Integrand size = 28, antiderivative size = 215 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=-\frac {22 a^4 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^4 (e \sec (c+d x))^{3/2}}{9 d}+\frac {22 a^4 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3}{9 d}+\frac {10 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )^2}{21 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{21 d} \] Output:
-22/3*a^4*e^2*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/cos(d*x+c)^(1/2)/(e* sec(d*x+c))^(1/2)+22/9*I*a^4*(e*sec(d*x+c))^(3/2)/d+22/3*a^4*e*(e*sec(d*x+ c))^(1/2)*sin(d*x+c)/d+2/9*I*a*(e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^3/d +10/21*I*(e*sec(d*x+c))^(3/2)*(a^2+I*a^2*tan(d*x+c))^2/d+22/21*I*(e*sec(d* x+c))^(3/2)*(a^4+I*a^4*tan(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.91 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.52 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=\frac {(e \sec (c+d x))^{3/2} \left (\frac {22 i \sqrt {2} e^{-i (3 c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right )}{-1+e^{2 i c}}+\frac {1}{56} \csc (c) \sec ^{\frac {9}{2}}(c+d x) (\cos (4 c)-i \sin (4 c)) (1260 \cos (d x)+1050 \cos (2 c+d x)+742 \cos (2 c+3 d x)+413 \cos (4 c+3 d x)+231 \cos (4 c+5 d x)-720 i \sin (d x)+720 i \sin (2 c+d x)-336 i \sin (2 c+3 d x)+336 i \sin (4 c+3 d x))\right ) (a+i a \tan (c+d x))^4}{9 d \sec ^{\frac {11}{2}}(c+d x) (\cos (d x)+i \sin (d x))^4} \] Input:
Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^4,x]
Output:
((e*Sec[c + d*x])^(3/2)*(((22*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I )*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-3*Sqrt[1 + E^((2*I)*(c + d* x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, - E^((2*I)*(c + d*x))]))/(E^(I*(3*c + d*x))*(-1 + E^((2*I)*c))) + (Csc[c]*Se c[c + d*x]^(9/2)*(Cos[4*c] - I*Sin[4*c])*(1260*Cos[d*x] + 1050*Cos[2*c + d *x] + 742*Cos[2*c + 3*d*x] + 413*Cos[4*c + 3*d*x] + 231*Cos[4*c + 5*d*x] - (720*I)*Sin[d*x] + (720*I)*Sin[2*c + d*x] - (336*I)*Sin[2*c + 3*d*x] + (3 36*I)*Sin[4*c + 3*d*x]))/56)*(a + I*a*Tan[c + d*x])^4)/(9*d*Sec[c + d*x]^( 11/2)*(Cos[d*x] + I*Sin[d*x])^4)
Time = 1.05 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3979, 3042, 3979, 3042, 3979, 3042, 3967, 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.
| \(\displaystyle \int (a+i a \tan (c+d x))^4 (e \sec (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (c+d x))^4 (e \sec (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle \frac {5}{3} a \int (e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)^3dx+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} a \int (e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)^3dx+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \int (e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)^2dx+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \int (e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)^2dx+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \int (e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)dx+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \int (e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)dx+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \int (e \sec (c+d x))^{3/2}dx+\frac {2 i a (e \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \int \left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 i a (e \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-e^2 \int \frac {1}{\sqrt {e \sec (c+d x)}}dx\right )+\frac {2 i a (e \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-e^2 \int \frac {1}{\sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 i a (e \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-\frac {e^2 \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\right )+\frac {2 i a (e \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-\frac {e^2 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\right )+\frac {2 i a (e \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d}+\frac {7}{5} a \left (a \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-\frac {2 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\right )+\frac {2 i a (e \sec (c+d x))^{3/2}}{3 d}\right )\right )+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d}\right )+\frac {2 i a (a+i a \tan (c+d x))^3 (e \sec (c+d x))^{3/2}}{9 d}\) |
