3.908 \(\int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^3 \, dx\)

Optimal. Leaf size=25 \[ \frac{i a (c-i c \tan (e+f x))^3}{3 f} \]

[Out]

((I/3)*a*(c - I*c*Tan[e + f*x])^3)/f

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Rubi [A]  time = 0.0705975, 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} \[ \frac{i a (c-i c \tan (e+f x))^3}{3 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3,x]

[Out]

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

Mathematica [B]  time = 0.168718, size = 55, normalized size = 2.2 \[ \frac{a c^3 \left (-\tan ^3(e+f x)-3 i \tan ^2(e+f x)-3 \tan ^{-1}(\tan (e+f x))+3 \tan (e+f x)+3 f x\right )}{3 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3,x]

[Out]

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

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Maple [A]  time = 0.004, size = 37, normalized size = 1.5 \begin{align*}{\frac{a{c}^{3}}{f} \left ( \tan \left ( fx+e \right ) -{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{3}}-i \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) } \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))^3,x)

[Out]

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

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Maxima [B]  time = 1.66708, size = 61, normalized size = 2.44 \begin{align*} -\frac{a c^{3} \tan \left (f x + e\right )^{3} + 3 i \, a c^{3} \tan \left (f x + e\right )^{2} - 3 \, a c^{3} \tan \left (f x + e\right )}{3 \, 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))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.31671, size = 126, normalized size = 5.04 \begin{align*} \frac{8 i \, a c^{3}}{3 \,{\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\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))^3,x, algorithm="fricas")

[Out]

8/3*I*a*c^3/(f*e^(6*I*f*x + 6*I*e) + 3*f*e^(4*I*f*x + 4*I*e) + 3*f*e^(2*I*f*x + 2*I*e) + f)

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Sympy [B]  time = 1.56655, size = 68, normalized size = 2.72 \begin{align*} \frac{8 i a c^{3} e^{- 6 i e}}{3 f \left (e^{6 i f x} + 3 e^{- 2 i e} e^{4 i f x} + 3 e^{- 4 i e} e^{2 i f x} + e^{- 6 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))**3,x)

[Out]

8*I*a*c**3*exp(-6*I*e)/(3*f*(exp(6*I*f*x) + 3*exp(-2*I*e)*exp(4*I*f*x) + 3*exp(-4*I*e)*exp(2*I*f*x) + exp(-6*I
*e)))

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Giac [B]  time = 1.53163, size = 65, normalized size = 2.6 \begin{align*} \frac{8 i \, a c^{3}}{3 \,{\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\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))^3,x, algorithm="giac")

[Out]

8/3*I*a*c^3/(f*e^(6*I*f*x + 6*I*e) + 3*f*e^(4*I*f*x + 4*I*e) + 3*f*e^(2*I*f*x + 2*I*e) + f)