Integrand size = 26, antiderivative size = 130 \[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}-\frac {6 a \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}} \] Output:
2/7*I*a/d/(e*cos(d*x+c))^(7/2)-6/5*a*cos(d*x+c)^(7/2)*EllipticE(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)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.97 (sec) , antiderivative size = 666, normalized size of antiderivative = 5.12 \[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\frac {\cos ^5(c+d x) \left (\csc (c) \sec (c) \left (\frac {6 \cos (c)}{5}-\frac {6}{5} i \sin (c)\right )+\sec ^4(c+d x) \left (\frac {2}{7} i \cos (c)+\frac {2 \sin (c)}{7}\right )+\sec (c) \sec ^3(c+d x) \left (\frac {2 \cos (c)}{5}-\frac {2}{5} i \sin (c)\right ) \sin (d x)+\sec (c) \sec (c+d x) \left (\frac {6 \cos (c)}{5}-\frac {6}{5} i \sin (c)\right ) \sin (d x)+\sec ^2(c+d x) \left (\frac {2 \cos (c)}{5}-\frac {2}{5} i \sin (c)\right ) \tan (c)\right ) (a+i a \tan (c+d x))}{d (e \cos (c+d x))^{7/2} (\cos (d x)+i \sin (d x))}-\frac {3 i \cos ^{\frac {9}{2}}(c+d x) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right ) (a+i a \tan (c+d x))}{5 d (e \cos (c+d x))^{7/2} (\cos (d x)+i \sin (d x))}+\frac {3 \cos ^{\frac {9}{2}}(c+d x) \cot (c) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right ) (a+i a \tan (c+d x))}{5 d (e \cos (c+d x))^{7/2} (\cos (d x)+i \sin (d x))} \] Input:
Integrate[(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(7/2),x]
Output:
(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]]]]*S qrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[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 ]))/(d*(e*Cos[c + d*x])^(7/2)*(Cos[d*x] + I*Sin[d*x])) + (3*Cos[c + d*x]^( 9/2)*Cot[c]*((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 + 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/...
Time = 0.76 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 3998, 3042, 3967, 3042, 4255, 3042, 4255, 3042, 4258, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3998 |
| \(\displaystyle \frac {\int (e \sec (c+d x))^{7/2} (i \tan (c+d x) a+a)dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e \sec (c+d x))^{7/2} (i \tan (c+d x) a+a)dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle \frac {a \int (e \sec (c+d x))^{7/2}dx+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}dx+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {a \left (\frac {3}{5} e^2 \int (e \sec (c+d x))^{3/2}dx+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{5/2}}{5 d}\right )+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {3}{5} e^2 \int \left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{5/2}}{5 d}\right )+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {a \left (\frac {3}{5} e^2 \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-e^2 \int \frac {1}{\sqrt {e \sec (c+d x)}}dx\right )+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{5/2}}{5 d}\right )+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {3}{5} e^2 \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-e^2 \int \frac {1}{\sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{5/2}}{5 d}\right )+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {a \left (\frac {3}{5} e^2 \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-\frac {e^2 \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\right )+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{5/2}}{5 d}\right )+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {3}{5} e^2 \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-\frac {e^2 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\right )+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{5/2}}{5 d}\right )+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {a \left (\frac {3}{5} e^2 \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-\frac {2 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\right )+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{5/2}}{5 d}\right )+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
Input:
Int[(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(7/2),x]
Output:
((((2*I)/7)*a*(e*Sec[c + d*x])^(7/2))/d + a*((2*e*(e*Sec[c + d*x])^(5/2)*S in[c + d*x])/(5*d) + (3*e^2*((-2*e^2*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Co s[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (2*e*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x] )/d))/5))/((e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2))
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]
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] + Simp[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])
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)])^(n_.), x_Symbol] :> Simp[(d*Cos[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]
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] + Simp[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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (113 ) = 226\).
Time = 3.50 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.96
| method | result | size |
| parts | \(-\frac {2 a \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (24 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-24 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} e +\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e}}{5 e^{4} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-12 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+6 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, d}+\frac {2 i a}{7 d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}\) | \(385\) |
| default | \(\frac {2 \left (336 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-168 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-504 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+252 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+280 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-126 \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-56 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+21 \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}-5 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{35 \left (8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-12 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+6 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, e^{3} d}\) | \(396\) |
Input:
int((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
Output:
-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*(sin(1/2*d* x+1/2*c)^2)^(1/2)*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)^4-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/ 2*c)+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*Elli pticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2*c )^2*cos(1/2*d*x+1/2*c)-3*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))*(-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)
Time = 0.08 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.71 \[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=-\frac {4 \, {\left (\sqrt {\frac {1}{2}} {\left (21 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 77 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 23 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 21 \, \sqrt {\frac {1}{2}} {\left (i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 4 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 6 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{35 \, {\left (d e^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )}} \] Input:
integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x, algorithm="fricas")
Output:
-4/35*(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*sqrt(1/2)*(I*a*e^(8*I*d*x + 8*I*c) + 4*I*a*e^(6*I*d*x + 6*I*c) + 6*I*a*e^(4*I*d*x + 4*I*c) + 4*I*a*e^( 2*I*d*x + 2*I*c) + I*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)
Timed out. \[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))**(7/2),x)
Output:
Timed out
\[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x, algorithm="maxima")
Output:
integrate((I*a*tan(d*x + c) + a)/(e*cos(d*x + c))^(7/2), x)
\[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x, algorithm="giac")
Output:
integrate((I*a*tan(d*x + c) + a)/(e*cos(d*x + c))^(7/2), x)
Timed out. \[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\int \frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \] Input:
int((a + a*tan(c + d*x)*1i)/(e*cos(c + d*x))^(7/2),x)
Output:
int((a + a*tan(c + d*x)*1i)/(e*cos(c + d*x))^(7/2), x)
\[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {e}\, a \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \tan \left (d x +c \right )}{\cos \left (d x +c \right )^{4}}d x \right ) i \right )}{e^{4}} \] Input:
int((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x)
Output:
(sqrt(e)*a*(int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x) + int((sqrt(cos(c + d*x))*tan(c + d*x))/cos(c + d*x)**4,x)*i))/e**4