3.915 \(\int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx\)

Optimal. Leaf size=77 \[ \frac{a^4 c^4 \tan ^7(e+f x)}{7 f}+\frac{3 a^4 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^4 c^4 \tan ^3(e+f x)}{f}+\frac{a^4 c^4 \tan (e+f x)}{f} \]

[Out]

(a^4*c^4*Tan[e + f*x])/f + (a^4*c^4*Tan[e + f*x]^3)/f + (3*a^4*c^4*Tan[e + f*x]^5)/(5*f) + (a^4*c^4*Tan[e + f*
x]^7)/(7*f)

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Rubi [A]  time = 0.0707983, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {3522, 3767} \[ \frac{a^4 c^4 \tan ^7(e+f x)}{7 f}+\frac{3 a^4 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^4 c^4 \tan ^3(e+f x)}{f}+\frac{a^4 c^4 \tan (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*c^4*Tan[e + f*x])/f + (a^4*c^4*Tan[e + f*x]^3)/f + (3*a^4*c^4*Tan[e + f*x]^5)/(5*f) + (a^4*c^4*Tan[e + f*
x]^7)/(7*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]

Rubi steps

\begin{align*} \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx &=\left (a^4 c^4\right ) \int \sec ^8(e+f x) \, dx\\ &=-\frac{\left (a^4 c^4\right ) \operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (e+f x)\right )}{f}\\ &=\frac{a^4 c^4 \tan (e+f x)}{f}+\frac{a^4 c^4 \tan ^3(e+f x)}{f}+\frac{3 a^4 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^4 c^4 \tan ^7(e+f x)}{7 f}\\ \end{align*}

Mathematica [A]  time = 0.219376, size = 49, normalized size = 0.64 \[ \frac{a^4 c^4 \left (\frac{1}{7} \tan ^7(e+f x)+\frac{3}{5} \tan ^5(e+f x)+\tan ^3(e+f x)+\tan (e+f x)\right )}{f} \]

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*c^4*(Tan[e + f*x] + Tan[e + f*x]^3 + (3*Tan[e + f*x]^5)/5 + Tan[e + f*x]^7/7))/f

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Maple [A]  time = 0.004, size = 46, normalized size = 0.6 \begin{align*}{\frac{{a}^{4}{c}^{4}}{f} \left ({\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{7}}{7}}+{\frac{3\, \left ( \tan \left ( fx+e \right ) \right ) ^{5}}{5}}+ \left ( \tan \left ( fx+e \right ) \right ) ^{3}+\tan \left ( fx+e \right ) \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*(1/7*tan(f*x+e)^7+3/5*tan(f*x+e)^5+tan(f*x+e)^3+tan(f*x+e))

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Maxima [A]  time = 1.7413, size = 92, normalized size = 1.19 \begin{align*} \frac{5 \, a^{4} c^{4} \tan \left (f x + e\right )^{7} + 21 \, a^{4} c^{4} \tan \left (f x + e\right )^{5} + 35 \, a^{4} c^{4} \tan \left (f x + e\right )^{3} + 35 \, a^{4} c^{4} \tan \left (f x + e\right )}{35 \, 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="maxima")

[Out]

1/35*(5*a^4*c^4*tan(f*x + e)^7 + 21*a^4*c^4*tan(f*x + e)^5 + 35*a^4*c^4*tan(f*x + e)^3 + 35*a^4*c^4*tan(f*x +
e))/f

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Fricas [C]  time = 1.27123, size = 437, normalized size = 5.68 \begin{align*} \frac{1120 i \, a^{4} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 672 i \, a^{4} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 224 i \, a^{4} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 32 i \, a^{4} c^{4}}{35 \,{\left (f e^{\left (14 i \, f x + 14 i \, e\right )} + 7 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 21 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 35 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 35 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 21 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 7 \, 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))^4*(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")

[Out]

1/35*(1120*I*a^4*c^4*e^(6*I*f*x + 6*I*e) + 672*I*a^4*c^4*e^(4*I*f*x + 4*I*e) + 224*I*a^4*c^4*e^(2*I*f*x + 2*I*
e) + 32*I*a^4*c^4)/(f*e^(14*I*f*x + 14*I*e) + 7*f*e^(12*I*f*x + 12*I*e) + 21*f*e^(10*I*f*x + 10*I*e) + 35*f*e^
(8*I*f*x + 8*I*e) + 35*f*e^(6*I*f*x + 6*I*e) + 21*f*e^(4*I*f*x + 4*I*e) + 7*f*e^(2*I*f*x + 2*I*e) + f)