Input:
Int[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^4,x]
Output:
(((2*I)/9)*a*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^3)/d + (5*a*((( (2*I)/7)*a*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^2)/d + (11*a*((7* a*((((2*I)/3)*a*(e*Sec[c + d*x])^(3/2))/d + a*((-2*e^2*EllipticE[(c + d*x) /2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (2*e*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/d)))/5 + (((2*I)/5)*(e*Sec[c + d*x])^(3/2)*(a^2 + I*a^ 2*Tan[c + d*x]))/d))/7))/3
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[((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 - 1)/(f*(m + n - 1))), x] + Simp[a*((m + 2*n - 2)/(m + n - 1)) Int[(d*Se c[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] && GtQ[n, 0] && NeQ[m + n - 1, 0] && IntegersQ [2*m, 2*n]
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]
Time = 17.10 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.25
| method | result | size |
| default | \(-\frac {2 a^{4} e \sqrt {e \sec \left (d x +c \right )}\, \left (\tan \left (d x +c \right ) \sec \left (d x +c \right )^{3} \left (-231 \cos \left (d x +c \right )^{4}+91 \cos \left (d x +c \right )^{3}+91 \cos \left (d x +c \right )^{2}-7 \cos \left (d x +c \right )-7\right )+i \left (-168-168 \sec \left (d x +c \right )+36 \sec \left (d x +c \right )^{2}+36 \sec \left (d x +c \right )^{3}\right )+i \left (231 \cos \left (d x +c \right )^{2}+462 \cos \left (d x +c \right )+231\right ) \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+i \left (-231 \cos \left (d x +c \right )^{2}-462 \cos \left (d x +c \right )-231\right ) \operatorname {EllipticE}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )}{63 d \left (\cos \left (d x +c \right )+1\right )}\) | \(269\) |
| parts | \(\frac {2 a^{4} \left (i \left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )+i \left (-\cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )-1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+\sin \left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}\, e}{d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 a^{4} \left (\tan \left (d x +c \right ) \sec \left (d x +c \right )^{3} \left (12 \cos \left (d x +c \right )^{4}-11 \cos \left (d x +c \right )^{3}-11 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )+5\right )+12 i \left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )+12 i \left (-\cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )-1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\right ) \sqrt {e \sec \left (d x +c \right )}\, e}{45 d \left (\cos \left (d x +c \right )+1\right )}-\frac {8 i a^{4} \left (\frac {\left (e \sec \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {e^{2} \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,e^{2}}+\frac {8 i a^{4} \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {12 a^{4} \sqrt {e \sec \left (d x +c \right )}\, e \left (2 \sin \left (d x +c \right )-\tan \left (d x +c \right )-\sec \left (d x +c \right ) \tan \left (d x +c \right )+i \left (2 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+2\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )+i \left (-2 \cos \left (d x +c \right )^{2}-4 \cos \left (d x +c \right )-2\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )\right )}{5 d \left (\cos \left (d x +c \right )+1\right )}\) | \(688\) |
Input:
int((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
-2/63*a^4/d*e*(e*sec(d*x+c))^(1/2)/(cos(d*x+c)+1)*(tan(d*x+c)*sec(d*x+c)^3 *(-231*cos(d*x+c)^4+91*cos(d*x+c)^3+91*cos(d*x+c)^2-7*cos(d*x+c)-7)+I*(-16 8-168*sec(d*x+c)+36*sec(d*x+c)^2+36*sec(d*x+c)^3)+I*(231*cos(d*x+c)^2+462* cos(d*x+c)+231)*EllipticF(I*(cot(d*x+c)-csc(d*x+c)),I)*(1/(cos(d*x+c)+1))^ (1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+I*(-231*cos(d*x+c)^2-462*cos(d*x+c )-231)*EllipticE(I*(cot(d*x+c)-csc(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)*(co s(d*x+c)/(cos(d*x+c)+1))^(1/2))
