3.393 \(\int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\)
Optimal. Leaf size=307 \[ \frac {3003 i \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{16384 \sqrt {2} a^{7/2} d}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}-\frac {3003 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16384 a^4 d}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}} \]
[Out]
3003/32768*I*arctanh(1/2*sec(d*x+c)*a^(1/2)*2^(1/2)/(a+I*a*tan(d*x+c))^(1/2))/a^(7/2)/d*2^(1/2)+1001/8192*I*co
s(d*x+c)/a^3/d/(a+I*a*tan(d*x+c))^(1/2)+429/5120*I*cos(d*x+c)^3/a^3/d/(a+I*a*tan(d*x+c))^(1/2)-3003/16384*I*co
s(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a^4/d-1001/10240*I*cos(d*x+c)^3*(a+I*a*tan(d*x+c))^(1/2)/a^4/d+1/10*I*cos(d*
x+c)^3/d/(a+I*a*tan(d*x+c))^(7/2)+13/160*I*cos(d*x+c)^3/a/d/(a+I*a*tan(d*x+c))^(5/2)+143/1920*I*cos(d*x+c)^3/a
^2/d/(a+I*a*tan(d*x+c))^(3/2)
________________________________________________________________________________________
Rubi [A] time = 0.52, antiderivative size = 307, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used =
{3502, 3497, 3490, 3489, 206} \[ -\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac {3003 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16384 a^4 d}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {3003 i \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{16384 \sqrt {2} a^{7/2} d}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(7/2),x]
[Out]
(((3003*I)/16384)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(7/2)*d) +
((I/10)*Cos[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (((13*I)/160)*Cos[c + d*x]^3)/(a*d*(a + I*a*Tan[c +
d*x])^(5/2)) + (((143*I)/1920)*Cos[c + d*x]^3)/(a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) + (((1001*I)/8192)*Cos[c
+ d*x])/(a^3*d*Sqrt[a + I*a*Tan[c + d*x]]) + (((429*I)/5120)*Cos[c + d*x]^3)/(a^3*d*Sqrt[a + I*a*Tan[c + d*x]]
) - (((3003*I)/16384)*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a^4*d) - (((1001*I)/10240)*Cos[c + d*x]^3*Sqrt
[a + I*a*Tan[c + d*x]])/(a^4*d)
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 3489
Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(-2*a)/(b*f), Subst[
Int[1/(2 - a*x^2), x], x, Sec[e + f*x]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 + b^
2, 0]
Rule 3490
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] + Dist[a/(2*d^2), Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan
[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && EqQ[m/2 + n, 0] && GtQ[n, 0]
Rule 3497
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d*
Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] + Dist[(a*(m + n))/(m*d^2), Int[(d*Sec[e + f*x])^(m + 2)*(
a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && LtQ[m, -
1] && IntegersQ[2*m, 2*n]
Rule 3502
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(b*f*(m + 2*n)), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), Int[(d*Sec[
e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx &=\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx}{20 a}\\ &=\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx}{320 a^2}\\ &=\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {429 \int \frac {\cos ^3(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{1280 a^3}\\ &=\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {3003 \int \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx}{10240 a^4}\\ &=\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}+\frac {1001 \int \frac {\cos (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{4096 a^3}\\ &=\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}+\frac {3003 \int \cos (c+d x) \sqrt {a+i a \tan (c+d x)} \, dx}{16384 a^4}\\ &=\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {3003 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16384 a^4 d}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}+\frac {3003 \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{32768 a^3}\\ &=\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {3003 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16384 a^4 d}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}+\frac {(3003 i) \operatorname {Subst}\left (\int \frac {1}{2-a x^2} \, dx,x,\frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}}\right )}{16384 a^3 d}\\ &=\frac {3003 i \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{16384 \sqrt {2} a^{7/2} d}+\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {3003 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16384 a^4 d}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.49, size = 175, normalized size = 0.57 \[ -\frac {\sec ^3(c+d x) \left (20048 e^{-2 i (c+d x)}+71190 e^{2 i (c+d x)}+5856 e^{-4 i (c+d x)}-48640 e^{4 i (c+d x)}+768 e^{-6 i (c+d x)}-2560 e^{6 i (c+d x)}+\frac {90090 e^{4 i (c+d x)} \tanh ^{-1}\left (\sqrt {1+e^{2 i (c+d x)}}\right )}{\sqrt {1+e^{2 i (c+d x)}}}+42140\right )}{491520 a^3 d (\tan (c+d x)-i)^3 \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(7/2),x]
