3.921 \(\int \frac{(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=114 \[ \frac{6 i c^4}{f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{i c^4 \log (\cos (e+f x))}{a^3 f}-\frac{c^4 x}{a^3}-\frac{6 i c^4}{a f (a+i a \tan (e+f x))^2}+\frac{8 i c^4}{3 f (a+i a \tan (e+f x))^3} \]

[Out]

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

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Rubi [A]  time = 0.133208, antiderivative size = 114, 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} \[ \frac{6 i c^4}{f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{i c^4 \log (\cos (e+f x))}{a^3 f}-\frac{c^4 x}{a^3}-\frac{6 i c^4}{a f (a+i a \tan (e+f x))^2}+\frac{8 i c^4}{3 f (a+i a \tan (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c^4*x)/a^3) - (I*c^4*Log[Cos[e + f*x]])/(a^3*f) + (((8*I)/3)*c^4)/(f*(a + I*a*Tan[e + f*x])^3) - ((6*I)*c^4
)/(a*f*(a + I*a*Tan[e + f*x])^2) + ((6*I)*c^4)/(f*(a^3 + I*a^3*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{(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx &=\left (a^4 c^4\right ) \int \frac{\sec ^8(e+f x)}{(a+i a \tan (e+f x))^7} \, dx\\ &=-\frac{\left (i c^4\right ) \operatorname{Subst}\left (\int \frac{(a-x)^3}{(a+x)^4} \, dx,x,i a \tan (e+f x)\right )}{a^3 f}\\ &=-\frac{\left (i c^4\right ) \operatorname{Subst}\left (\int \left (\frac{1}{-a-x}+\frac{8 a^3}{(a+x)^4}-\frac{12 a^2}{(a+x)^3}+\frac{6 a}{(a+x)^2}\right ) \, dx,x,i a \tan (e+f x)\right )}{a^3 f}\\ &=-\frac{c^4 x}{a^3}-\frac{i c^4 \log (\cos (e+f x))}{a^3 f}+\frac{8 i c^4}{3 f (a+i a \tan (e+f x))^3}-\frac{6 i c^4}{a f (a+i a \tan (e+f x))^2}+\frac{6 i c^4}{f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.58133, size = 121, normalized size = 1.06 \[ \frac{c^4 \sec ^3(e+f x) (-9 i \sin (e+f x)+6 f x \sin (3 (e+f x))+2 i \sin (3 (e+f x))-3 \cos (e+f x)+\cos (3 (e+f x)) (6 \log (\cos (e+f x))-6 i f x-2)+6 i \sin (3 (e+f x)) \log (\cos (e+f x)))}{6 a^3 f (\tan (e+f x)-i)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.031, size = 91, normalized size = 0.8 \begin{align*} -{\frac{8\,{c}^{4}}{3\,f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}+{\frac{i{c}^{4}\ln \left ( \tan \left ( fx+e \right ) -i \right ) }{f{a}^{3}}}+6\,{\frac{{c}^{4}}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{6\,i{c}^{4}}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.4894, size = 266, normalized size = 2.33 \begin{align*} -\frac{{\left (12 \, c^{4} f x e^{\left (6 i \, f x + 6 i \, e\right )} + 6 i \, c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 6 i \, c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 3 i \, c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - 2 i \, c^{4}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{6 \, a^{3} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/6*(12*c^4*f*x*e^(6*I*f*x + 6*I*e) + 6*I*c^4*e^(6*I*f*x + 6*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) - 6*I*c^4*e^(4
*I*f*x + 4*I*e) + 3*I*c^4*e^(2*I*f*x + 2*I*e) - 2*I*c^4)*e^(-6*I*f*x - 6*I*e)/(a^3*f)

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Sympy [A]  time = 7.78101, size = 158, normalized size = 1.39 \begin{align*} - \frac{i c^{4} \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a^{3} f} - \frac{\left (\begin{cases} 2 c^{4} x e^{6 i e} - \frac{i c^{4} e^{4 i e} e^{- 2 i f x}}{f} + \frac{i c^{4} e^{2 i e} e^{- 4 i f x}}{2 f} - \frac{i c^{4} e^{- 6 i f x}}{3 f} & \text{for}\: f \neq 0 \\x \left (2 c^{4} e^{6 i e} - 2 c^{4} e^{4 i e} + 2 c^{4} e^{2 i e} - 2 c^{4}\right ) & \text{otherwise} \end{cases}\right ) e^{- 6 i e}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))**4/(a+I*a*tan(f*x+e))**3,x)

[Out]

-I*c**4*log(exp(2*I*f*x) + exp(-2*I*e))/(a**3*f) - Piecewise((2*c**4*x*exp(6*I*e) - I*c**4*exp(4*I*e)*exp(-2*I
*f*x)/f + I*c**4*exp(2*I*e)*exp(-4*I*f*x)/(2*f) - I*c**4*exp(-6*I*f*x)/(3*f), Ne(f, 0)), (x*(2*c**4*exp(6*I*e)
- 2*c**4*exp(4*I*e) + 2*c**4*exp(2*I*e) - 2*c**4), True))*exp(-6*I*e)/a**3

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Giac [A]  time = 1.57987, size = 263, normalized size = 2.31 \begin{align*} -\frac{-\frac{60 i \, c^{4} \log \left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}{a^{3}} + \frac{30 i \, c^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} + \frac{30 i \, c^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} + \frac{147 i \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 1002 \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 2445 i \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 3820 \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 2445 i \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1002 \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 147 i \, c^{4}}{a^{3}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}^{6}}}{30 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

-1/30*(-60*I*c^4*log(tan(1/2*f*x + 1/2*e) - I)/a^3 + 30*I*c^4*log(abs(tan(1/2*f*x + 1/2*e) + 1))/a^3 + 30*I*c^
4*log(abs(tan(1/2*f*x + 1/2*e) - 1))/a^3 + (147*I*c^4*tan(1/2*f*x + 1/2*e)^6 + 1002*c^4*tan(1/2*f*x + 1/2*e)^5
- 2445*I*c^4*tan(1/2*f*x + 1/2*e)^4 - 3820*c^4*tan(1/2*f*x + 1/2*e)^3 + 2445*I*c^4*tan(1/2*f*x + 1/2*e)^2 + 1
002*c^4*tan(1/2*f*x + 1/2*e) - 147*I*c^4)/(a^3*(tan(1/2*f*x + 1/2*e) - I)^6))/f