Integrand size = 28, antiderivative size = 152 \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {6 e^6 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d}+\frac {6 e^5 (e \sec (c+d x))^{3/2} \sin (c+d x)}{7 a^2 d}+\frac {18 e^3 (e \sec (c+d x))^{7/2} \sin (c+d x)}{35 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )} \] Output:
6/7*e^6*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*(e*sec(d*x +c))^(1/2)/a^2/d+6/7*e^5*(e*sec(d*x+c))^(3/2)*sin(d*x+c)/a^2/d+18/35*e^3*( e*sec(d*x+c))^(7/2)*sin(d*x+c)/a^2/d-4/5*I*e^2*(e*sec(d*x+c))^(9/2)/d/(a^2 +I*a^2*tan(d*x+c))
Time = 1.46 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.56 \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {e^6 \sec ^3(c+d x) \sqrt {e \sec (c+d x)} \left (-56 i \cos (c+d x)+60 \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-5 \sin (c+d x)+15 \sin (3 (c+d x))\right )}{70 a^2 d} \] Input:
Integrate[(e*Sec[c + d*x])^(13/2)/(a + I*a*Tan[c + d*x])^2,x]
Output:
(e^6*Sec[c + d*x]^3*Sqrt[e*Sec[c + d*x]]*((-56*I)*Cos[c + d*x] + 60*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] - 5*Sin[c + d*x] + 15*Sin[3*(c + d* x)]))/(70*a^2*d)
Time = 0.67 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 3981, 3042, 4255, 3042, 4255, 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.
\(\displaystyle \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2}dx\) |
\(\Big \downarrow \) 3981 |
\(\displaystyle \frac {9 e^2 \int (e \sec (c+d x))^{9/2}dx}{5 a^2}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 e^2 \int \left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{9/2}dx}{5 a^2}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \int (e \sec (c+d x))^{5/2}dx+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{7/2}}{7 d}\right )}{5 a^2}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \int \left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{7/2}}{7 d}\right )}{5 a^2}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \sqrt {e \sec (c+d x)}dx+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{7/2}}{7 d}\right )}{5 a^2}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{7/2}}{7 d}\right )}{5 a^2}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{7/2}}{7 d}\right )}{5 a^2}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{7/2}}{7 d}\right )}{5 a^2}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{3 d}+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \sec (c+d x))^{7/2}}{7 d}\right )}{5 a^2}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
Input:
Int[(e*Sec[c + d*x])^(13/2)/(a + I*a*Tan[c + d*x])^2,x]
Output:
(9*e^2*((2*e*(e*Sec[c + d*x])^(7/2)*Sin[c + d*x])/(7*d) + (5*e^2*((2*e^2*S qrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(3*d) + (2*e*(e*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)))/7))/(5*a^2) - (((4*I)/5) *e^2*(e*Sec[c + d*x])^(9/2))/(d*(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[(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 = 5.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {e^{6} \left (-\frac {6 \tan \left (d x +c \right )}{7}+\frac {2 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}}{7}+\frac {4 i \sec \left (d x +c \right )^{2}}{5}+\frac {6 i \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )}{7}\right ) \sqrt {e \sec \left (d x +c \right )}}{a^{2} d}\) | \(121\) |
Input:
int((e*sec(d*x+c))^(13/2)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-e^6/a^2/d*(-6/7*tan(d*x+c)+2/7*tan(d*x+c)*sec(d*x+c)^2+4/5*I*sec(d*x+c)^2 +6/7*I*(cos(d*x+c)+1)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1)) ^(1/2)*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),I))*(e*sec(d*x+c))^(1/2)
Time = 0.09 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.32 \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (15 i \, e^{6} e^{\left (6 i \, d x + 6 i \, c\right )} + 51 i \, e^{6} e^{\left (4 i \, d x + 4 i \, c\right )} + 61 i \, e^{6} e^{\left (2 i \, d x + 2 i \, c\right )} - 15 i \, e^{6}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 15 \, \sqrt {2} {\left (i \, e^{6} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 i \, e^{6} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, e^{6} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{6}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{35 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \] Input:
integrate((e*sec(d*x+c))^(13/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas" )
Output:
-2/35*(sqrt(2)*(15*I*e^6*e^(6*I*d*x + 6*I*c) + 51*I*e^6*e^(4*I*d*x + 4*I*c ) + 61*I*e^6*e^(2*I*d*x + 2*I*c) - 15*I*e^6)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) + 15*sqrt(2)*(I*e^6*e^(6*I*d*x + 6*I*c) + 3*I *e^6*e^(4*I*d*x + 4*I*c) + 3*I*e^6*e^(2*I*d*x + 2*I*c) + I*e^6)*sqrt(e)*we ierstrassPInverse(-4, 0, e^(I*d*x + I*c)))/(a^2*d*e^(6*I*d*x + 6*I*c) + 3* a^2*d*e^(4*I*d*x + 4*I*c) + 3*a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)
Timed out. \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate((e*sec(d*x+c))**(13/2)/(a+I*a*tan(d*x+c))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*sec(d*x+c))^(13/2)/(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.
\[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {13}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((e*sec(d*x+c))^(13/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
integrate((e*sec(d*x + c))^(13/2)/(I*a*tan(d*x + c) + a)^2, x)
Timed out. \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{13/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \] Input:
int((e/cos(c + d*x))^(13/2)/(a + a*tan(c + d*x)*1i)^2,x)
Output:
int((e/cos(c + d*x))^(13/2)/(a + a*tan(c + d*x)*1i)^2, x)
\[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{6}}{\tan \left (d x +c \right )^{2}-2 \tan \left (d x +c \right ) i -1}d x \right ) e^{6}}{a^{2}} \] Input:
int((e*sec(d*x+c))^(13/2)/(a+I*a*tan(d*x+c))^2,x)
Output:
( - sqrt(e)*int((sqrt(sec(c + d*x))*sec(c + d*x)**6)/(tan(c + d*x)**2 - 2* tan(c + d*x)*i - 1),x)*e**6)/a**2