[Out]
-1/491520*((42140 + 20048/E^((2*I)*(c + d*x)) + 71190*E^((2*I)*(c + d*x)) + 5856/E^((4*I)*(c + d*x)) - 48640*E
^((4*I)*(c + d*x)) + 768/E^((6*I)*(c + d*x)) - 2560*E^((6*I)*(c + d*x)) + (90090*E^((4*I)*(c + d*x))*ArcTanh[S
qrt[1 + E^((2*I)*(c + d*x))]])/Sqrt[1 + E^((2*I)*(c + d*x))])*Sec[c + d*x]^3)/(a^3*d*(-I + Tan[c + d*x])^3*Sqr
t[a + I*a*Tan[c + d*x]])
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fricas [A] time = 0.61, size = 322, normalized size = 1.05 \[ \frac {{\left (45045 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (24600576 i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + 24600576 i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} + 24600576 i\right )} e^{\left (-i \, d x - i \, c\right )}}{67108864 \, a^{3} d}\right ) - 45045 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-24600576 i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - 24600576 i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} + 24600576 i\right )} e^{\left (-i \, d x - i \, c\right )}}{67108864 \, a^{3} d}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-1280 i \, e^{\left (14 i \, d x + 14 i \, c\right )} - 25600 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 11275 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 56665 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 31094 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 12952 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3312 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 384 i\right )}\right )} e^{\left (-10 i \, d x - 10 i \, c\right )}}{491520 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^(7/2),x, algorithm="fricas")
[Out]
1/491520*(45045*I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(10*I*d*x + 10*I*c)*log(1/67108864*(sqrt(2)*sqrt(1/2)*(2
4600576*I*a^3*d*e^(2*I*d*x + 2*I*c) + 24600576*I*a^3*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(1/(a^7*d^2)) +
24600576*I)*e^(-I*d*x - I*c)/(a^3*d)) - 45045*I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(10*I*d*x + 10*I*c)*log(1/
67108864*(sqrt(2)*sqrt(1/2)*(-24600576*I*a^3*d*e^(2*I*d*x + 2*I*c) - 24600576*I*a^3*d)*sqrt(a/(e^(2*I*d*x + 2*
I*c) + 1))*sqrt(1/(a^7*d^2)) + 24600576*I)*e^(-I*d*x - I*c)/(a^3*d)) + sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1
))*(-1280*I*e^(14*I*d*x + 14*I*c) - 25600*I*e^(12*I*d*x + 12*I*c) + 11275*I*e^(10*I*d*x + 10*I*c) + 56665*I*e^
(8*I*d*x + 8*I*c) + 31094*I*e^(6*I*d*x + 6*I*c) + 12952*I*e^(4*I*d*x + 4*I*c) + 3312*I*e^(2*I*d*x + 2*I*c) + 3
84*I))*e^(-10*I*d*x - 10*I*c)/(a^4*d)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{3}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^(7/2),x, algorithm="giac")
[Out]
integrate(cos(d*x + c)^3/(I*a*tan(d*x + c) + a)^(7/2), x)
________________________________________________________________________________________
maple [A] time = 1.45, size = 427, normalized size = 1.39 \[ \frac {\sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (786432 i \left (\cos ^{11}\left (d x +c \right )\right )+786432 \sin \left (d x +c \right ) \left (\cos ^{10}\left (d x +c \right )\right )-466944 i \left (\cos ^{9}\left (d x +c \right )\right )-73728 \sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )+5120 i \left (\cos ^{7}\left (d x +c \right )\right )+66560 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )+9152 i \left (\cos ^{5}\left (d x +c \right )\right )+45045 i \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+82368 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+45045 i \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+24024 i \left (\cos ^{3}\left (d x +c \right )\right )+45045 \sqrt {2}\, \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+120120 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-180180 i \cos \left (d x +c \right )\right )}{983040 d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^(7/2),x)
[Out]
1/983040/d*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(786432*I*cos(d*x+c)^11+786432*sin(d*x+c)*cos(d*x+c)
^10-466944*I*cos(d*x+c)^9-73728*sin(d*x+c)*cos(d*x+c)^8+5120*I*cos(d*x+c)^7+66560*sin(d*x+c)*cos(d*x+c)^6+9152
