\(\int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx\) [185]

Optimal result

Integrand size = 26, antiderivative size = 123 \[ \int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=-\frac {6 a e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}+\frac {6 a e^3 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a e (e \sec (c+d x))^{5/2} \sin (c+d x)}{5 d} \]

[Out]

2/7*I*a*(e*sec(d*x+c))^(7/2)/d+2/5*a*e*(e*sec(d*x+c))^(5/2)*sin(d*x+c)/d-6/5*a*e^4*(cos(1/2*d*x+1/2*c)^2)^(1/2
)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/cos(d*x+c)^(1/2)/(e*sec(d*x+c))^(1/2)+6/5*a*e^3*s
in(d*x+c)*(e*sec(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3567, 3853, 3856, 2719} \[ \int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=-\frac {6 a e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {6 a e^3 \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 d}+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}+\frac {2 a e \sin (c+d x) (e \sec (c+d x))^{5/2}}{5 d} \]

[In]

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

[Out]

(-6*a*e^4*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((2*I)/7)*a*(e*Sec[c + d
*x])^(7/2))/d + (6*a*e^3*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*e*(e*Sec[c + d*x])^(5/2)*Sin[c + d*x]
)/(5*d)

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 3567

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*((d*Sec[
e + f*x])^m/(f*m)), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

Rule 3853

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] + Dist[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]

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 a (e \sec (c+d x))^{7/2}}{7 d}+a \int (e \sec (c+d x))^{7/2} \, dx \\ & = \frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}+\frac {2 a e (e \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {1}{5} \left (3 a e^2\right ) \int (e \sec (c+d x))^{3/2} \, dx \\ & = \frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}+\frac {6 a e^3 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a e (e \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac {1}{5} \left (3 a e^4\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx \\ & = \frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}+\frac {6 a e^3 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a e (e \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac {\left (3 a e^4\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = -\frac {6 a e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i a (e \sec (c+d x))^{7/2}}{7 d}+\frac {6 a e^3 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a e (e \sec (c+d x))^{5/2} \sin (c+d x)}{5 d} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 2.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.27 \[ \int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\frac {a e e^{-i d x} (e \sec (c+d x))^{5/2} (\cos (d x)-i \sin (d x)) (\cos (c+3 d x)+i \sin (c+3 d x)) \left (-36 i-28 i \cos (2 (c+d x))+7 i e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+7 \sec (c+d x) \sin (3 (c+d x))+27 \tan (c+d x)\right )}{70 d} \]

[In]

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

[Out]

(a*e*(e*Sec[c + d*x])^(5/2)*(Cos[d*x] - I*Sin[d*x])*(Cos[c + 3*d*x] + I*Sin[c + 3*d*x])*(-36*I - (28*I)*Cos[2*
(c + d*x)] + ((7*I)*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^
((2*I)*(c + d*x)) + 7*Sec[c + d*x]*Sin[3*(c + d*x)] + 27*Tan[c + d*x]))/(70*d*E^(I*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. 433 vs. \(2 (130 ) = 260\).

Time = 27.13 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.53

method	result	size
default	\(\frac {2 a \sqrt {e \sec \left (d x +c \right )}\, e^{3} \left (3 i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )-3 i E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )+6 i \cos \left (d x +c \right ) F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-6 i \cos \left (d x +c \right ) E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+3 i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-3 i E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+3 \sin \left (d x +c \right )+\tan \left (d x +c \right )+\sec \left (d x +c \right ) \tan \left (d x +c \right )\right )}{5 d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 i a \left (e \sec \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d}\)	\(434\)
parts	\(\frac {2 a \sqrt {e \sec \left (d x +c \right )}\, e^{3} \left (3 i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )-3 i E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )+6 i \cos \left (d x +c \right ) F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-6 i \cos \left (d x +c \right ) E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+3 i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-3 i E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+3 \sin \left (d x +c \right )+\tan \left (d x +c \right )+\sec \left (d x +c \right ) \tan \left (d x +c \right )\right )}{5 d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 i a \left (e \sec \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d}\)	\(434\)

[In]

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

[Out]

2/5*a/d*(e*sec(d*x+c))^(1/2)*e^3/(cos(d*x+c)+1)*(3*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)*cos(d*x+c)^2-3*I*EllipticE(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)*cos(d*x+c)^2+6*I*cos(d*x+c)*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)-6*I*cos(d*x+c)*EllipticE(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)+3*I*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),I)*(c
os(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)-3*I*EllipticE(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)+3*sin(d*x+c)+tan(d*x+c)+sec(d*x+c)*tan(d*x+c))+2/7*I*a*(e*sec
(d*x+c))^(7/2)/d

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.69 \[ \int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (21 i \, a e^{3} e^{\left (7 i \, d x + 7 i \, c\right )} + 77 i \, a e^{3} e^{\left (5 i \, d x + 5 i \, c\right )} + 23 i \, a e^{3} e^{\left (3 i \, d x + 3 i \, c\right )} + 7 i \, a e^{3} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 21 \, \sqrt {2} {\left (i \, a e^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 i \, a e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, a e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a e^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{35 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

-2/35*(sqrt(2)*(21*I*a*e^3*e^(7*I*d*x + 7*I*c) + 77*I*a*e^3*e^(5*I*d*x + 5*I*c) + 23*I*a*e^3*e^(3*I*d*x + 3*I*
c) + 7*I*a*e^3*e^(I*d*x + I*c))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) + 21*sqrt(2)*(I*a*e^
3*e^(6*I*d*x + 6*I*c) + 3*I*a*e^3*e^(4*I*d*x + 4*I*c) + 3*I*a*e^3*e^(2*I*d*x + 2*I*c) + I*a*e^3)*sqrt(e)*weier
strassZeta(-4, 0, weierstrassPInverse(-4, 0, e^(I*d*x + I*c))))/(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*
c) + 3*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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