3.951 \(\int \frac{(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^4} \, dx\)

Optimal. Leaf size=62 \[ \frac{i a^2 c^2}{3 f \left (c^2-i c^2 \tan (e+f x)\right )^3}-\frac{i a^2}{2 f (c-i c \tan (e+f x))^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.112079, antiderivative size = 62, 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} \[ \frac{i a^2 c^2}{3 f \left (c^2-i c^2 \tan (e+f x)\right )^3}-\frac{i a^2}{2 f (c-i c \tan (e+f x))^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 1.61233, size = 75, normalized size = 1.21 \[ \frac{a^2 (-3 i \sin (2 (e+f x))+9 \cos (2 (e+f x))+8) (\sin (6 e+8 f x)-i \cos (6 e+8 f x))}{96 c^4 f (\cos (f x)+i \sin (f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(8 + 9*Cos[2*(e + f*x)] - (3*I)*Sin[2*(e + f*x)])*((-I)*Cos[6*e + 8*f*x] + Sin[6*e + 8*f*x]))/(96*c^4*f*(
Cos[f*x] + I*Sin[f*x])^2)

________________________________________________________________________________________

Maple [A]  time = 0.028, size = 39, normalized size = 0.6 \begin{align*}{\frac{{a}^{2}}{f{c}^{4}} \left ({\frac{-{\frac{i}{2}}}{ \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}+{\frac{1}{3\, \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))^2/(c-I*c*tan(f*x+e))^4,x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 1.42029, size = 142, normalized size = 2.29 \begin{align*} \frac{-3 i \, a^{2} e^{\left (8 i \, f x + 8 i \, e\right )} - 8 i \, a^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 6 i \, a^{2} e^{\left (4 i \, f x + 4 i \, e\right )}}{96 \, c^{4} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(-3*I*a^2*e^(8*I*f*x + 8*I*e) - 8*I*a^2*e^(6*I*f*x + 6*I*e) - 6*I*a^2*e^(4*I*f*x + 4*I*e))/(c^4*f)

________________________________________________________________________________________

Sympy [A]  time = 0.920543, size = 141, normalized size = 2.27 \begin{align*} \begin{cases} \frac{- 192 i a^{2} c^{8} f^{2} e^{8 i e} e^{8 i f x} - 512 i a^{2} c^{8} f^{2} e^{6 i e} e^{6 i f x} - 384 i a^{2} c^{8} f^{2} e^{4 i e} e^{4 i f x}}{6144 c^{12} f^{3}} & \text{for}\: 6144 c^{12} f^{3} \neq 0 \\\frac{x \left (a^{2} e^{8 i e} + 2 a^{2} e^{6 i e} + a^{2} e^{4 i e}\right )}{4 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))**2/(c-I*c*tan(f*x+e))**4,x)

[Out]

Piecewise(((-192*I*a**2*c**8*f**2*exp(8*I*e)*exp(8*I*f*x) - 512*I*a**2*c**8*f**2*exp(6*I*e)*exp(6*I*f*x) - 384
*I*a**2*c**8*f**2*exp(4*I*e)*exp(4*I*f*x))/(6144*c**12*f**3), Ne(6144*c**12*f**3, 0)), (x*(a**2*exp(8*I*e) + 2
*a**2*exp(6*I*e) + a**2*exp(4*I*e))/(4*c**4), True))

________________________________________________________________________________________

Giac [B]  time = 1.29046, size = 189, normalized size = 3.05 \begin{align*} -\frac{2 \,{\left (3 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 6 i \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 17 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 16 i \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 17 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 6 i \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 3 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{3 \, 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))^2/(c-I*c*tan(f*x+e))^4,x, algorithm="giac")

[Out]

-2/3*(3*a^2*tan(1/2*f*x + 1/2*e)^7 + 6*I*a^2*tan(1/2*f*x + 1/2*e)^6 - 17*a^2*tan(1/2*f*x + 1/2*e)^5 - 16*I*a^2
*tan(1/2*f*x + 1/2*e)^4 + 17*a^2*tan(1/2*f*x + 1/2*e)^3 + 6*I*a^2*tan(1/2*f*x + 1/2*e)^2 - 3*a^2*tan(1/2*f*x +
1/2*e))/(c^4*f*(tan(1/2*f*x + 1/2*e) + I)^8)