*I*cos(d*x+c)^5+45045*I*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2)*arctan(1/2*(I*cos(d*x+c)-I+sin(d*x+c))/si
n(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*cos(d*x+c)+82368*sin(d*x+c)*cos(d*x+c)^4+45045*I*(-2*co
s(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2)*arctan(1/2*(I*cos(d*x+c)-I+sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos
(d*x+c)))^(1/2)*2^(1/2))+24024*I*cos(d*x+c)^3+45045*2^(1/2)*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ar
ctan(1/2*(I*cos(d*x+c)-I+sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))+120120*cos(d*x+c
)^2*sin(d*x+c)-180180*I*cos(d*x+c))/a^4
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maxima [B] time = 1.64, size = 5803, normalized size = 18.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^(7/2),x, algorithm="maxima")
[Out]
1/1966080*((cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), c
os(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)^(3/4)*(((3160*I*sqrt(2
)*cos(10*d*x + 10*c) + 3160*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)
))^2 + (3160*I*sqrt(2)*cos(10*d*x + 10*c) + 3160*sqrt(2)*sin(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c
), cos(10*d*x + 10*c)))^2 + (33480*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 3348
0*I*sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 66960*I*sqrt(2)*cos(1/5*arctan2(sin(1
0*d*x + 10*c), cos(10*d*x + 10*c))) + 33480*I*sqrt(2))*cos(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))
) + (6320*I*sqrt(2)*cos(10*d*x + 10*c) + 6320*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c),
cos(10*d*x + 10*c))) + 33480*(sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sqrt(2)*sin
(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10
*d*x + 10*c))) + sqrt(2))*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 3160*I*sqrt(2)*cos(10*d*x
+ 10*c) + 3160*sqrt(2)*sin(10*d*x + 10*c))*cos(7/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 1
0*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) + (1960*I*sqrt(2)*cos(10*d*x + 10*c) +
46200*I*sqrt(2)*cos(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 130560*I*sqrt(2)*cos(3/5*arctan2(si
n(10*d*x + 10*c), cos(10*d*x + 10*c))) + 24960*I*sqrt(2)*cos(2/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c
))) + 1960*sqrt(2)*sin(10*d*x + 10*c) + 46200*sqrt(2)*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))
- 130560*sqrt(2)*sin(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 24960*sqrt(2)*sin(2/5*arctan2(sin
(10*d*x + 10*c), cos(10*d*x + 10*c))) - 5120*I*sqrt(2))*cos(3/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), co
s(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) - (3160*(sqrt(2)*cos(10*d*x
+ 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 3160*(sq
rt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c
)))^2 + 33480*(sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sqrt(2)*sin(1/5*arctan2(si
n(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))
+ sqrt(2))*cos(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 6320*(sqrt(2)*cos(10*d*x + 10*c) - I*sqr
t(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - (33480*I*sqrt(2)*cos(1/5*a
rctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 33480*I*sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10
*d*x + 10*c)))^2 + 66960*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 33480*I*sqrt(2))
*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 3160*sqrt(2)*cos(10*d*x + 10*c) - 3160*I*sqrt(2)*s
in(10*d*x + 10*c))*sin(7/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(s
in(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) - (1960*sqrt(2)*cos(10*d*x + 10*c) + 46200*sqrt(2)*cos(4/5*arcta
n2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 130560*sqrt(2)*cos(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x +
10*c))) + 24960*sqrt(2)*cos(2/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 1960*I*sqrt(2)*sin(10*d*x +
10*c) - 46200*I*sqrt(2)*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 130560*I*sqrt(2)*sin(3/5*a
rctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 24960*I*sqrt(2)*sin(2/5*arctan2(sin(10*d*x + 10*c), cos(10*d
*x + 10*c))) - 5120*sqrt(2))*sin(3/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5
*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)))*sqrt(a) + (cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10
*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x +
10*c), cos(10*d*x + 10*c))) + 1)^(1/4)*(((420*I*sqrt(2)*cos(10*d*x + 10*c) + 420*sqrt(2)*sin(10*d*x + 10*c))*
cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^4 + (420*I*sqrt(2)*cos(10*d*x + 10*c) + 420*sqrt(2)*s
in(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^4 + (1680*I*sqrt(2)*cos(10*d*x + 1
0*c) + 1680*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^3 + (2520*I*s
qrt(2)*cos(10*d*x + 10*c) + 2520*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x +
10*c)))^2 + ((840*I*sqrt(2)*cos(10*d*x + 10*c) + 840*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x +
10*c), cos(10*d*x + 10*c)))^2 + (1680*I*sqrt(2)*cos(10*d*x + 10*c) + 1680*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*
arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 840*I*sqrt(2)*cos(10*d*x + 10*c) + 840*sqrt(2)*sin(10*d*x +
10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + (1680*I*sqrt(2)*cos(10*d*x + 10*c) + 1680
*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 420*I*sqrt(2)*cos(10*d
*x + 10*c) + 420*sqrt(2)*sin(10*d*x + 10*c))*cos(9/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x +
10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) + ((-3584*I*sqrt(2)*cos(10*d*x + 10*c)
- 3584*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + (-3584*I*sqrt
(2)*cos(10*d*x + 10*c) - 3584*sqrt(2)*sin(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*
c)))^2 + (-61320*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 - 61320*I*sqrt(2)*sin(1/
5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 - 122640*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), co
s(10*d*x + 10*c))) - 61320*I*sqrt(2))*cos(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + (83520*I*sqrt
(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 83520*I*sqrt(2)*sin(1/5*arctan2(sin(10*d*x +
10*c), cos(10*d*x + 10*c)))^2 + 167040*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 83
520*I*sqrt(2))*cos(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + (-7168*I*sqrt(2)*cos(10*d*x + 10*c)
- 7168*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 61320*(sqrt(2)*c
os(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10
*d*x + 10*c)))^2 + 2*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2))*sin(4/5*arcta
n2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 83520*(sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x +
10*c)))^2 + sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*sqrt(2)*cos(1/5*arctan2(sin
(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2))*sin(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 35
84*I*sqrt(2)*cos(10*d*x + 10*c) - 3584*sqrt(2)*sin(10*d*x + 10*c))*cos(5/2*arctan2(sin(1/5*arctan2(sin(10*d*x
+ 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) + (-420*I*sqrt(2
)*cos(10*d*x + 10*c) - 12600*I*sqrt(2)*cos(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 54720*I*sqrt
(2)*cos(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 21120*I*sqrt(2)*cos(2/5*arctan2(sin(10*d*x + 10
*c), cos(10*d*x + 10*c))) - 420*sqrt(2)*sin(10*d*x + 10*c) - 12600*sqrt(2)*sin(4/5*arctan2(sin(10*d*x + 10*c),
cos(10*d*x + 10*c))) + 54720*sqrt(2)*sin(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 21120*sqrt(2)
*sin(2/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 92160*I*sqrt(2))*cos(1/2*arctan2(sin(1/5*arctan2(s
in(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) - 420*
((sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x +
10*c)))^4 + (sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), co
s(10*d*x + 10*c)))^4 + 4*(sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*
x + 10*c), cos(10*d*x + 10*c)))^3 + 6*(sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arct
an2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*((sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c)
)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*(sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(1
0*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(
2)*sin(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 4*(sqrt(2)*cos(10*d*x + 10
*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2)*cos(10*
d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*sin(9/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 1
0*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) + (3584*(sqrt(2)*cos(10*d*x + 10*c) - I
*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 3584*(sqrt(2)*cos(10
*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 6132
0*(sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 1
0*c), cos(10*d*x + 10*c)))^2 + 2*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2))*c
