3.914 \(\int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx\)

Optimal. Leaf size=100 \[ \frac{a^5 c^4 \tan ^7(e+f x)}{7 f}+\frac{3 a^5 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^5 c^4 \tan ^3(e+f x)}{f}+\frac{a^5 c^4 \tan (e+f x)}{f}+\frac{i a^5 c^4 \sec ^8(e+f x)}{8 f} \]

[Out]

((I/8)*a^5*c^4*Sec[e + f*x]^8)/f + (a^5*c^4*Tan[e + f*x])/f + (a^5*c^4*Tan[e + f*x]^3)/f + (3*a^5*c^4*Tan[e +
f*x]^5)/(5*f) + (a^5*c^4*Tan[e + f*x]^7)/(7*f)

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Rubi [A]  time = 0.100032, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 3486, 3767} \[ \frac{a^5 c^4 \tan ^7(e+f x)}{7 f}+\frac{3 a^5 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^5 c^4 \tan ^3(e+f x)}{f}+\frac{a^5 c^4 \tan (e+f x)}{f}+\frac{i a^5 c^4 \sec ^8(e+f x)}{8 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^5*(c - I*c*Tan[e + f*x])^4,x]

[Out]

((I/8)*a^5*c^4*Sec[e + f*x]^8)/f + (a^5*c^4*Tan[e + f*x])/f + (a^5*c^4*Tan[e + f*x]^3)/f + (3*a^5*c^4*Tan[e +
f*x]^5)/(5*f) + (a^5*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 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

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

Mathematica [A]  time = 4.93492, size = 74, normalized size = 0.74 \[ \frac{a^5 c^4 \sec (e) \sec ^8(e+f x) (56 \sin (e+2 f x)+28 \sin (3 e+4 f x)+8 \sin (5 e+6 f x)+\sin (7 e+8 f x)-35 \sin (e)+35 i \cos (e))}{280 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])^5*(c - I*c*Tan[e + f*x])^4,x]

[Out]

(a^5*c^4*Sec[e]*Sec[e + f*x]^8*((35*I)*Cos[e] - 35*Sin[e] + 56*Sin[e + 2*f*x] + 28*Sin[3*e + 4*f*x] + 8*Sin[5*
e + 6*f*x] + Sin[7*e + 8*f*x]))/(280*f)

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Maple [A]  time = 0.003, size = 90, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}{c}^{4}}{f} \left ( \tan \left ( fx+e \right ) +{\frac{i}{8}} \left ( \tan \left ( fx+e \right ) \right ) ^{8}+{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{7}}{7}}+{\frac{i}{2}} \left ( \tan \left ( fx+e \right ) \right ) ^{6}+{\frac{3\, \left ( \tan \left ( fx+e \right ) \right ) ^{5}}{5}}+{\frac{3\,i}{4}} \left ( \tan \left ( fx+e \right ) \right ) ^{4}+ \left ( \tan \left ( fx+e \right ) \right ) ^{3}+{\frac{i}{2}} \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))^5*(c-I*c*tan(f*x+e))^4,x)

[Out]

1/f*a^5*c^4*(tan(f*x+e)+1/8*I*tan(f*x+e)^8+1/7*tan(f*x+e)^7+1/2*I*tan(f*x+e)^6+3/5*tan(f*x+e)^5+3/4*I*tan(f*x+
e)^4+tan(f*x+e)^3+1/2*I*tan(f*x+e)^2)

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Maxima [A]  time = 2.39473, size = 178, normalized size = 1.78 \begin{align*} \frac{105 i \, a^{5} c^{4} \tan \left (f x + e\right )^{8} + 120 \, a^{5} c^{4} \tan \left (f x + e\right )^{7} + 420 i \, a^{5} c^{4} \tan \left (f x + e\right )^{6} + 504 \, a^{5} c^{4} \tan \left (f x + e\right )^{5} + 630 i \, a^{5} c^{4} \tan \left (f x + e\right )^{4} + 840 \, a^{5} c^{4} \tan \left (f x + e\right )^{3} + 420 i \, a^{5} c^{4} \tan \left (f x + e\right )^{2} + 840 \, a^{5} c^{4} \tan \left (f x + e\right )}{840 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^5*(c-I*c*tan(f*x+e))^4,x, algorithm="maxima")

[Out]

1/840*(105*I*a^5*c^4*tan(f*x + e)^8 + 120*a^5*c^4*tan(f*x + e)^7 + 420*I*a^5*c^4*tan(f*x + e)^6 + 504*a^5*c^4*
tan(f*x + e)^5 + 630*I*a^5*c^4*tan(f*x + e)^4 + 840*a^5*c^4*tan(f*x + e)^3 + 420*I*a^5*c^4*tan(f*x + e)^2 + 84
0*a^5*c^4*tan(f*x + e))/f

