\(\int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx\) [229]

Optimal result

Integrand size = 28, antiderivative size = 80 \[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i}{5 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \]

[Out]

6/5*(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)+2/5*I/d/(e*sec(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3583, 3856, 2719} \[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i}{5 d (a+i a \tan (c+d x)) \sqrt {e \sec (c+d x)}} \]

[In]

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

[Out]

(6*EllipticE[(c + d*x)/2, 2])/(5*a*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + ((2*I)/5)/(d*Sqrt[e*Sec[c + d*
x]]*(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 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}{5 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))}+\frac {3 \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 a} \\ & = \frac {2 i}{5 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))}+\frac {3 \int \sqrt {\cos (c+d x)} \, dx}{5 a \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = \frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i}{5 d \sqrt {e \sec (c+d x)} (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 = 1.57 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {\left (4+4 \cos (2 (c+d x))-2 e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+3 i \sin (2 (c+d x))\right ) (i+\tan (c+d x))}{5 a d \sqrt {e \sec (c+d x)}} \]

[In]

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

[Out]

((4 + 4*Cos[2*(c + d*x)] - 2*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4
, -E^((2*I)*(c + d*x))] + (3*I)*Sin[2*(c + d*x)])*(I + Tan[c + d*x]))/(5*a*d*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. 439 vs. \(2 (94 ) = 188\).

Time = 7.54 (sec) , antiderivative size = 440, normalized size of antiderivative = 5.50

method	result	size
default	\(-\frac {2 i \left (i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-3 \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+i \cos \left (d x +c \right ) \sin \left (d x +c \right )-\left (\cos ^{3}\left (d x +c \right )\right )+6 \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-6 F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+3 i \sin \left (d x +c \right )-\left (\cos ^{2}\left (d x +c \right )\right )+3 \sec \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-3 \sec \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )}{5 a d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}}\)	\(440\)

[In]

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

[Out]

-2/5*I/a/d/(cos(d*x+c)+1)/(e*sec(d*x+c))^(1/2)*(I*cos(d*x+c)^2*sin(d*x+c)+3*(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)-3*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2
)*EllipticF(I*(-csc(d*x+c)+cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+I*sin(d*x+c)*cos(d*x+c)-cos(d*x+
c)^3+6*(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)-6*El
lipticF(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)+3*I*sin(d*x+c
)-cos(d*x+c)^2+3*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)-3*sec(d*x+c)*EllipticF(I*(-csc(d*x+c)+cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(co
s(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 = 107, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {{\left (\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (7 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 8 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 12 i \, \sqrt {2} \sqrt {e} e^{\left (3 i \, d x + 3 i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{10 \, a d e} \]

[In]

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

[Out]

1/10*(sqrt(2)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*(7*I*e^(4*I*d*x + 4*I*c) + 8*I*e^(2*I*d*x + 2*I*c) + I)*e^(1/2
*I*d*x + 1/2*I*c) + 12*I*sqrt(2)*sqrt(e)*e^(3*I*d*x + 3*I*c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0,
e^(I*d*x + I*c))))*e^(-3*I*d*x - 3*I*c)/(a*d*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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