)/d-6/7*I*a^2*cot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^(1/
2)/d-2/7*a^2*cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(1/2)/d+104/21*I*a^2*cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^(1/2
)/d
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Rubi [A]
time = 0.43, antiderivative size = 222, normalized size of antiderivative = 1.00, 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 = {4326, 3634,
3679, 12, 3625, 211} \begin {gather*} \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}-\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} \end {gather*}
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 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 3625
Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-2*a*(b/f), Subst[Int[1/(a*c - b*d - 2*a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Rule 3634
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-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] /;
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] && GtQ[m, 1] && Lt
Q[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Rule 3679
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(A*d - B*c)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(f*
(n + 1)*(c^2 + d^2))), x] - Dist[1/(a*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n
+ 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c*m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]
Rule 4326
Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ
[u, x]
Rubi steps
\begin {align*} \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 ) \text {Subst}\left (\int \frac {1}{-i a-2 a^2 x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}\\ &=\frac {(4-4 i) a^{5/2} \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*}
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Mathematica [A]
time = 2.68, size = 167, normalized size = 0.75 \begin {gather*} \frac {4 i a^2 e^{-i (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 {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{21 d \left (-1+e^{2 i (c+d x)}\right )^3} \end {gather*}
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] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1556 vs. \(2 (179 ) = 358\).
time = 46.27, size = 1557, normalized size = 7.01
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result |
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\(\text {Expression too large to display}\) |
\(1557\) |
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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,method=_RETURNVERBOSE)
[Out]
-1/21/d*(84*I*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-129*2^(1
/2)*cos(d*x+c)^2+84*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+84
*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)-1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-168*arctan(((-1+cos(d
*x+c))/sin(d*x+c))^(1/2)*2^(1/2)+1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^2-168*arctan(((-1+cos(d*x+c)
)/sin(d*x+c))^(1/2)*2^(1/2)-1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^2-19*cos(d*x+c)^3*2^(1/2)+84*I*ar
ctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)-1)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+42*I*ln((((-1+cos(d*x+c)
)/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+sin(d*x+c)+cos(d*x+c)-1)/(-((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*s
in(d*x+c)+cos(d*x+c)+sin(d*x+c)-1))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+52*I*sin(d*x+c)*2^(1/2)+84*arctan(((-1+
cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+84*arctan(((-1+cos(d*
x+c))/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+42*ln((-((-1+cos(d*x+c))/si
n(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*
x+c)+sin(d*x+c)+cos(d*x+c)-1))*cos(d*x+c)^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)-84*ln((-((-1+cos(d*x+c))/sin(d*
x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)
+sin(d*x+c)+cos(d*x+c)-1))*cos(d*x+c)^2*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+16*cos(d*x+c)*2^(1/2)+42*ln((-((-1+
cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+cos(d*x+c)+sin(d*x+c)-1)/(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*
2^(1/2)*sin(d*x+c)+sin(d*x+c)+cos(d*x+c)-1))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+80*cos(d*x+c)^4*2^(1/2)+84*I*a
rctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+84*I*arcta
n(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^4*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+42*I*ln((((-1+
cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+sin(d*x+c)+cos(d*x+c)-1)/(-((-1+cos(d*x+c))/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*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)+80*I*cos(d*x+c)^
3*sin(d*x+c)*2^(1/2)-168*I*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)+1)*cos(d*x+c)^2*((-1+cos(d*x+c))/
sin(d*x+c))^(1/2)-168*I*arctan(((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)-1)*cos(d*x+c)^2*((-1+cos(d*x+c))/sin
(d*x+c))^(1/2)-84*I*ln((((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+sin(d*x+c)+cos(d*x+c)-1)/(-((-1+
cos(d*x+c))/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*((-1+cos(d*x+c))/sin(d
*x+c))^(1/2)-61*I*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)-68*I*cos(d*x+c)*sin(d*x+c)*2^(1/2)+52*2^(1/2))*sin(d*x+c)*(c
os(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)/(I*sin(d*x+c)+cos(d*x+c)-1)/cos(d*x
+c)^4*2^(1/2)*a^2
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 3165 vs. \(2 (168) = 336\).
