\(\int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx\) [662]

Optimal result

Integrand size = 26, antiderivative size = 130 \[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}-\frac {6 a \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}} \]

[Out]

2/7*I*a/d/(e*cos(d*x+c))^(7/2)-6/5*a*cos(d*x+c)^(7/2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elliptic
E(sin(1/2*d*x+1/2*c),2^(1/2))/d/(e*cos(d*x+c))^(7/2)+2/5*a*cos(d*x+c)*sin(d*x+c)/d/(e*cos(d*x+c))^(7/2)+6/5*a*
cos(d*x+c)^3*sin(d*x+c)/d/(e*cos(d*x+c))^(7/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3596, 3567, 3853, 3856, 2719} \[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}-\frac {6 a \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \sin (c+d x) \cos ^3(c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {2 a \sin (c+d x) \cos (c+d x)}{5 d (e \cos (c+d x))^{7/2}} \]

[In]

Int[(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(7/2),x]

[Out]

(((2*I)/7)*a)/(d*(e*Cos[c + d*x])^(7/2)) - (6*a*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2])/(5*d*(e*Cos[c +
d*x])^(7/2)) + (2*a*Cos[c + d*x]*Sin[c + d*x])/(5*d*(e*Cos[c + d*x])^(7/2)) + (6*a*Cos[c + d*x]^3*Sin[c + d*x]
)/(5*d*(e*Cos[c + d*x])^(7/2))

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 3567

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*((d*Sec[
e + f*x])^m/(f*m)), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

Rule 3596

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(d*Co
s[e + f*x])^m*(d*Sec[e + f*x])^m, Int[(a + b*Tan[e + f*x])^n/(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e,
f, m, n}, x] &&  !IntegerQ[m]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}} \\ & = \frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {a \int (e \sec (c+d x))^{7/2} \, dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}} \\ & = \frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {\left (3 a e^2\right ) \int (e \sec (c+d x))^{3/2} \, dx}{5 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}} \\ & = \frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}-\frac {\left (3 a e^4\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}} \\ & = \frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}-\frac {\left (3 a \cos ^{\frac {7}{2}}(c+d x)\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 (e \cos (c+d x))^{7/2}} \\ & = \frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}-\frac {6 a \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 7.52 (sec) , antiderivative size = 666, normalized size of antiderivative = 5.12 \[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\frac {\cos ^5(c+d x) \left (\csc (c) \sec (c) \left (\frac {6 \cos (c)}{5}-\frac {6}{5} i \sin (c)\right )+\sec ^4(c+d x) \left (\frac {2}{7} i \cos (c)+\frac {2 \sin (c)}{7}\right )+\sec (c) \sec ^3(c+d x) \left (\frac {2 \cos (c)}{5}-\frac {2}{5} i \sin (c)\right ) \sin (d x)+\sec (c) \sec (c+d x) \left (\frac {6 \cos (c)}{5}-\frac {6}{5} i \sin (c)\right ) \sin (d x)+\sec ^2(c+d x) \left (\frac {2 \cos (c)}{5}-\frac {2}{5} i \sin (c)\right ) \tan (c)\right ) (a+i a \tan (c+d x))}{d (e \cos (c+d x))^{7/2} (\cos (d x)+i \sin (d x))}-\frac {3 i \cos ^{\frac {9}{2}}(c+d x) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right ) (a+i a \tan (c+d x))}{5 d (e \cos (c+d x))^{7/2} (\cos (d x)+i \sin (d x))}+\frac {3 \cos ^{\frac {9}{2}}(c+d x) \cot (c) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right ) (a+i a \tan (c+d x))}{5 d (e \cos (c+d x))^{7/2} (\cos (d x)+i \sin (d x))} \]

[In]

Integrate[(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(7/2),x]

[Out]

