3.1068 \(\int \frac{c+d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx\)

Optimal. Leaf size=47 \[ \frac{-d+i c}{2 f (a+i a \tan (e+f x))}+\frac{x (c-i d)}{2 a} \]

[Out]

((c - I*d)*x)/(2*a) + (I*c - d)/(2*f*(a + I*a*Tan[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.042418, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3526, 8} \[ \frac{-d+i c}{2 f (a+i a \tan (e+f x))}+\frac{x (c-i d)}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c - I*d)*x)/(2*a) + (I*c - d)/(2*f*(a + I*a*Tan[e + f*x]))

Rule 3526

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

Rule 8

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

Rubi steps

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

Mathematica [B]  time = 0.549965, size = 102, normalized size = 2.17 \[ \frac{\cos (e+f x) (c+d \tan (e+f x)) ((c (2 f x-i)-2 i d f x+d) \tan (e+f x)-2 i c f x+c+d (-2 f x+i))}{4 a f (\tan (e+f x)-i) (c \cos (e+f x)+d \sin (e+f x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[e + f*x]*(c + d*Tan[e + f*x])*(c - (2*I)*c*f*x + d*(I - 2*f*x) + (d - (2*I)*d*f*x + c*(-I + 2*f*x))*Tan[e
+ f*x]))/(4*a*f*(c*Cos[e + f*x] + d*Sin[e + f*x])*(-I + Tan[e + f*x]))

________________________________________________________________________________________

Maple [B]  time = 0.028, size = 121, normalized size = 2.6 \begin{align*}{\frac{-{\frac{i}{4}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) c}{af}}-{\frac{\ln \left ( \tan \left ( fx+e \right ) -i \right ) d}{4\,af}}+{\frac{c}{2\,af \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{i}{2}}d}{af \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{\ln \left ( \tan \left ( fx+e \right ) +i \right ) d}{4\,af}}+{\frac{{\frac{i}{4}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) c}{af}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*I/f/a*ln(tan(f*x+e)-I)*c-1/4/f/a*ln(tan(f*x+e)-I)*d+1/2/f*c/a/(tan(f*x+e)-I)+1/2*I/f/a/(tan(f*x+e)-I)*d+1
/4/f/a*ln(tan(f*x+e)+I)*d+1/4*I/f/a*ln(tan(f*x+e)+I)*c

________________________________________________________________________________________

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 1.58086, size = 108, normalized size = 2.3 \begin{align*} \frac{{\left (2 \,{\left (c - i \, d\right )} f x e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.3799, size = 88, normalized size = 1.87 \begin{align*} \begin{cases} \frac{\left (i c - d\right ) e^{- 2 i e} e^{- 2 i f x}}{4 a f} & \text{for}\: 4 a f e^{2 i e} \neq 0 \\x \left (- \frac{c - i d}{2 a} + \frac{\left (c e^{2 i e} + c - i d e^{2 i e} + i d\right ) e^{- 2 i e}}{2 a}\right ) & \text{otherwise} \end{cases} + \frac{x \left (c - i d\right )}{2 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((I*c - d)*exp(-2*I*e)*exp(-2*I*f*x)/(4*a*f), Ne(4*a*f*exp(2*I*e), 0)), (x*(-(c - I*d)/(2*a) + (c*ex
p(2*I*e) + c - I*d*exp(2*I*e) + I*d)*exp(-2*I*e)/(2*a)), True)) + x*(c - I*d)/(2*a)

________________________________________________________________________________________

Giac [B]  time = 1.3945, size = 122, normalized size = 2.6 \begin{align*} -\frac{\frac{{\left (i \, c + d\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a} + \frac{{\left (-i \, c - d\right )} \log \left (-i \, \tan \left (f x + e\right ) + 1\right )}{a} + \frac{-i \, c \tan \left (f x + e\right ) - d \tan \left (f x + e\right ) - 3 \, c - i \, d}{a{\left (\tan \left (f x + e\right ) - i\right )}}}{4 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((I*c + d)*log(tan(f*x + e) - I)/a + (-I*c - d)*log(-I*tan(f*x + e) + 1)/a + (-I*c*tan(f*x + e) - d*tan(f
*x + e) - 3*c - I*d)/(a*(tan(f*x + e) - I)))/f