time = 0.79, size = 3165, normalized size = 14.26 \begin {gather*} \text {Too large to display} \end {gather*}
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]
-2/105*(2*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*(3*(-(35*I - 35)*a^2*cos(7*d*
x + 7*c) + (35*I - 35)*a^2*cos(5*d*x + 5*c) - (21*I - 21)*a^2*cos(3*d*x + 3*c) + (I - 1)*a^2*cos(d*x + c) + (3
5*I + 35)*a^2*sin(7*d*x + 7*c) - (35*I + 35)*a^2*sin(5*d*x + 5*c) + (21*I + 21)*a^2*sin(3*d*x + 3*c) - (I + 1)
*a^2*sin(d*x + c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 5*(13*((I - 1)*a^2*cos(d*x + c)
- (I + 1)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (13*I - 13)*a^2*cos(d*x + c) + 13*((I - 1)*a^2*cos(d*x + c) -
(I + 1)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 - (13*I + 13)*a^2*sin(d*x + c) + 21*(-(I - 1)*a^2*cos(2*d*x + 2*
c)^2 - (I - 1)*a^2*sin(2*d*x + 2*c)^2 + (2*I - 2)*a^2*cos(2*d*x + 2*c) - (I - 1)*a^2)*cos(3*d*x + 3*c) + 26*(-
(I - 1)*a^2*cos(d*x + c) + (I + 1)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + 21*((I + 1)*a^2*cos(2*d*x + 2*c)^2 + (
I + 1)*a^2*sin(2*d*x + 2*c)^2 - (2*I + 2)*a^2*cos(2*d*x + 2*c) + (I + 1)*a^2)*sin(3*d*x + 3*c))*cos(3/2*arctan
2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 3*(-(35*I + 35)*a^2*cos(7*d*x + 7*c) + (35*I + 35)*a^2*cos(5*d*x
+ 5*c) - (21*I + 21)*a^2*cos(3*d*x + 3*c) + (I + 1)*a^2*cos(d*x + c) - (35*I - 35)*a^2*sin(7*d*x + 7*c) + (35*
I - 35)*a^2*sin(5*d*x + 5*c) - (21*I - 21)*a^2*sin(3*d*x + 3*c) + (I - 1)*a^2*sin(d*x + c))*sin(7/2*arctan2(si
n(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 5*(13*((I + 1)*a^2*cos(d*x + c) + (I - 1)*a^2*sin(d*x + c))*cos(2*d*x
+ 2*c)^2 + (13*I + 13)*a^2*cos(d*x + c) + 13*((I + 1)*a^2*cos(d*x + c) + (I - 1)*a^2*sin(d*x + c))*sin(2*d*x
+ 2*c)^2 + (13*I - 13)*a^2*sin(d*x + c) + 21*(-(I + 1)*a^2*cos(2*d*x + 2*c)^2 - (I + 1)*a^2*sin(2*d*x + 2*c)^2
+ (2*I + 2)*a^2*cos(2*d*x + 2*c) - (I + 1)*a^2)*cos(3*d*x + 3*c) + 26*(-(I + 1)*a^2*cos(d*x + c) - (I - 1)*a^
2*sin(d*x + c))*cos(2*d*x + 2*c) + 21*(-(I - 1)*a^2*cos(2*d*x + 2*c)^2 - (I - 1)*a^2*sin(2*d*x + 2*c)^2 + (2*I
- 2)*a^2*cos(2*d*x + 2*c) - (I - 1)*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) + 105*(2*(-(I + 1)*a^2*cos(2*d*x + 2*c)^4 - (I + 1)*a^2*sin(2*d*x + 2*c)^4 + (4*I + 4)*a^2*cos
(2*d*x + 2*c)^3 - (6*I + 6)*a^2*cos(2*d*x + 2*c)^2 + (4*I + 4)*a^2*cos(2*d*x + 2*c) + 2*(-(I + 1)*a^2*cos(2*d*
x + 2*c)^2 + (2*I + 2)*a^2*cos(2*d*x + 2*c) - (I + 1)*a^2)*sin(2*d*x + 2*c)^2 - (I + 1)*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)) + ((I - 1)*a^2*cos(2*d*x + 2*c)^4 + (I
- 1)*a^2*sin(2*d*x + 2*c)^4 - (4*I - 4)*a^2*cos(2*d*x + 2*c)^3 + (6*I - 6)*a^2*cos(2*d*x + 2*c)^2 - (4*I - 4)*
a^2*cos(2*d*x + 2*c) + 2*((I - 1)*a^2*cos(2*d*x + 2*c)^2 - (2*I - 2)*a^2*cos(2*d*x + 2*c) + (I - 1)*a^2)*sin(2
*d*x + 2*c)^2 + (I - 1)*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*arctan
2(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*arc
tan2(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) + 2*((152*(-(I - 1)*a^2*cos(d*x + c) + (I + 1)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 - (152
*I - 152)*a^2*cos(d*x + c) + 152*(-(I - 1)*a^2*cos(d*x + c) + (I + 1)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + (
152*I + 152)*a^2*sin(d*x + c) + 105*(-(I - 1)*a^2*cos(2*d*x + 2*c)^2 - (I - 1)*a^2*sin(2*d*x + 2*c)^2 + (2*I -
