\(\int \frac {(e \sec (c+d x))^{7/2}}{a+i a \tan (c+d x)} \, dx\) [225]

Optimal result

Integrand size = 28, antiderivative size = 101 \[ \int \frac {(e \sec (c+d x))^{7/2}}{a+i a \tan (c+d x)} \, dx=-\frac {2 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i e^2 (e \sec (c+d x))^{3/2}}{3 a d}+\frac {2 e^3 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a d} \]

[Out]

-2/3*I*e^2*(e*sec(d*x+c))^(3/2)/a/d-2*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))/a/d/cos(d*x+c)^(1/2)/(e*sec(d*x+c))^(1/2)+2*e^3*sin(d*x+c)*(e*sec(d*x+c))^(1/2)/a/d

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 101, 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 = {3582, 3853, 3856, 2719} \[ \int \frac {(e \sec (c+d x))^{7/2}}{a+i a \tan (c+d x)} \, dx=-\frac {2 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 e^3 \sin (c+d x) \sqrt {e \sec (c+d x)}}{a d}-\frac {2 i e^2 (e \sec (c+d x))^{3/2}}{3 a d} \]

[In]

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

[Out]

(-2*e^4*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (((2*I)/3)*e^2*(e*Sec[c + d
*x])^(3/2))/(a*d) + (2*e^3*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(a*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 3582

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d^2*(
d*Sec[e + f*x])^(m - 2)*((a + b*Tan[e + f*x])^(n + 1)/(b*f*(m + n - 1))), x] + Dist[d^2*((m - 2)/(a*(m + n - 1
))), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2
+ b^2, 0] && LtQ[n, 0] && GtQ[m, 1] &&  !ILtQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

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

Mathematica [C] (verified)

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

Time = 1.75 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.01 \[ \int \frac {(e \sec (c+d x))^{7/2}}{a+i a \tan (c+d x)} \, dx=\frac {2 i e^3 \sqrt {e \sec (c+d x)} (\cos (c)+i \sin (c)) (\cos (d x)+i \sin (d x)) \left (-4+\sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+i \tan (c+d x)\right )}{3 a d} \]

[In]

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

[Out]

(((2*I)/3)*e^3*Sqrt[e*Sec[c + d*x]]*(Cos[c] + I*Sin[c])*(Cos[d*x] + I*Sin[d*x])*(-4 + Sqrt[1 + E^((2*I)*(c + d
*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + I*Tan[c + d*x]))/(a*d)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (116 ) = 232\).

Time = 6.54 (sec) , antiderivative size = 410, normalized size of antiderivative = 4.06

method	result	size
default	\(\frac {2 e^{3} \sqrt {e \sec \left (d x +c \right )}\, \left (3 i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )-3 i E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )+6 i \cos \left (d x +c \right ) F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-6 i \cos \left (d x +c \right ) E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+3 i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-3 i E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-i+3 \sin \left (d x +c \right )-i \sec \left (d x +c \right )\right )}{3 a d \left (\cos \left (d x +c \right )+1\right )}\)	\(410\)

[In]

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

[Out]

2/3*e^3/a/d*(e*sec(d*x+c))^(1/2)/(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)-I+3*sin(d*x+c)-I*sec(d*x+c))

Fricas [C] (verification not implemented)

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

Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.22 \[ \int \frac {(e \sec (c+d x))^{7/2}}{a+i a \tan (c+d x)} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (3 i \, e^{3} e^{\left (3 i \, d x + 3 i \, c\right )} + 5 i \, e^{3} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 3 \, \sqrt {2} {\left (i \, e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{3 \, {\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]

[In]

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

[Out]

-2/3*(sqrt(2)*(3*I*e^3*e^(3*I*d*x + 3*I*c) + 5*I*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) + 3*sqrt(2)*(I*e^3*e^(2*I*d*x + 2*I*c) + I*e^3)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPI
nverse(-4, 0, e^(I*d*x + I*c))))/(a*d*e^(2*I*d*x + 2*I*c) + a*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((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]

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

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

[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)