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Fricas [A]  time = 1.39721, size = 527, normalized size = 5.27 \begin{align*} \frac{2240 i \, a^{5} c^{4} e^{\left (8 i \, f x + 8 i \, e\right )} + 1792 i \, a^{5} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 896 i \, a^{5} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 256 i \, a^{5} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 32 i \, a^{5} c^{4}}{35 \,{\left (f e^{\left (16 i \, f x + 16 i \, e\right )} + 8 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 28 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 56 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 70 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 56 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 28 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, 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))^5*(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")

[Out]

1/35*(2240*I*a^5*c^4*e^(8*I*f*x + 8*I*e) + 1792*I*a^5*c^4*e^(6*I*f*x + 6*I*e) + 896*I*a^5*c^4*e^(4*I*f*x + 4*I
*e) + 256*I*a^5*c^4*e^(2*I*f*x + 2*I*e) + 32*I*a^5*c^4)/(f*e^(16*I*f*x + 16*I*e) + 8*f*e^(14*I*f*x + 14*I*e) +
28*f*e^(12*I*f*x + 12*I*e) + 56*f*e^(10*I*f*x + 10*I*e) + 70*f*e^(8*I*f*x + 8*I*e) + 56*f*e^(6*I*f*x + 6*I*e)
+ 28*f*e^(4*I*f*x + 4*I*e) + 8*f*e^(2*I*f*x + 2*I*e) + f)

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Sympy [B]  time = 24.7359, size = 269, normalized size = 2.69 \begin{align*} \frac{\frac{64 i a^{5} c^{4} e^{- 8 i e} e^{8 i f x}}{f} + \frac{256 i a^{5} c^{4} e^{- 10 i e} e^{6 i f x}}{5 f} + \frac{128 i a^{5} c^{4} e^{- 12 i e} e^{4 i f x}}{5 f} + \frac{256 i a^{5} c^{4} e^{- 14 i e} e^{2 i f x}}{35 f} + \frac{32 i a^{5} c^{4} e^{- 16 i e}}{35 f}}{e^{16 i f x} + 8 e^{- 2 i e} e^{14 i f x} + 28 e^{- 4 i e} e^{12 i f x} + 56 e^{- 6 i e} e^{10 i f x} + 70 e^{- 8 i e} e^{8 i f x} + 56 e^{- 10 i e} e^{6 i f x} + 28 e^{- 12 i e} e^{4 i f x} + 8 e^{- 14 i e} e^{2 i f x} + e^{- 16 i e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))**5*(c-I*c*tan(f*x+e))**4,x)

[Out]

(64*I*a**5*c**4*exp(-8*I*e)*exp(8*I*f*x)/f + 256*I*a**5*c**4*exp(-10*I*e)*exp(6*I*f*x)/(5*f) + 128*I*a**5*c**4
*exp(-12*I*e)*exp(4*I*f*x)/(5*f) + 256*I*a**5*c**4*exp(-14*I*e)*exp(2*I*f*x)/(35*f) + 32*I*a**5*c**4*exp(-16*I
*e)/(35*f))/(exp(16*I*f*x) + 8*exp(-2*I*e)*exp(14*I*f*x) + 28*exp(-4*I*e)*exp(12*I*f*x) + 56*exp(-6*I*e)*exp(1
0*I*f*x) + 70*exp(-8*I*e)*exp(8*I*f*x) + 56*exp(-10*I*e)*exp(6*I*f*x) + 28*exp(-12*I*e)*exp(4*I*f*x) + 8*exp(-
14*I*e)*exp(2*I*f*x) + exp(-16*I*e))

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Giac [B]  time = 2.23496, size = 257, normalized size = 2.57 \begin{align*} \frac{2240 i \, a^{5} c^{4} e^{\left (8 i \, f x + 8 i \, e\right )} + 1792 i \, a^{5} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 896 i \, a^{5} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 256 i \, a^{5} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 32 i \, a^{5} c^{4}}{35 \,{\left (f e^{\left (16 i \, f x + 16 i \, e\right )} + 8 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 28 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 56 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 70 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 56 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 28 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, 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))^5*(c-I*c*tan(f*x+e))^4,x, algorithm="giac")

[Out]

1/35*(2240*I*a^5*c^4*e^(8*I*f*x + 8*I*e) + 1792*I*a^5*c^4*e^(6*I*f*x + 6*I*e) + 896*I*a^5*c^4*e^(4*I*f*x + 4*I
*e) + 256*I*a^5*c^4*e^(2*I*f*x + 2*I*e) + 32*I*a^5*c^4)/(f*e^(16*I*f*x + 16*I*e) + 8*f*e^(14*I*f*x + 14*I*e) +
28*f*e^(12*I*f*x + 12*I*e) + 56*f*e^(10*I*f*x + 10*I*e) + 70*f*e^(8*I*f*x + 8*I*e) + 56*f*e^(6*I*f*x + 6*I*e)
+ 28*f*e^(4*I*f*x + 4*I*e) + 8*f*e^(2*I*f*x + 2*I*e) + f)