3.932 \(\int \frac{(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^2} \, dx\)

Optimal. Leaf size=83 \[ \frac{4 i a^3}{f \left (c^2-i c^2 \tan (e+f x)\right )}-\frac{i a^3 \log (\cos (e+f x))}{c^2 f}+\frac{a^3 x}{c^2}-\frac{2 i a^3}{f (c-i c \tan (e+f x))^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.122977, antiderivative size = 83, 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} \[ \frac{4 i a^3}{f \left (c^2-i c^2 \tan (e+f x)\right )}-\frac{i a^3 \log (\cos (e+f x))}{c^2 f}+\frac{a^3 x}{c^2}-\frac{2 i a^3}{f (c-i c \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 1.92661, size = 113, normalized size = 1.36 \[ \frac{a^3 (\cos (2 e+5 f x)+i \sin (2 e+5 f x)) \left (\cos (2 (e+f x)) \left (-i \log \left (\cos ^2(e+f x)\right )+2 f x-i\right )+\sin (2 (e+f x)) \left (-\log \left (\cos ^2(e+f x)\right )-2 i f x+1\right )+2 i\right )}{2 c^2 f (\cos (f x)+i \sin (f x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(2*I + Cos[2*(e + f*x)]*(-I + 2*f*x - I*Log[Cos[e + f*x]^2]) + (1 - (2*I)*f*x - Log[Cos[e + f*x]^2])*Sin[
2*(e + f*x)])*(Cos[2*e + 5*f*x] + I*Sin[2*e + 5*f*x]))/(2*c^2*f*(Cos[f*x] + I*Sin[f*x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.026, size = 69, normalized size = 0.8 \begin{align*} -4\,{\frac{{a}^{3}}{f{c}^{2} \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{2\,i{a}^{3}}{f{c}^{2} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}+{\frac{i{a}^{3}\ln \left ( \tan \left ( fx+e \right ) +i \right ) }{f{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 1.38698, size = 150, normalized size = 1.81 \begin{align*} \frac{-i \, a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 i \, a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 2 i \, a^{3} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{2 \, c^{2} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 2.09635, size = 102, normalized size = 1.23 \begin{align*} - \frac{i a^{3} \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{2} f} + \frac{\begin{cases} - \frac{i a^{3} e^{4 i e} e^{4 i f x}}{2 f} + \frac{i a^{3} e^{2 i e} e^{2 i f x}}{f} & \text{for}\: f \neq 0 \\x \left (2 a^{3} e^{4 i e} - 2 a^{3} e^{2 i e}\right ) & \text{otherwise} \end{cases}}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))**3/(c-I*c*tan(f*x+e))**2,x)

[Out]

-I*a**3*log(exp(2*I*f*x) + exp(-2*I*e))/(c**2*f) + Piecewise((-I*a**3*exp(4*I*e)*exp(4*I*f*x)/(2*f) + I*a**3*e
xp(2*I*e)*exp(2*I*f*x)/f, Ne(f, 0)), (x*(2*a**3*exp(4*I*e) - 2*a**3*exp(2*I*e)), True))/c**2

________________________________________________________________________________________

Giac [B]  time = 1.41421, size = 217, normalized size = 2.61 \begin{align*} -\frac{-\frac{12 i \, a^{3} \log \left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}{c^{2}} + \frac{6 i \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{c^{2}} + \frac{6 i \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{c^{2}} + \frac{25 i \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 100 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 198 i \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 100 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 25 i \, a^{3}}{c^{2}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{4}}}{6 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(-12*I*a^3*log(tan(1/2*f*x + 1/2*e) + I)/c^2 + 6*I*a^3*log(abs(tan(1/2*f*x + 1/2*e) + 1))/c^2 + 6*I*a^3*l
og(abs(tan(1/2*f*x + 1/2*e) - 1))/c^2 + (25*I*a^3*tan(1/2*f*x + 1/2*e)^4 - 100*a^3*tan(1/2*f*x + 1/2*e)^3 - 19
8*I*a^3*tan(1/2*f*x + 1/2*e)^2 + 100*a^3*tan(1/2*f*x + 1/2*e) + 25*I*a^3)/(c^2*(tan(1/2*f*x + 1/2*e) + I)^4))/
f