3.426 \(\int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=80 \[ \frac{4 i \sqrt{e \sec (c+d x)}}{5 a d \sqrt{a+i a \tan (c+d x)}}+\frac{2 i \sqrt{e \sec (c+d x)}}{5 d (a+i a \tan (c+d x))^{3/2}} \]

[Out]

(((2*I)/5)*Sqrt[e*Sec[c + d*x]])/(d*(a + I*a*Tan[c + d*x])^(3/2)) + (((4*I)/5)*Sqrt[e*Sec[c + d*x]])/(a*d*Sqrt
[a + I*a*Tan[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.144943, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3502, 3488} \[ \frac{4 i \sqrt{e \sec (c+d x)}}{5 a d \sqrt{a+i a \tan (c+d x)}}+\frac{2 i \sqrt{e \sec (c+d x)}}{5 d (a+i a \tan (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[e*Sec[c + d*x]]/(a + I*a*Tan[c + d*x])^(3/2),x]

[Out]

(((2*I)/5)*Sqrt[e*Sec[c + d*x]])/(d*(a + I*a*Tan[c + d*x])^(3/2)) + (((4*I)/5)*Sqrt[e*Sec[c + d*x]])/(a*d*Sqrt
[a + I*a*Tan[c + d*x]])

Rule 3502

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 3488

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

Rubi steps

\begin{align*} \int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^{3/2}} \, dx &=\frac{2 i \sqrt{e \sec (c+d x)}}{5 d (a+i a \tan (c+d x))^{3/2}}+\frac{2 \int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{5 a}\\ &=\frac{2 i \sqrt{e \sec (c+d x)}}{5 d (a+i a \tan (c+d x))^{3/2}}+\frac{4 i \sqrt{e \sec (c+d x)}}{5 a d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.120254, size = 63, normalized size = 0.79 \[ \frac{2 (3+2 i \tan (c+d x)) \sqrt{e \sec (c+d x)}}{5 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[e*Sec[c + d*x]]/(a + I*a*Tan[c + d*x])^(3/2),x]

[Out]

(2*Sqrt[e*Sec[c + d*x]]*(3 + (2*I)*Tan[c + d*x]))/(5*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])

________________________________________________________________________________________

Maple [A]  time = 0.308, size = 101, normalized size = 1.3 \begin{align*}{\frac{-{\frac{2\,i}{5}}\cos \left ( dx+c \right ) \left ( 2\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -2\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+2\,i\sin \left ( dx+c \right ) -\cos \left ( dx+c \right ) \right ) }{{a}^{2}d}\sqrt{{\frac{e}{\cos \left ( dx+c \right ) }}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*sec(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(3/2),x)

[Out]

-2/5*I/d/a^2*(e/cos(d*x+c))^(1/2)*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*cos(d*x+c)*(2*I*cos(d*x+c)^2*
sin(d*x+c)-2*cos(d*x+c)^3+2*I*sin(d*x+c)-cos(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 1.91847, size = 108, normalized size = 1.35 \begin{align*} \frac{\sqrt{e}{\left (i \, \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 i \, \cos \left (\frac{1}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right ) + \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, \sin \left (\frac{1}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right )\right )}}{5 \, a^{\frac{3}{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

1/5*sqrt(e)*(I*cos(5/2*d*x + 5/2*c) + 5*I*cos(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + sin(5
/2*d*x + 5/2*c) + 5*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))))/(a^(3/2)*d)

________________________________________________________________________________________

Fricas [A]  time = 2.10288, size = 219, normalized size = 2.74 \begin{align*} \frac{\sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (5 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 6 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-\frac{5}{2} i \, d x - \frac{5}{2} i \, c\right )}}{5 \, a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/5*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*(5*I*e^(4*I*d*x + 4*I*c) + 6*I*e^(2*I*
d*x + 2*I*c) + I)*e^(-5/2*I*d*x - 5/2*I*c)/(a^2*d)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e \sec{\left (c + d x \right )}}}{\left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))**(1/2)/(a+I*a*tan(d*x+c))**(3/2),x)

[Out]

Integral(sqrt(e*sec(c + d*x))/(a*(I*tan(c + d*x) + 1))**(3/2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e \sec \left (d x + c\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(e*sec(d*x + c))/(I*a*tan(d*x + c) + a)^(3/2), x)