3.949 \(\int \frac{(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx\)

Optimal. Leaf size=50 \[ -\frac{i a^4 \left (c^2+i c^2 \tan (e+f x)\right )^4}{8 f \left (c^3-i c^3 \tan (e+f x)\right )^4} \]

[Out]

((-I/8)*a^4*(c^2 + I*c^2*Tan[e + f*x])^4)/(f*(c^3 - I*c^3*Tan[e + f*x])^4)

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Rubi [A]  time = 0.104294, 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} \[ -\frac{i a^4 \left (c^2+i c^2 \tan (e+f x)\right )^4}{8 f \left (c^3-i c^3 \tan (e+f x)\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x])^4,x]

[Out]

((-I/8)*a^4*(c^2 + I*c^2*Tan[e + f*x])^4)/(f*(c^3 - I*c^3*Tan[e + f*x])^4)

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{(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx &=\left (a^4 c^4\right ) \int \frac{\sec ^8(e+f x)}{(c-i c \tan (e+f x))^8} \, dx\\ &=\frac{\left (i a^4\right ) \operatorname{Subst}\left (\int \frac{(c-x)^3}{(c+x)^5} \, dx,x,-i c \tan (e+f x)\right )}{c^3 f}\\ &=-\frac{i a^4 (c+i c \tan (e+f x))^4}{8 f \left (c^2-i c^2 \tan (e+f x)\right )^4}\\ \end{align*}

Mathematica [A]  time = 0.347583, size = 34, normalized size = 0.68 \[ \frac{a^4 (\sin (8 (e+f x))-i \cos (8 (e+f x)))}{8 c^4 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x])^4,x]

[Out]

(a^4*((-I)*Cos[8*(e + f*x)] + Sin[8*(e + f*x)]))/(8*c^4*f)

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Maple [A]  time = 0.03, size = 66, normalized size = 1.3 \begin{align*}{\frac{{a}^{4}}{f{c}^{4}} \left ({\frac{-2\,i}{ \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}- \left ( \tan \left ( fx+e \right ) +i \right ) ^{-1}+{\frac{3\,i}{ \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}+4\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{-3} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(f*x+e))^4/(c-I*c*tan(f*x+e))^4,x)

[Out]

1/f*a^4/c^4*(-2*I/(tan(f*x+e)+I)^4-1/(tan(f*x+e)+I)+3*I/(tan(f*x+e)+I)^2+4/(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((a+I*a*tan(f*x+e))^4/(c-I*c*tan(f*x+e))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.29027, size = 54, normalized size = 1.08 \begin{align*} -\frac{i \, a^{4} e^{\left (8 i \, f x + 8 i \, e\right )}}{8 \, c^{4} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^4/(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.974253, size = 48, normalized size = 0.96 \begin{align*} \begin{cases} - \frac{i a^{4} e^{8 i e} e^{8 i f x}}{8 c^{4} f} & \text{for}\: 8 c^{4} f \neq 0 \\\frac{a^{4} x e^{8 i e}}{c^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))**4/(c-I*c*tan(f*x+e))**4,x)

[Out]

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

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Giac [B]  time = 1.60398, size = 119, normalized size = 2.38 \begin{align*} -\frac{2 \,{\left (a^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 7 \, a^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 7 \, a^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - a^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{c^{4} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^4/(c-I*c*tan(f*x+e))^4,x, algorithm="giac")

[Out]

-2*(a^4*tan(1/2*f*x + 1/2*e)^7 - 7*a^4*tan(1/2*f*x + 1/2*e)^5 + 7*a^4*tan(1/2*f*x + 1/2*e)^3 - a^4*tan(1/2*f*x
+ 1/2*e))/(c^4*f*(tan(1/2*f*x + 1/2*e) + I)^8)