\(\int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \, dx\) [231]

Optimal result

Integrand size = 28, antiderivative size = 114 \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \, dx=\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \sec (c+d x))^{3/2}}+\frac {2 i}{9 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \]

[Out]

14/45*sin(d*x+c)/a/d/e/(e*sec(d*x+c))^(3/2)+14/15*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(si
n(1/2*d*x+1/2*c),2^(1/2))/a/d/e^2/cos(d*x+c)^(1/2)/(e*sec(d*x+c))^(1/2)+2/9*I/d/(e*sec(d*x+c))^(5/2)/(a+I*a*ta
n(d*x+c))

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3583, 3854, 3856, 2719} \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \, dx=\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \sec (c+d x))^{3/2}}+\frac {2 i}{9 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{5/2}} \]

[In]

Int[1/((e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])),x]

[Out]

(14*EllipticE[(c + d*x)/2, 2])/(15*a*d*e^2*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (14*Sin[c + d*x])/(45*a*
d*e*(e*Sec[c + d*x])^(3/2)) + ((2*I)/9)/(d*(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x]))

Rule 2719

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]

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}{9 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))}+\frac {7 \int \frac {1}{(e \sec (c+d x))^{5/2}} \, dx}{9 a} \\ & = \frac {14 \sin (c+d x)}{45 a d e (e \sec (c+d x))^{3/2}}+\frac {2 i}{9 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))}+\frac {7 \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{15 a e^2} \\ & = \frac {14 \sin (c+d x)}{45 a d e (e \sec (c+d x))^{3/2}}+\frac {2 i}{9 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))}+\frac {7 \int \sqrt {\cos (c+d x)} \, dx}{15 a e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = \frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \sec (c+d x))^{3/2}}+\frac {2 i}{9 d (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 2.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \, dx=\frac {\left (106+104 \cos (2 (c+d x))-2 \cos (4 (c+d x))-56 e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+70 i \sin (2 (c+d x))-7 i \sin (4 (c+d x))\right ) (i+\tan (c+d x))}{180 a d e^2 \sqrt {e \sec (c+d x)}} \]

[In]

Integrate[1/((e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])),x]

[Out]

((106 + 104*Cos[2*(c + d*x)] - 2*Cos[4*(c + d*x)] - 56*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hyper
geometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + (70*I)*Sin[2*(c + d*x)] - (7*I)*Sin[4*(c + d*x)])*(I + Tan
[c + d*x]))/(180*a*d*e^2*Sqrt[e*Sec[c + d*x]])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (124 ) = 248\).

Time = 10.56 (sec) , antiderivative size = 477, normalized size of antiderivative = 4.18

method	result	size
default	\(-\frac {2 i \left (5 i \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+5 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )-5 \left (\cos ^{5}\left (d x +c \right )\right )+7 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-5 \left (\cos ^{4}\left (d x +c \right )\right )-21 \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, 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}}\, \cos \left (d x +c \right )+21 \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\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}}\, \cos \left (d x +c \right )+7 i \cos \left (d x +c \right ) \sin \left (d x +c \right )-42 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}}+42 \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\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}}+21 i \sin \left (d x +c \right )-21 \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}}+21 \sec \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\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 )}{45 a d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, e^{2}}\)	\(477\)

[In]

int(1/(e*sec(d*x+c))^(5/2)/(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-2/45*I/a/d/(cos(d*x+c)+1)/(e*sec(d*x+c))^(1/2)/e^2*(5*I*sin(d*x+c)*cos(d*x+c)^4+5*I*cos(d*x+c)^3*sin(d*x+c)-5
*cos(d*x+c)^5+7*I*cos(d*x+c)^2*sin(d*x+c)-5*cos(d*x+c)^4-21*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(I*(-cs
c(d*x+c)+cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+21*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(I*(
-csc(d*x+c)+cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+7*I*cos(d*x+c)*sin(d*x+c)-42*EllipticF(I*(-csc(
d*x+c)+cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+42*(cos(d*x+c)/(cos(d*x+c)+1)
)^(1/2)*EllipticE(I*(-csc(d*x+c)+cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)+21*I*sin(d*x+c)-21*sec(d*x+c)*Ellipti
cF(I*(-csc(d*x+c)+cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+21*sec(d*x+c)*(cos
(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(I*(-csc(d*x+c)+cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2))

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \, dx=\frac {{\left (\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-9 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 174 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 212 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 34 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 336 i \, \sqrt {2} \sqrt {e} e^{\left (5 i \, d x + 5 i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{360 \, a d e^{3}} \]

[In]

integrate(1/(e*sec(d*x+c))^(5/2)/(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/360*(sqrt(2)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*(-9*I*e^(8*I*d*x + 8*I*c) + 174*I*e^(6*I*d*x + 6*I*c) + 212*I
*e^(4*I*d*x + 4*I*c) + 34*I*e^(2*I*d*x + 2*I*c) + 5*I)*e^(1/2*I*d*x + 1/2*I*c) + 336*I*sqrt(2)*sqrt(e)*e^(5*I*
d*x + 5*I*c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, e^(I*d*x + I*c))))*e^(-5*I*d*x - 5*I*c)/(a*d*e^
3)

Sympy [F]

\[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \, dx=- \frac {i \int \frac {1}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \tan {\left (c + d x \right )} - i \left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx}{a} \]

[In]

integrate(1/(e*sec(d*x+c))**(5/2)/(a+I*a*tan(d*x+c)),x)

[Out]

-I*Integral(1/((e*sec(c + d*x))**(5/2)*tan(c + d*x) - I*(e*sec(c + d*x))**(5/2)), x)/a

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/(e*sec(d*x+c))^(5/2)/(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]

\[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \, dx=\int { \frac {1}{\left (e \sec \left (d x + c\right )\right )^{\frac {5}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(e*sec(d*x+c))^(5/2)/(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate(1/((e*sec(d*x + c))^(5/2)*(I*a*tan(d*x + c) + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))} \, dx=\int \frac {1}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

int(1/((e/cos(c + d*x))^(5/2)*(a + a*tan(c + d*x)*1i)),x)

[Out]

int(1/((e/cos(c + d*x))^(5/2)*(a + a*tan(c + d*x)*1i)), x)