3.928 \(\int \frac{1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx\)

Optimal. Leaf size=37 \[ \frac{\sin (e+f x) \cos (e+f x)}{2 a c f}+\frac{x}{2 a c} \]

[Out]

x/(2*a*c) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*c*f)

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Rubi [A]  time = 0.0725315, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 2635, 8} \[ \frac{\sin (e+f x) \cos (e+f x)}{2 a c f}+\frac{x}{2 a c} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])),x]

[Out]

x/(2*a*c) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*c*f)

Rule 3522

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Sec[e + f*x]^(2*m)*(c + d*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0] && IntegerQ[m] &&  !(IGtQ[n, 0] && (LtQ[m, 0] || GtQ[m, n]))

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx &=\frac{\int \cos ^2(e+f x) \, dx}{a c}\\ &=\frac{\cos (e+f x) \sin (e+f x)}{2 a c f}+\frac{\int 1 \, dx}{2 a c}\\ &=\frac{x}{2 a c}+\frac{\cos (e+f x) \sin (e+f x)}{2 a c f}\\ \end{align*}

Mathematica [A]  time = 0.0283078, size = 29, normalized size = 0.78 \[ \frac{2 (e+f x)+\sin (2 (e+f x))}{4 a c f} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])),x]

[Out]

(2*(e + f*x) + Sin[2*(e + f*x)])/(4*a*c*f)

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Maple [C]  time = 0.036, size = 90, normalized size = 2.4 \begin{align*}{\frac{-{\frac{i}{4}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) }{fac}}+{\frac{1}{4\,fac \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{i}{4}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) }{fac}}+{\frac{1}{4\,fac \left ( \tan \left ( fx+e \right ) +i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+I*a*tan(f*x+e))/(c-I*c*tan(f*x+e)),x)

[Out]

-1/4*I/f/a/c*ln(tan(f*x+e)-I)+1/4/f/a/c/(tan(f*x+e)-I)+1/4*I/f/a/c*ln(tan(f*x+e)+I)+1/4/f/a/c/(tan(f*x+e)+I)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))/(c-I*c*tan(f*x+e)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [C]  time = 1.37227, size = 122, normalized size = 3.3 \begin{align*} \frac{{\left (4 \, f x e^{\left (2 i \, f x + 2 i \, e\right )} - i \, e^{\left (4 i \, f x + 4 i \, e\right )} + i\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a c f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))/(c-I*c*tan(f*x+e)),x, algorithm="fricas")

[Out]

1/8*(4*f*x*e^(2*I*f*x + 2*I*e) - I*e^(4*I*f*x + 4*I*e) + I)*e^(-2*I*f*x - 2*I*e)/(a*c*f)

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Sympy [A]  time = 0.623378, size = 119, normalized size = 3.22 \begin{align*} \begin{cases} \frac{\left (- 8 i a c f e^{4 i e} e^{2 i f x} + 8 i a c f e^{- 2 i f x}\right ) e^{- 2 i e}}{64 a^{2} c^{2} f^{2}} & \text{for}\: 64 a^{2} c^{2} f^{2} e^{2 i e} \neq 0 \\x \left (\frac{\left (e^{4 i e} + 2 e^{2 i e} + 1\right ) e^{- 2 i e}}{4 a c} - \frac{1}{2 a c}\right ) & \text{otherwise} \end{cases} + \frac{x}{2 a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))/(c-I*c*tan(f*x+e)),x)

[Out]

Piecewise(((-8*I*a*c*f*exp(4*I*e)*exp(2*I*f*x) + 8*I*a*c*f*exp(-2*I*f*x))*exp(-2*I*e)/(64*a**2*c**2*f**2), Ne(
64*a**2*c**2*f**2*exp(2*I*e), 0)), (x*((exp(4*I*e) + 2*exp(2*I*e) + 1)*exp(-2*I*e)/(4*a*c) - 1/(2*a*c)), True)
) + x/(2*a*c)

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Giac [A]  time = 1.43097, size = 62, normalized size = 1.68 \begin{align*} \frac{\frac{f x + e}{a c} + \frac{\tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )} a c}}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))/(c-I*c*tan(f*x+e)),x, algorithm="giac")

[Out]

1/2*((f*x + e)/(a*c) + tan(f*x + e)/((tan(f*x + e)^2 + 1)*a*c))/f