3.904 \(\int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^3 \, dx\)

Optimal. Leaf size=88 \[ -\frac{i c^3 (a+i a \tan (e+f x))^7}{7 a^2 f}+\frac{2 i c^3 (a+i a \tan (e+f x))^6}{3 a f}-\frac{4 i c^3 (a+i a \tan (e+f x))^5}{5 f} \]

[Out]

(((-4*I)/5)*c^3*(a + I*a*Tan[e + f*x])^5)/f + (((2*I)/3)*c^3*(a + I*a*Tan[e + f*x])^6)/(a*f) - ((I/7)*c^3*(a +
I*a*Tan[e + f*x])^7)/(a^2*f)

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Rubi [A]  time = 0.117099, antiderivative size = 88, 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, 3487, 43} \[ -\frac{i c^3 (a+i a \tan (e+f x))^7}{7 a^2 f}+\frac{2 i c^3 (a+i a \tan (e+f x))^6}{3 a f}-\frac{4 i c^3 (a+i a \tan (e+f x))^5}{5 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^5*(c - I*c*Tan[e + f*x])^3,x]

[Out]

(((-4*I)/5)*c^3*(a + I*a*Tan[e + f*x])^5)/f + (((2*I)/3)*c^3*(a + I*a*Tan[e + f*x])^6)/(a*f) - ((I/7)*c^3*(a +
I*a*Tan[e + f*x])^7)/(a^2*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 4.41042, size = 93, normalized size = 1.06 \[ \frac{a^5 c^3 \sec (e) \sec ^7(e+f x) (-35 \sin (2 e+f x)+42 \sin (2 e+3 f x)+14 \sin (4 e+5 f x)+2 \sin (6 e+7 f x)+35 i \cos (2 e+f x)+35 \sin (f x)+35 i \cos (f x))}{210 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])^5*(c - I*c*Tan[e + f*x])^3,x]

[Out]

(a^5*c^3*Sec[e]*Sec[e + f*x]^7*((35*I)*Cos[f*x] + (35*I)*Cos[2*e + f*x] + 35*Sin[f*x] - 35*Sin[2*e + f*x] + 42
*Sin[2*e + 3*f*x] + 14*Sin[4*e + 5*f*x] + 2*Sin[6*e + 7*f*x]))/(210*f)

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Maple [A]  time = 0.003, size = 81, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}{c}^{3}}{f} \left ( \tan \left ( fx+e \right ) -{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{7}}{7}}+{\frac{i}{3}} \left ( \tan \left ( fx+e \right ) \right ) ^{6}-{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{5}}{5}}+i \left ( \tan \left ( fx+e \right ) \right ) ^{4}+{\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))^5*(c-I*c*tan(f*x+e))^3,x)

[Out]

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

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Maxima [A]  time = 1.71668, size = 157, normalized size = 1.78 \begin{align*} -\frac{30 \, a^{5} c^{3} \tan \left (f x + e\right )^{7} - 70 i \, a^{5} c^{3} \tan \left (f x + e\right )^{6} + 42 \, a^{5} c^{3} \tan \left (f x + e\right )^{5} - 210 i \, a^{5} c^{3} \tan \left (f x + e\right )^{4} - 70 \, a^{5} c^{3} \tan \left (f x + e\right )^{3} - 210 i \, a^{5} c^{3} \tan \left (f x + e\right )^{2} - 210 \, a^{5} c^{3} \tan \left (f x + e\right )}{210 \, 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))^3,x, algorithm="maxima")

[Out]

-1/210*(30*a^5*c^3*tan(f*x + e)^7 - 70*I*a^5*c^3*tan(f*x + e)^6 + 42*a^5*c^3*tan(f*x + e)^5 - 210*I*a^5*c^3*ta
n(f*x + e)^4 - 70*a^5*c^3*tan(f*x + e)^3 - 210*I*a^5*c^3*tan(f*x + e)^2 - 210*a^5*c^3*tan(f*x + e))/f

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Fricas [B]  time = 1.27749, size = 491, normalized size = 5.58 \begin{align*} \frac{4480 i \, a^{5} c^{3} e^{\left (8 i \, f x + 8 i \, e\right )} + 4480 i \, a^{5} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 2688 i \, a^{5} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 896 i \, a^{5} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 128 i \, a^{5} c^{3}}{105 \,{\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))^5*(c-I*c*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/105*(4480*I*a^5*c^3*e^(8*I*f*x + 8*I*e) + 4480*I*a^5*c^3*e^(6*I*f*x + 6*I*e) + 2688*I*a^5*c^3*e^(4*I*f*x + 4
*I*e) + 896*I*a^5*c^3*e^(2*I*f*x + 2*I*e) + 128*I*a^5*c^3)/(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 [B]  time = 19.6171, size = 253, normalized size = 2.88 \begin{align*} \frac{\frac{128 i a^{5} c^{3} e^{- 6 i e} e^{8 i f x}}{3 f} + \frac{128 i a^{5} c^{3} e^{- 8 i e} e^{6 i f x}}{3 f} + \frac{128 i a^{5} c^{3} e^{- 10 i e} e^{4 i f x}}{5 f} + \frac{128 i a^{5} c^{3} e^{- 12 i e} e^{2 i f x}}{15 f} + \frac{128 i a^{5} c^{3} e^{- 14 i e}}{105 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))**5*(c-I*c*tan(f*x+e))**3,x)

[Out]

(128*I*a**5*c**3*exp(-6*I*e)*exp(8*I*f*x)/(3*f) + 128*I*a**5*c**3*exp(-8*I*e)*exp(6*I*f*x)/(3*f) + 128*I*a**5*
c**3*exp(-10*I*e)*exp(4*I*f*x)/(5*f) + 128*I*a**5*c**3*exp(-12*I*e)*exp(2*I*f*x)/(15*f) + 128*I*a**5*c**3*exp(
-14*I*e)/(105*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 = 1.99174, size = 239, normalized size = 2.72 \begin{align*} \frac{4480 i \, a^{5} c^{3} e^{\left (8 i \, f x + 8 i \, e\right )} + 4480 i \, a^{5} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 2688 i \, a^{5} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 896 i \, a^{5} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 128 i \, a^{5} c^{3}}{105 \,{\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))^5*(c-I*c*tan(f*x+e))^3,x, algorithm="giac")

[Out]

1/105*(4480*I*a^5*c^3*e^(8*I*f*x + 8*I*e) + 4480*I*a^5*c^3*e^(6*I*f*x + 6*I*e) + 2688*I*a^5*c^3*e^(4*I*f*x + 4
*I*e) + 896*I*a^5*c^3*e^(2*I*f*x + 2*I*e) + 128*I*a^5*c^3)/(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)