3.892 \(\int (a+i a \tan (e+f x)) (c-i c \tan (e+f x)) \, dx\)

Optimal. Leaf size=12 \[ \frac{a c \tan (e+f x)}{f} \]

[Out]

(a*c*Tan[e + f*x])/f

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Rubi [A]  time = 0.0239701, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3522, 3767, 8} \[ \frac{a c \tan (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*c*Tan[e + f*x])/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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.0091411, size = 12, normalized size = 1. \[ \frac{a c \tan (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*c*Tan[e + f*x])/f

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Maple [A]  time = 0.003, size = 13, normalized size = 1.1 \begin{align*}{\frac{ac\tan \left ( fx+e \right ) }{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*tan(f*x+e)/f

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Maxima [A]  time = 1.57676, size = 16, normalized size = 1.33 \begin{align*} \frac{a c \tan \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*tan(f*x + e)/f

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Fricas [C]  time = 1.06589, size = 50, normalized size = 4.17 \begin{align*} \frac{2 i \, a c}{f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I*a*c/(f*e^(2*I*f*x + 2*I*e) + f)

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Sympy [C]  time = 0.368875, size = 31, normalized size = 2.58 \begin{align*} \frac{2 i a c e^{- 2 i e}}{f \left (e^{2 i f x} + e^{- 2 i e}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I*a*c*exp(-2*I*e)/(f*(exp(2*I*f*x) + exp(-2*I*e)))

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Giac [A]  time = 1.36223, size = 18, normalized size = 1.5 \begin{align*} \frac{a c \tan \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*tan(f*x + e)/f