(Cos[c + d*x]^5*(Csc[c]*Sec[c]*((6*Cos[c])/5 - ((6*I)/5)*Sin[c]) + Sec[c + d*x]^4*(((2*I)/7)*Cos[c] + (2*Sin[c
])/7) + Sec[c]*Sec[c + d*x]^3*((2*Cos[c])/5 - ((2*I)/5)*Sin[c])*Sin[d*x] + Sec[c]*Sec[c + d*x]*((6*Cos[c])/5 -
((6*I)/5)*Sin[c])*Sin[d*x] + Sec[c + d*x]^2*((2*Cos[c])/5 - ((2*I)/5)*Sin[c])*Tan[c])*(a + I*a*Tan[c + d*x]))
/(d*(e*Cos[c + d*x])^(7/2)*(Cos[d*x] + I*Sin[d*x])) - (((3*I)/5)*Cos[c + d*x]^(9/2)*((HypergeometricPFQ[{-1/2,
-1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c
]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Ta
n[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt
[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]])*(a + I*a*Tan
[c + d*x]))/(d*(e*Cos[c + d*x])^(7/2)*(Cos[d*x] + I*Sin[d*x])) + (3*Cos[c + d*x]^(9/2)*Cot[c]*((Hypergeometric
PFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + Ar
cTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*S
qrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan
[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]])*(a
+ I*a*Tan[c + d*x]))/(5*d*(e*Cos[c + d*x])^(7/2)*(Cos[d*x] + I*Sin[d*x]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (137 ) = 274\).

Time = 11.27 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.96

method	result	size
parts	\(-\frac {2 a \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e}}{5 e^{4} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {2 i a}{7 d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}\)	\(385\)
default	\(\frac {2 \left (336 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-168 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-504 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+252 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+280 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-126 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}-5 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{35 \left (8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{3} d}\)	\(396\)

[In]

int((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/5*a*(e*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)/e^4/sin(1/2*d*x+1/2*c)^3/(8*sin(1/2*d*x+1/2*c
)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*EllipticE
(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^
4-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-
1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+8*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-3*Ellipti
cE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2))*(-2*sin(1/2*d*x+
1/2*c)^4*e+sin(1/2*d*x+1/2*c)^2*e)^(1/2)/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/d+2/7*I*a/d/(e*cos(d*x+c))^(7/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.81 \[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} {\left (21 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 77 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 23 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 21 \, {\left (i \, \sqrt {2} a e^{\left (8 i \, d x + 8 i \, c\right )} + 4 i \, \sqrt {2} a e^{\left (6 i \, d x + 6 i \, c\right )} + 6 i \, \sqrt {2} a e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, \sqrt {2} a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {2} a\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{35 \, {\left (d e^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )}} \]

[In]

integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

-2/35*(2*sqrt(1/2)*(21*I*a*e^(8*I*d*x + 8*I*c) + 77*I*a*e^(6*I*d*x + 6*I*c) + 23*I*a*e^(4*I*d*x + 4*I*c) + 7*I
*a*e^(2*I*d*x + 2*I*c))*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*e^(-1/2*I*d*x - 1/2*I*c) + 21*(I*sqrt(2)*a*e^(8*I*d*x
+ 8*I*c) + 4*I*sqrt(2)*a*e^(6*I*d*x + 6*I*c) + 6*I*sqrt(2)*a*e^(4*I*d*x + 4*I*c) + 4*I*sqrt(2)*a*e^(2*I*d*x +
2*I*c) + I*sqrt(2)*a)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, e^(I*d*x + I*c))))/(d*e^4*e^(8
*I*d*x + 8*I*c) + 4*d*e^4*e^(6*I*d*x + 6*I*c) + 6*d*e^4*e^(4*I*d*x + 4*I*c) + 4*d*e^4*e^(2*I*d*x + 2*I*c) + d*
e^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((I*a*tan(d*x + c) + a)/(e*cos(d*x + c))^(7/2), x)

Giac [F]

\[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)/(e*cos(d*x + c))^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\int \frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]

[In]

int((a + a*tan(c + d*x)*1i)/(e*cos(c + d*x))^(7/2),x)

[Out]

int((a + a*tan(c + d*x)*1i)/(e*cos(c + d*x))^(7/2), x)