2)*a^2*cos(2*d*x + 2*c) - (I - 1)*a^2)*cos(5*d*x + 5*c) + 245*((I - 1)*a^2*cos(2*d*x + 2*c)^2 + (I - 1)*a^2*s
in(2*d*x + 2*c)^2 - (2*I - 2)*a^2*cos(2*d*x + 2*c) + (I - 1)*a^2)*cos(3*d*x + 3*c) + 304*((I - 1)*a^2*cos(d*x
+ c) - (I + 1)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + 105*((I + 1)*a^2*cos(2*d*x + 2*c)^2 + (I + 1)*a^2*sin(2*d*
x + 2*c)^2 - (2*I + 2)*a^2*cos(2*d*x + 2*c) + (I + 1)*a^2)*sin(5*d*x + 5*c) + 245*(-(I + 1)*a^2*cos(2*d*x + 2*
c)^2 - (I + 1)*a^2*sin(2*d*x + 2*c)^2 + (2*I + 2)*a^2*cos(2*d*x + 2*c) - (I + 1)*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)) + 115*(((I - 1)*a^2*cos(d*x + c) - (I + 1)*a^2*sin(d*x + c)
)*cos(2*d*x + 2*c)^4 + ((I - 1)*a^2*cos(d*x + c) - (I + 1)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^4 + 4*(-(I - 1)*
a^2*cos(d*x + c) + (I + 1)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^3 + 6*((I - 1)*a^2*cos(d*x + c) - (I + 1)*a^2*si
n(d*x + c))*cos(2*d*x + 2*c)^2 + (I - 1)*a^2*cos(d*x + c) + 2*(((I - 1)*a^2*cos(d*x + c) - (I + 1)*a^2*sin(d*x
+ c))*cos(2*d*x + 2*c)^2 + (I - 1)*a^2*cos(d*x + c) - (I + 1)*a^2*sin(d*x + c) + 2*(-(I - 1)*a^2*cos(d*x + c)
+ (I + 1)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 - (I + 1)*a^2*sin(d*x + c) + 4*(-(I - 1)*a^2
*cos(d*x + c) + (I + 1)*a^2*sin(d*x + c))*cos(2...
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 466 vs. \(2 (168) = 336\).
time = 1.08, size = 466, normalized size = 2.10 \begin {gather*} -\frac {16 \, \sqrt {2} {\left (-40 i \, a^{2} e^{\left (7 i \, d x + 7 i \, c\right )} + 77 i \, a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} - 70 i \, a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + 21 i \, a^{2} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} - 21 \, \sqrt {-\frac {128 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 (16 i \, a^{3} e^{\left (i \, d x + i \, c\right )} + \sqrt {2} \sqrt {-\frac {128 i \, a^{5}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{4 \, a^{2}}\right ) + 21 \, \sqrt {-\frac {128 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 (16 i \, a^{3} e^{\left (i \, d x + i \, c\right )} + \sqrt {2} \sqrt {-\frac {128 i \, a^{5}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{4 \, a^{2}}\right )}{84 \, {\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 {gather*}
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]
-1/84*(16*sqrt(2)*(-40*I*a^2*e^(7*I*d*x + 7*I*c) + 77*I*a^2*e^(5*I*d*x + 5*I*c) - 70*I*a^2*e^(3*I*d*x + 3*I*c)
+ 21*I*a^2*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x +
2*I*c) - 1)) - 21*sqrt(-128*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*(16*I*a^3*e^(I*d*x + I*c) + sqrt(2)*sqrt(-128*I*a^5/d^2)*(I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt
(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-I*d*x - I*c)/a^
2) + 21*sqrt(-128*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)*l
og(1/4*(16*I*a^3*e^(I*d*x + I*c) + sqrt(2)*sqrt(-128*I*a^5/d^2)*(-I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt(a/(e^(2*
I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-I*d*x - I*c)/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)
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(c + d*x)^(9/2)*(a + a*tan(c + d*x)*1i)^(5/2),x)
[Out]
int(cot(c + d*x)^(9/2)*(a + a*tan(c + d*x)*1i)^(5/2), x)
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