\(\int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx\) [232]

Optimal result

Integrand size = 28, antiderivative size = 145 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=\frac {30 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{77 a d e^4}+\frac {18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac {30 \sin (c+d x)}{77 a d e^3 \sqrt {e \sec (c+d x)}}+\frac {2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \]

[Out]

18/77*sin(d*x+c)/a/d/e/(e*sec(d*x+c))^(5/2)+30/77*sin(d*x+c)/a/d/e^3/(e*sec(d*x+c))^(1/2)+30/77*(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/d/e^4+2/11*I/d/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3583, 3854, 3856, 2720} \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=\frac {30 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{77 a d e^4}+\frac {30 \sin (c+d x)}{77 a d e^3 \sqrt {e \sec (c+d x)}}+\frac {18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac {2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}} \]

[In]

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

[Out]

(30*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(77*a*d*e^4) + (18*Sin[c + d*x])/(77*a*
d*e*(e*Sec[c + d*x])^(5/2)) + (30*Sin[c + d*x])/(77*a*d*e^3*Sqrt[e*Sec[c + d*x]]) + ((2*I)/11)/(d*(e*Sec[c + d
*x])^(7/2)*(a + I*a*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 3583

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(d*
Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(b*f*(m + 2*n))), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), Int[(d*Sec[
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] && LtQ[n
, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]

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 {2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac {9 \int \frac {1}{(e \sec (c+d x))^{7/2}} \, dx}{11 a} \\ & = \frac {18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac {2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac {45 \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{77 a e^2} \\ & = \frac {18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac {30 \sin (c+d x)}{77 a d e^3 \sqrt {e \sec (c+d x)}}+\frac {2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac {15 \int \sqrt {e \sec (c+d x)} \, dx}{77 a e^4} \\ & = \frac {18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac {30 \sin (c+d x)}{77 a d e^3 \sqrt {e \sec (c+d x)}}+\frac {2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac {\left (15 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 a e^4} \\ & = \frac {30 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{77 a d e^4}+\frac {18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac {30 \sin (c+d x)}{77 a d e^3 \sqrt {e \sec (c+d x)}}+\frac {2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.84 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=-\frac {(e \sec (c+d x))^{3/2} \left (-148 \cos (c+d x)+34 \cos (3 (c+d x))+2 \cos (5 (c+d x))+240 i \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (c+d x)+i \sin (c+d x))+78 i \sin (c+d x)+87 i \sin (3 (c+d x))+9 i \sin (5 (c+d x))\right )}{616 a d e^5 (-i+\tan (c+d x))} \]

[In]

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

[Out]

-1/616*((e*Sec[c + d*x])^(3/2)*(-148*Cos[c + d*x] + 34*Cos[3*(c + d*x)] + 2*Cos[5*(c + d*x)] + (240*I)*Sqrt[Co
s[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] + I*Sin[c + d*x]) + (78*I)*Sin[c + d*x] + (87*I)*Sin[3*(c
+ d*x)] + (9*I)*Sin[5*(c + d*x)]))/(a*d*e^5*(-I + Tan[c + d*x]))

Maple [A] (verified)

Time = 8.92 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.31

method	result	size
default	\(\frac {\frac {2 i \left (\cos ^{5}\left (d x +c \right )\right )}{11}+\frac {2 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{11}+\frac {30 i F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}{77}+\frac {18 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{77}+\frac {30 i \sec \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}{77}+\frac {30 \sin \left (d x +c \right )}{77}}{a d \sqrt {e \sec \left (d x +c \right )}\, e^{3}}\)	\(190\)

[In]

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

[Out]

2/77/a/d/(e*sec(d*x+c))^(1/2)/e^3*(7*I*cos(d*x+c)^5+7*sin(d*x+c)*cos(d*x+c)^4+15*I*EllipticF(I*(-csc(d*x+c)+co
t(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*(-csc(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 = 137, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=\frac {{\left (\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-11 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 121 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 70 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 226 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 53 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} - 480 i \, \sqrt {2} \sqrt {e} e^{\left (6 i \, d x + 6 i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{1232 \, a d e^{4}} \]

[In]

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

[Out]

1/1232*(sqrt(2)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*(-11*I*e^(10*I*d*x + 10*I*c) - 121*I*e^(8*I*d*x + 8*I*c) + 7
0*I*e^(6*I*d*x + 6*I*c) + 226*I*e^(4*I*d*x + 4*I*c) + 53*I*e^(2*I*d*x + 2*I*c) + 7*I)*e^(1/2*I*d*x + 1/2*I*c)
- 480*I*sqrt(2)*sqrt(e)*e^(6*I*d*x + 6*I*c)*weierstrassPInverse(-4, 0, e^(I*d*x + I*c)))*e^(-6*I*d*x - 6*I*c)/
(a*d*e^4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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