os(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 83520*(sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), c
os(10*d*x + 10*c)))^2 + sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*sqrt(2)*cos(1/5
*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2))*cos(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x +
10*c))) + 7168*(sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c),
cos(10*d*x + 10*c))) + (-61320*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 - 61320*I
*sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 - 122640*I*sqrt(2)*cos(1/5*arctan2(sin(10*
d*x + 10*c), cos(10*d*x + 10*c))) - 61320*I*sqrt(2))*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))
+ (83520*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 83520*I*sqrt(2)*sin(1/5*arctan
2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 167040*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x
+ 10*c))) + 83520*I*sqrt(2))*sin(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 3584*sqrt(2)*cos(10*d
*x + 10*c) - 3584*I*sqrt(2)*sin(10*d*x + 10*c))*sin(5/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x
+ 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) + (420*sqrt(2)*cos(10*d*x + 10*c) +
12600*sqrt(2)*cos(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 54720*sqrt(2)*cos(3/5*arctan2(sin(10
*d*x + 10*c), cos(10*d*x + 10*c))) + 21120*sqrt(2)*cos(2/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) -
420*I*sqrt(2)*sin(10*d*x + 10*c) - 12600*I*sqrt(2)*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) +
54720*I*sqrt(2)*sin(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 21120*I*sqrt(2)*sin(2/5*arctan2(sin
(10*d*x + 10*c), cos(10*d*x + 10*c))) + 92160*sqrt(2))*sin(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos
(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)))*sqrt(a) - (90090*sqrt(2)*ar
ctan2((cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10
*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)^(1/4)*sin(1/2*arctan2(sin(1
/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))
+ 1)), (cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(1
0*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(
1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))
+ 1)) + 1) - 90090*sqrt(2)*arctan2((cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arct
an2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) +
1)^(1/4)*sin(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x
+ 10*c), cos(10*d*x + 10*c))) + 1)), (cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arc
tan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) +
1)^(1/4)*cos(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x
+ 10*c), cos(10*d*x + 10*c))) + 1)) - 1) - 45045*I*sqrt(2)*log(sqrt(cos(1/5*arctan2(sin(10*d*x + 10*c), cos(1
0*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x
+ 10*c), cos(10*d*x + 10*c))) + 1)*cos(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), c
os(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1))^2 + sqrt(cos(1/5*arctan2(sin(10*d*x + 10*c), cos
(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*
x + 10*c), cos(10*d*x + 10*c))) + 1)*sin(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))),
cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1))^2 + 2*(cos(1/5*arctan2(sin(10*d*x + 10*c), cos
(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*
x + 10*c), cos(10*d*x + 10*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10
*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) + 1) + 45045*I*sqrt(2)*log(sqrt(cos(1/5*
arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^
2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)*cos(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x
+ 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1))^2 + sqrt(cos(1/
5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))
)^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)*sin(1/2*arctan2(sin(1/5*arctan2(sin(10*d
*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1))^2 - 2*(cos(1/
5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))
)^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/5*arctan2(si
n(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) + 1))*s
qrt(a))/(a^4*d)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^3}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(c + d*x)^3/(a + a*tan(c + d*x)*1i)^(7/2),x)
[Out]
int(cos(c + d*x)^3/(a + a*tan(c + d*x)*1i)^(7/2), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**3/(a+I*a*tan(d*x+c))**(7/2),x)
[Out]
Timed out
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