\(\int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))^2} \, dx\) [244]

Optimal result

Integrand size = 28, antiderivative size = 181 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))^2} \, dx=\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d e^4}+\frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}}+\frac {2 \sin (c+d x)}{7 a^2 d e^3 \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )} \]

[Out]

2/15*e*sin(d*x+c)/a^2/d/(e*sec(d*x+c))^(9/2)+6/35*sin(d*x+c)/a^2/d/e/(e*sec(d*x+c))^(5/2)+2/7*sin(d*x+c)/a^2/d
/e^3/(e*sec(d*x+c))^(1/2)+2/7*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(
1/2))*cos(d*x+c)^(1/2)*(e*sec(d*x+c))^(1/2)/a^2/d/e^4+4/15*I*e^2/d/(e*sec(d*x+c))^(11/2)/(a^2+I*a^2*tan(d*x+c)
)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3581, 3854, 3856, 2720} \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))^2} \, dx=\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d e^4}+\frac {2 \sin (c+d x)}{7 a^2 d e^3 \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}+\frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}} \]

[In]

Int[1/((e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])^2),x]

[Out]

(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(7*a^2*d*e^4) + (2*e*Sin[c + d*x])/(15*a
^2*d*(e*Sec[c + d*x])^(9/2)) + (6*Sin[c + d*x])/(35*a^2*d*e*(e*Sec[c + d*x])^(5/2)) + (2*Sin[c + d*x])/(7*a^2*
d*e^3*Sqrt[e*Sec[c + d*x]]) + (((4*I)/15)*e^2)/(d*(e*Sec[c + d*x])^(11/2)*(a^2 + I*a^2*Tan[c + d*x]))

Rule 2720

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]

Rule 3581

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] - Dist[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] || (Integers
Q[n, m + 1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]

Rule 3854

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
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (11 e^2\right ) \int \frac {1}{(e \sec (c+d x))^{11/2}} \, dx}{15 a^2} \\ & = \frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {3 \int \frac {1}{(e \sec (c+d x))^{7/2}} \, dx}{5 a^2} \\ & = \frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}}+\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {3 \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{7 a^2 e^2} \\ & = \frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}}+\frac {2 \sin (c+d x)}{7 a^2 d e^3 \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\int \sqrt {e \sec (c+d x)} \, dx}{7 a^2 e^4} \\ & = \frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}}+\frac {2 \sin (c+d x)}{7 a^2 d e^3 \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 a^2 e^4} \\ & = \frac {2 \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 e^4}+\frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}}+\frac {2 \sin (c+d x)}{7 a^2 d e^3 \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.90 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))^2} \, dx=-\frac {(e \sec (c+d x))^{5/2} \left (296 i+228 i \cos (2 (c+d x))-72 i \cos (4 (c+d x))-4 i \cos (6 (c+d x))+480 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-17 \sin (2 (c+d x))+128 \sin (4 (c+d x))+11 \sin (6 (c+d x))\right )}{1680 a^2 d e^6 (-i+\tan (c+d x))^2} \]

[In]

Integrate[1/((e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])^2),x]

[Out]

-1/1680*((e*Sec[c + d*x])^(5/2)*(296*I + (228*I)*Cos[2*(c + d*x)] - (72*I)*Cos[4*(c + d*x)] - (4*I)*Cos[6*(c +
d*x)] + 480*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - 17*Sin[2*(
c + d*x)] + 128*Sin[4*(c + d*x)] + 11*Sin[6*(c + d*x)]))/(a^2*d*e^6*(-I + Tan[c + d*x])^2)

Maple [A] (verified)

Time = 9.96 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.14

method	result	size
default	\(\frac {\frac {4 i \left (\cos ^{7}\left (d x +c \right )\right )}{15}+\frac {4 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{15}+\frac {2 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{15}+\frac {2 i F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}{7}+\frac {6 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35}+\frac {2 i \sec \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}{7}+\frac {2 \sin \left (d x +c \right )}{7}}{a^{2} d \sqrt {e \sec \left (d x +c \right )}\, e^{3}}\)	\(206\)

[In]

int(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

2/105/a^2/d/(e*sec(d*x+c))^(1/2)/e^3*(14*I*cos(d*x+c)^7+14*sin(d*x+c)*cos(d*x+c)^6+7*sin(d*x+c)*cos(d*x+c)^4+1
5*I*EllipticF(I*(-csc(d*x+c)+cot(d*x+c)),I)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)+9*cos(d
*x+c)^2*sin(d*x+c)+15*I*sec(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticF(I*(-cs
c(d*x+c)+cot(d*x+c)),I)+15*sin(d*x+c))

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))^2} \, dx=\frac {{\left (\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-15 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 200 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 245 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 592 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 211 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 56 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} - 960 i \, \sqrt {2} \sqrt {e} e^{\left (8 i \, d x + 8 i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{3360 \, a^{2} d e^{4}} \]

[In]

integrate(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

1/3360*(sqrt(2)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*(-15*I*e^(12*I*d*x + 12*I*c) - 200*I*e^(10*I*d*x + 10*I*c) +
245*I*e^(8*I*d*x + 8*I*c) + 592*I*e^(6*I*d*x + 6*I*c) + 211*I*e^(4*I*d*x + 4*I*c) + 56*I*e^(2*I*d*x + 2*I*c)
+ 7*I)*e^(1/2*I*d*x + 1/2*I*c) - 960*I*sqrt(2)*sqrt(e)*e^(8*I*d*x + 8*I*c)*weierstrassPInverse(-4, 0, e^(I*d*x
+ I*c)))*e^(-8*I*d*x - 8*I*c)/(a^2*d*e^4)

Sympy [F(-1)]

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

[In]

integrate(1/(e*sec(d*x+c))**(7/2)/(a+I*a*tan(d*x+c))**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

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

[In]

integrate(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")

[Out]

integrate(1/((e*sec(d*x + c))^(7/2)*(I*a*tan(d*x + c) + a)^2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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