3.894 \(\int \frac{c-i c \tan (e+f x)}{(a+i a \tan (e+f x))^2} \, dx\)

Optimal. Leaf size=25 \[ \frac{i c}{2 f (a+i a \tan (e+f x))^2} \]

[Out]

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

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Rubi [A]  time = 0.074293, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3522, 3487, 32} \[ \frac{i c}{2 f (a+i a \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{c-i c \tan (e+f x)}{(a+i a \tan (e+f x))^2} \, dx &=(a c) \int \frac{\sec ^2(e+f x)}{(a+i a \tan (e+f x))^3} \, dx\\ &=-\frac{(i c) \operatorname{Subst}\left (\int \frac{1}{(a+x)^3} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac{i c}{2 f (a+i a \tan (e+f x))^2}\\ \end{align*}

Mathematica [A]  time = 0.795433, size = 45, normalized size = 1.8 \[ \frac{(\tan (e+f x)-3 i) (c-i c \tan (e+f x))}{8 a^2 f (\tan (e+f x)-i)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*I + Tan[e + f*x])*(c - I*c*Tan[e + f*x]))/(8*a^2*f*(-I + Tan[e + f*x])^2)

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Maple [A]  time = 0.024, size = 22, normalized size = 0.9 \begin{align*}{\frac{-{\frac{i}{2}}c}{f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-I*c*tan(f*x+e))/(a+I*a*tan(f*x+e))^2,x)

[Out]

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

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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((c-I*c*tan(f*x+e))/(a+I*a*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.09435, size = 92, normalized size = 3.68 \begin{align*} \frac{{\left (2 i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{8 \, a^{2} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))/(a+I*a*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.411576, size = 102, normalized size = 4.08 \begin{align*} \begin{cases} \frac{\left (8 i a^{2} c f e^{4 i e} e^{- 2 i f x} + 4 i a^{2} c f e^{2 i e} e^{- 4 i f x}\right ) e^{- 6 i e}}{32 a^{4} f^{2}} & \text{for}\: 32 a^{4} f^{2} e^{6 i e} \neq 0 \\\frac{x \left (c e^{2 i e} + c\right ) e^{- 4 i e}}{2 a^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))/(a+I*a*tan(f*x+e))**2,x)

[Out]

Piecewise(((8*I*a**2*c*f*exp(4*I*e)*exp(-2*I*f*x) + 4*I*a**2*c*f*exp(2*I*e)*exp(-4*I*f*x))*exp(-6*I*e)/(32*a**
4*f**2), Ne(32*a**4*f**2*exp(6*I*e), 0)), (x*(c*exp(2*I*e) + c)*exp(-4*I*e)/(2*a**2), True))

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Giac [B]  time = 1.37751, size = 88, normalized size = 3.52 \begin{align*} -\frac{2 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - i \, c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{2} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))/(a+I*a*tan(f*x+e))^2,x, algorithm="giac")

[Out]

-2*(c*tan(1/2*f*x + 1/2*e)^3 - I*c*tan(1/2*f*x + 1/2*e)^2 - c*tan(1/2*f*x + 1/2*e))/(a^2*f*(tan(1/2*f*x + 1/2*
e) - I)^4)