Integrand size = 28, antiderivative size = 190 \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {2 (e \cos (c+d x))^{7/2} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 a^2 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \cos (c+d x) (e \cos (c+d x))^{7/2} \sin (c+d x)}{15 a^2 d}+\frac {6 (e \cos (c+d x))^{7/2} \tan (c+d x)}{35 a^2 d}+\frac {2 (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{7 a^2 d}+\frac {4 i \cos ^2(c+d x) (e \cos (c+d x))^{7/2}}{15 d \left (a^2+i a^2 \tan (c+d x)\right )} \] Output:
2/7*(e*cos(d*x+c))^(7/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/a^2/d/cos( d*x+c)^(7/2)+2/15*cos(d*x+c)*(e*cos(d*x+c))^(7/2)*sin(d*x+c)/a^2/d+6/35*(e *cos(d*x+c))^(7/2)*tan(d*x+c)/a^2/d+2/7*(e*cos(d*x+c))^(7/2)*sec(d*x+c)^2* tan(d*x+c)/a^2/d+4/15*I*cos(d*x+c)^2*(e*cos(d*x+c))^(7/2)/d/(a^2+I*a^2*tan (d*x+c))
Time = 1.99 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.82 \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {e^3 \sqrt {e \cos (c+d x)} \left (-240 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+\sqrt {\cos (c+d x)} (-296 i \cos (c+d x)+68 i \cos (3 (c+d x))+4 i \cos (5 (c+d x))+134 \sin (c+d x)-117 \sin (3 (c+d x))-11 \sin (5 (c+d x)))\right )}{840 a^2 d \cos ^{\frac {5}{2}}(c+d x) (-i+\tan (c+d x))^2} \] Input:
Integrate[(e*Cos[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^2,x]
Output:
(e^3*Sqrt[e*Cos[c + d*x]]*(-240*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x) ] + I*Sin[2*(c + d*x)]) + Sqrt[Cos[c + d*x]]*((-296*I)*Cos[c + d*x] + (68* I)*Cos[3*(c + d*x)] + (4*I)*Cos[5*(c + d*x)] + 134*Sin[c + d*x] - 117*Sin[ 3*(c + d*x)] - 11*Sin[5*(c + d*x)])))/(840*a^2*d*Cos[c + d*x]^(5/2)*(-I + Tan[c + d*x])^2)
Time = 1.04 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3998, 3042, 3981, 3042, 4256, 3042, 4256, 3042, 4256, 3042, 4258, 3042, 3120}
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.
| \(\displaystyle \int \frac {(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3998 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \int \frac {1}{(e \sec (c+d x))^{7/2} (i \tan (c+d x) a+a)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \int \frac {1}{(e \sec (c+d x))^{7/2} (i \tan (c+d x) a+a)^2}dx\) |
| \(\Big \downarrow \) 3981 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (\frac {11 e^2 \int \frac {1}{(e \sec (c+d x))^{11/2}}dx}{15 a^2}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (\frac {11 e^2 \int \frac {1}{\left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{11/2}}dx}{15 a^2}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (\frac {11 e^2 \left (\frac {9 \int \frac {1}{(e \sec (c+d x))^{7/2}}dx}{11 e^2}+\frac {2 \sin (c+d x)}{11 d e (e \sec (c+d x))^{9/2}}\right )}{15 a^2}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (\frac {11 e^2 \left (\frac {9 \int \frac {1}{\left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{11 e^2}+\frac {2 \sin (c+d x)}{11 d e (e \sec (c+d x))^{9/2}}\right )}{15 a^2}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (\frac {11 e^2 \left (\frac {9 \left (\frac {5 \int \frac {1}{(e \sec (c+d x))^{3/2}}dx}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 e^2}+\frac {2 \sin (c+d x)}{11 d e (e \sec (c+d x))^{9/2}}\right )}{15 a^2}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (\frac {11 e^2 \left (\frac {9 \left (\frac {5 \int \frac {1}{\left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 e^2}+\frac {2 \sin (c+d x)}{11 d e (e \sec (c+d x))^{9/2}}\right )}{15 a^2}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (\frac {11 e^2 \left (\frac {9 \left (\frac {5 \left (\frac {\int \sqrt {e \sec (c+d x)}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 e^2}+\frac {2 \sin (c+d x)}{11 d e (e \sec (c+d x))^{9/2}}\right )}{15 a^2}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (\frac {11 e^2 \left (\frac {9 \left (\frac {5 \left (\frac {\int \sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 e^2}+\frac {2 \sin (c+d x)}{11 d e (e \sec (c+d x))^{9/2}}\right )}{15 a^2}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (\frac {11 e^2 \left (\frac {9 \left (\frac {5 \left (\frac {\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 e^2}+\frac {2 \sin (c+d x)}{11 d e (e \sec (c+d x))^{9/2}}\right )}{15 a^2}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (\frac {11 e^2 \left (\frac {9 \left (\frac {5 \left (\frac {\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 e^2}+\frac {2 \sin (c+d x)}{11 d e (e \sec (c+d x))^{9/2}}\right )}{15 a^2}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \left (\frac {11 e^2 \left (\frac {9 \left (\frac {5 \left (\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{3 d e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 e^2}+\frac {2 \sin (c+d x)}{11 d e (e \sec (c+d x))^{9/2}}\right )}{15 a^2}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}\right )\) |
Input:
Int[(e*Cos[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^2,x]
Output:
(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*((11*e^2*((2*Sin[c + d*x])/( 11*d*e*(e*Sec[c + d*x])^(9/2)) + (9*((2*Sin[c + d*x])/(7*d*e*(e*Sec[c + d* x])^(5/2)) + (5*((2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Se c[c + d*x]])/(3*d*e^2) + (2*Sin[c + d*x])/(3*d*e*Sqrt[e*Sec[c + d*x]])))/( 7*e^2)))/(11*e^2)))/(15*a^2) + (((4*I)/15)*e^2)/(d*(e*Sec[c + d*x])^(11/2) *(a^2 + I*a^2*Tan[c + d*x])))
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 _)])^(n_), x_Symbol] :> Simp[2*d^2*(d*Sec[e + f*x])^(m - 2)*((a + b*Tan[e + f*x])^(n + 1)/(b*f*(m + 2*n))), x] - Simp[d^2*((m - 2)/(b^2*(m + 2*n))) Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[ {a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, -1] && ((ILtQ[n/2, 0] && IGtQ[m - 1/2, 0]) || EqQ[n, -2] || IGtQ[m + n, 0] || (IntegersQ[n, m + 1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]
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[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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. 386 vs. \(2 (167 ) = 334\).
Time = 14.99 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.04
| method | result | size |
| default | \(-\frac {2 e^{4} \left (-14 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+3584 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+25088 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-12544 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{14} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+6272 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+19264 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+14336 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}-16800 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+224 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+9104 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-25088 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}-3128 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3584 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}+700 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-15680 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-90 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1568 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}\right )}{105 a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\) | \(387\) |
Input:
int((e*cos(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-2/105/a^2/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^4*(-14 *I*sin(1/2*d*x+1/2*c)+3584*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^16+25088* I*sin(1/2*d*x+1/2*c)^11-12544*sin(1/2*d*x+1/2*c)^14*cos(1/2*d*x+1/2*c)+627 2*I*sin(1/2*d*x+1/2*c)^7+19264*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+14 336*I*sin(1/2*d*x+1/2*c)^15-16800*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c) +224*I*sin(1/2*d*x+1/2*c)^3+9104*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-2 5088*I*sin(1/2*d*x+1/2*c)^13-3128*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)- 3584*I*sin(1/2*d*x+1/2*c)^17+700*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-1 5680*I*sin(1/2*d*x+1/2*c)^9-90*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+15* (sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(co s(1/2*d*x+1/2*c),2^(1/2))-1568*I*sin(1/2*d*x+1/2*c)^5)/d
Time = 0.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.79 \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {{\left (-960 i \, \sqrt {\frac {1}{2}} e^{\frac {7}{2}} e^{\left (7 i \, d x + 7 i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \sqrt {\frac {1}{2}} {\left (-15 i \, e^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 185 i \, e^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 430 i \, e^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 162 i \, e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 49 i \, e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, e^{3}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{1680 \, a^{2} d} \] Input:
integrate((e*cos(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
1/1680*(-960*I*sqrt(1/2)*e^(7/2)*e^(7*I*d*x + 7*I*c)*weierstrassPInverse(- 4, 0, e^(I*d*x + I*c)) + sqrt(1/2)*(-15*I*e^3*e^(10*I*d*x + 10*I*c) - 185* I*e^3*e^(8*I*d*x + 8*I*c) + 430*I*e^3*e^(6*I*d*x + 6*I*c) + 162*I*e^3*e^(4 *I*d*x + 4*I*c) + 49*I*e^3*e^(2*I*d*x + 2*I*c) + 7*I*e^3)*sqrt(e*e^(2*I*d* x + 2*I*c) + e)*e^(-1/2*I*d*x - 1/2*I*c))*e^(-7*I*d*x - 7*I*c)/(a^2*d)
Timed out. \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(7/2)/(a+I*a*tan(d*x+c))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*cos(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((e*cos(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
integrate((e*cos(d*x + c))^(7/2)/(I*a*tan(d*x + c) + a)^2, x)
Timed out. \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \] Input:
int((e*cos(c + d*x))^(7/2)/(a + a*tan(c + d*x)*1i)^2,x)
Output:
int((e*cos(c + d*x))^(7/2)/(a + a*tan(c + d*x)*1i)^2, x)
\[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\tan \left (d x +c \right )^{2}-2 \tan \left (d x +c \right ) i -1}d x \right ) e^{3}}{a^{2}} \] Input:
int((e*cos(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x)
Output:
( - sqrt(e)*int((sqrt(cos(c + d*x))*cos(c + d*x)**3)/(tan(c + d*x)**2 - 2* tan(c + d*x)*i - 1),x)*e**3)/a**2