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Sympy [C]  time = 15.8109, size = 223, normalized size = 2.9 \begin{align*} \frac{\frac{32 i a^{4} c^{4} e^{- 8 i e} e^{6 i f x}}{f} + \frac{96 i a^{4} c^{4} e^{- 10 i e} e^{4 i f x}}{5 f} + \frac{32 i a^{4} c^{4} e^{- 12 i e} e^{2 i f x}}{5 f} + \frac{32 i a^{4} c^{4} e^{- 14 i e}}{35 f}}{e^{14 i f x} + 7 e^{- 2 i e} e^{12 i f x} + 21 e^{- 4 i e} e^{10 i f x} + 35 e^{- 6 i e} e^{8 i f x} + 35 e^{- 8 i e} e^{6 i f x} + 21 e^{- 10 i e} e^{4 i f x} + 7 e^{- 12 i e} e^{2 i f x} + e^{- 14 i e}} \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]

(32*I*a**4*c**4*exp(-8*I*e)*exp(6*I*f*x)/f + 96*I*a**4*c**4*exp(-10*I*e)*exp(4*I*f*x)/(5*f) + 32*I*a**4*c**4*e
xp(-12*I*e)*exp(2*I*f*x)/(5*f) + 32*I*a**4*c**4*exp(-14*I*e)/(35*f))/(exp(14*I*f*x) + 7*exp(-2*I*e)*exp(12*I*f
*x) + 21*exp(-4*I*e)*exp(10*I*f*x) + 35*exp(-6*I*e)*exp(8*I*f*x) + 35*exp(-8*I*e)*exp(6*I*f*x) + 21*exp(-10*I*
e)*exp(4*I*f*x) + 7*exp(-12*I*e)*exp(2*I*f*x) + exp(-14*I*e))

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Giac [B]  time = 3.50071, size = 878, normalized size = 11.4 \begin{align*} \text{result too large to display} \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]

-1/35*(35*a^4*c^4*tan(f*x)^7*tan(e)^6 + 35*a^4*c^4*tan(f*x)^6*tan(e)^7 + 35*a^4*c^4*tan(f*x)^7*tan(e)^4 - 105*
a^4*c^4*tan(f*x)^6*tan(e)^5 - 105*a^4*c^4*tan(f*x)^5*tan(e)^6 + 35*a^4*c^4*tan(f*x)^4*tan(e)^7 + 21*a^4*c^4*ta
n(f*x)^7*tan(e)^2 - 35*a^4*c^4*tan(f*x)^6*tan(e)^3 + 315*a^4*c^4*tan(f*x)^5*tan(e)^4 + 315*a^4*c^4*tan(f*x)^4*
tan(e)^5 - 35*a^4*c^4*tan(f*x)^3*tan(e)^6 + 21*a^4*c^4*tan(f*x)^2*tan(e)^7 + 5*a^4*c^4*tan(f*x)^7 - 7*a^4*c^4*
tan(f*x)^6*tan(e) + 105*a^4*c^4*tan(f*x)^5*tan(e)^2 - 315*a^4*c^4*tan(f*x)^4*tan(e)^3 - 315*a^4*c^4*tan(f*x)^3
*tan(e)^4 + 105*a^4*c^4*tan(f*x)^2*tan(e)^5 - 7*a^4*c^4*tan(f*x)*tan(e)^6 + 5*a^4*c^4*tan(e)^7 + 21*a^4*c^4*ta
n(f*x)^5 - 35*a^4*c^4*tan(f*x)^4*tan(e) + 315*a^4*c^4*tan(f*x)^3*tan(e)^2 + 315*a^4*c^4*tan(f*x)^2*tan(e)^3 -
35*a^4*c^4*tan(f*x)*tan(e)^4 + 21*a^4*c^4*tan(e)^5 + 35*a^4*c^4*tan(f*x)^3 - 105*a^4*c^4*tan(f*x)^2*tan(e) - 1
05*a^4*c^4*tan(f*x)*tan(e)^2 + 35*a^4*c^4*tan(e)^3 + 35*a^4*c^4*tan(f*x) + 35*a^4*c^4*tan(e))/(f*tan(f*x)^7*ta
n(e)^7 - 7*f*tan(f*x)^6*tan(e)^6 + 21*f*tan(f*x)^5*tan(e)^5 - 35*f*tan(f*x)^4*tan(e)^4 + 35*f*tan(f*x)^3*tan(e
)^3 - 21*f*tan(f*x)^2*tan(e)^2 + 7*f*tan(f*x)*tan(e) - f)