3.911 \(\int \frac{(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=50 \[ \frac{i c^3 \left (a^2-i a^2 \tan (e+f x)\right )^3}{6 f \left (a^3+i a^3 \tan (e+f x)\right )^3} \]

[Out]

((I/6)*c^3*(a^2 - I*a^2*Tan[e + f*x])^3)/(f*(a^3 + I*a^3*Tan[e + f*x])^3)

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Rubi [A]  time = 0.105004, antiderivative size = 50, 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, 3487, 37} \[ \frac{i c^3 \left (a^2-i a^2 \tan (e+f x)\right )^3}{6 f \left (a^3+i a^3 \tan (e+f x)\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/6)*c^3*(a^2 - I*a^2*Tan[e + f*x])^3)/(f*(a^3 + I*a^3*Tan[e + f*x])^3)

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx &=\left (a^3 c^3\right ) \int \frac{\sec ^6(e+f x)}{(a+i a \tan (e+f x))^6} \, dx\\ &=-\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{(a-x)^2}{(a+x)^4} \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=\frac{i c^3 (1-i \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}\\ \end{align*}

Mathematica [A]  time = 0.243045, size = 34, normalized size = 0.68 \[ \frac{c^3 (\sin (6 (e+f x))+i \cos (6 (e+f x)))}{6 a^3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(I*Cos[6*(e + f*x)] + Sin[6*(e + f*x)]))/(6*a^3*f)

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Maple [A]  time = 0.028, size = 50, normalized size = 1. \begin{align*}{\frac{{c}^{3}}{f{a}^{3}} \left ( \left ( \tan \left ( fx+e \right ) -i \right ) ^{-1}+{\frac{2\,i}{ \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{4}{3\, \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-I*c*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^3,x)

[Out]

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

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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))^3/(a+I*a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.42533, size = 54, normalized size = 1.08 \begin{align*} \frac{i \, c^{3} e^{\left (-6 i \, f x - 6 i \, e\right )}}{6 \, a^{3} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/6*I*c^3*e^(-6*I*f*x - 6*I*e)/(a^3*f)

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Sympy [A]  time = 0.640469, size = 53, normalized size = 1.06 \begin{align*} \begin{cases} \frac{i c^{3} e^{- 6 i e} e^{- 6 i f x}}{6 a^{3} f} & \text{for}\: 6 a^{3} f e^{6 i e} \neq 0 \\\frac{c^{3} x e^{- 6 i e}}{a^{3}} & \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))**3/(a+I*a*tan(f*x+e))**3,x)

[Out]

Piecewise((I*c**3*exp(-6*I*e)*exp(-6*I*f*x)/(6*a**3*f), Ne(6*a**3*f*exp(6*I*e), 0)), (c**3*x*exp(-6*I*e)/a**3,
True))

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Giac [A]  time = 1.52313, size = 97, normalized size = 1.94 \begin{align*} -\frac{2 \,{\left (3 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 10 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 3 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{3 \, a^{3} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

-2/3*(3*c^3*tan(1/2*f*x + 1/2*e)^5 - 10*c^3*tan(1/2*f*x + 1/2*e)^3 + 3*c^3*tan(1/2*f*x + 1/2*e))/(a^3*f*(tan(1
/2*f*x + 1/2*e) - I)^6)