Time = 0.11 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.16 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (231 i \, a^{4} e e^{\left (9 i \, d x + 9 i \, c\right )} + 406 i \, a^{4} e e^{\left (7 i \, d x + 7 i \, c\right )} + 540 i \, a^{4} e e^{\left (5 i \, d x + 5 i \, c\right )} + 330 i \, a^{4} e e^{\left (3 i \, d x + 3 i \, c\right )} + 77 i \, a^{4} e e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 231 \, \sqrt {2} {\left (i \, a^{4} e e^{\left (8 i \, d x + 8 i \, c\right )} + 4 i \, a^{4} e e^{\left (6 i \, d x + 6 i \, c\right )} + 6 i \, a^{4} e e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, a^{4} e e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{4} e\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{63 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:
integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")
Output:
-2/63*(sqrt(2)*(231*I*a^4*e*e^(9*I*d*x + 9*I*c) + 406*I*a^4*e*e^(7*I*d*x + 7*I*c) + 540*I*a^4*e*e^(5*I*d*x + 5*I*c) + 330*I*a^4*e*e^(3*I*d*x + 3*I*c ) + 77*I*a^4*e*e^(I*d*x + I*c))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I *d*x + 1/2*I*c) + 231*sqrt(2)*(I*a^4*e*e^(8*I*d*x + 8*I*c) + 4*I*a^4*e*e^( 6*I*d*x + 6*I*c) + 6*I*a^4*e*e^(4*I*d*x + 4*I*c) + 4*I*a^4*e*e^(2*I*d*x + 2*I*c) + I*a^4*e)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0 , e^(I*d*x + I*c))))/(d*e^(8*I*d*x + 8*I*c) + 4*d*e^(6*I*d*x + 6*I*c) + 6* d*e^(4*I*d*x + 4*I*c) + 4*d*e^(2*I*d*x + 2*I*c) + d)
\[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=a^{4} \left (\int \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (- 6 \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (c + d x \right )}\right )\, dx + \int \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{4}{\left (c + d x \right )}\, dx + \int 4 i \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan {\left (c + d x \right )}\, dx + \int \left (- 4 i \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{3}{\left (c + d x \right )}\right )\, dx\right ) \] Input:
integrate((e*sec(d*x+c))**(3/2)*(a+I*a*tan(d*x+c))**4,x)
Output:
a**4*(Integral((e*sec(c + d*x))**(3/2), x) + Integral(-6*(e*sec(c + d*x))* *(3/2)*tan(c + d*x)**2, x) + Integral((e*sec(c + d*x))**(3/2)*tan(c + d*x) **4, x) + Integral(4*I*(e*sec(c + d*x))**(3/2)*tan(c + d*x), x) + Integral (-4*I*(e*sec(c + d*x))**(3/2)*tan(c + d*x)**3, x))
\[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=\int { \left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} \,d x } \] Input:
integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")
Output:
integrate((e*sec(d*x + c))^(3/2)*(I*a*tan(d*x + c) + a)^4, x)
\[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=\int { \left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} \,d x } \] Input:
integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^4,x, algorithm="giac")
Output:
integrate((e*sec(d*x + c))^(3/2)*(I*a*tan(d*x + c) + a)^4, x)
Timed out. \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=\int {\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4 \,d x \] Input:
int((e/cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i)^4,x)
Output:
int((e/cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i)^4, x)
\[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx=\frac {\sqrt {e}\, a^{4} e \left (-24 \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right ) \tan \left (d x +c \right )^{2} i +88 \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right ) i +21 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right ) \tan \left (d x +c \right )^{4}d x \right ) d -126 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right ) \tan \left (d x +c \right )^{2}d x \right ) d +21 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) d \right )}{21 d} \] Input:
int((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^4,x)
Output:
(sqrt(e)*a**4*e*( - 24*sqrt(sec(c + d*x))*sec(c + d*x)*tan(c + d*x)**2*i + 88*sqrt(sec(c + d*x))*sec(c + d*x)*i + 21*int(sqrt(sec(c + d*x))*sec(c + d*x)*tan(c + d*x)**4,x)*d - 126*int(sqrt(sec(c + d*x))*sec(c + d*x)*tan(c + d*x)**2,x)*d + 21*int(sqrt(sec(c + d*x))*sec(c + d*x),x)*d))/(21*d)