3.10.22 \(\int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx\) [922]

Optimal. Leaf size=50 \[ \frac {i c^4 \left (a^2-i a^2 \tan (e+f x)\right )^4}{8 f \left (a^3+i a^3 \tan (e+f x)\right )^4} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 3568, 37} \begin {gather*} \frac {i c^4 \left (a^2-i a^2 \tan (e+f x)\right )^4}{8 f \left (a^3+i a^3 \tan (e+f x)\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 3568

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 3603

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]))

Rubi steps

\begin {align*} \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx &=\left (a^4 c^4\right ) \int \frac {\sec ^8(e+f x)}{(a+i a \tan (e+f x))^8} \, dx\\ &=-\frac {\left (i c^4\right ) \text {Subst}\left (\int \frac {(a-x)^3}{(a+x)^5} \, dx,x,i a \tan (e+f x)\right )}{a^3 f}\\ &=\frac {i c^4 (1-i \tan (e+f x))^4}{8 f (a+i a \tan (e+f x))^4}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 34, normalized size = 0.68 \begin {gather*} \frac {c^4 (i \cos (8 (e+f x))+\sin (8 (e+f x)))}{8 a^4 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^4*(I*Cos[8*(e + f*x)] + Sin[8*(e + f*x)]))/(8*a^4*f)

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Maple [A]
time = 0.23, size = 66, normalized size = 1.32

method	result	size
risch	\(\frac {i c^{4} {\mathrm e}^{-8 i \left (f x +e \right )}}{8 a^{4} f}\)	\(22\)
derivativedivides	\(\frac {c^{4} \left (-\frac {1}{\tan \left (f x +e \right )-i}+\frac {2 i}{\left (\tan \left (f x +e \right )-i\right )^{4}}-\frac {3 i}{\left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {4}{\left (\tan \left (f x +e \right )-i\right )^{3}}\right )}{f \,a^{4}}\)	\(66\)
default	\(\frac {c^{4} \left (-\frac {1}{\tan \left (f x +e \right )-i}+\frac {2 i}{\left (\tan \left (f x +e \right )-i\right )^{4}}-\frac {3 i}{\left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {4}{\left (\tan \left (f x +e \right )-i\right )^{3}}\right )}{f \,a^{4}}\)	\(66\)
norman	\(\frac {\frac {c^{4} \tan \left (f x +e \right )}{a f}-\frac {7 c^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{a f}+\frac {7 c^{4} \left (\tan ^{5}\left (f x +e \right )\right )}{a f}-\frac {c^{4} \left (\tan ^{7}\left (f x +e \right )\right )}{a f}-\frac {4 i c^{4} \left (\tan ^{2}\left (f x +e \right )\right )}{a f}-\frac {4 i c^{4} \left (\tan ^{6}\left (f x +e \right )\right )}{a f}+\frac {8 i c^{4} \left (\tan ^{4}\left (f x +e \right )\right )}{a f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{4} a^{3}}\)	\(151\)

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))^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [A]
time = 0.76, size = 21, normalized size = 0.42 \begin {gather*} \frac {i \, c^{4} e^{\left (-8 i \, f x - 8 i \, e\right )}}{8 \, a^{4} f} \end {gather*}

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))^4,x, algorithm="fricas")

[Out]

1/8*I*c^4*e^(-8*I*f*x - 8*I*e)/(a^4*f)

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Sympy [A]
time = 0.26, size = 51, normalized size = 1.02 \begin {gather*} \begin {cases} \frac {i c^{4} e^{- 8 i e} e^{- 8 i f x}}{8 a^{4} f} & \text {for}\: a^{4} f e^{8 i e} \neq 0 \\\frac {c^{4} x e^{- 8 i e}}{a^{4}} & \text {otherwise} \end {cases} \end {gather*}

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))**4,x)

[Out]

Piecewise((I*c**4*exp(-8*I*e)*exp(-8*I*f*x)/(8*a**4*f), Ne(a**4*f*exp(8*I*e), 0)), (c**4*x*exp(-8*I*e)/a**4, T
rue))

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Giac [A]
time = 0.96, size = 88, normalized size = 1.76 \begin {gather*} -\frac {2 \, {\left (c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 7 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 7 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{8}} \end {gather*}

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))^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.72, size = 76, normalized size = 1.52 \begin {gather*} -\frac {c^4\,\mathrm {tan}\left (e+f\,x\right )\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}-\mathrm {i}\right )}{a^4\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4\,1{}\mathrm {i}+4\,{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,6{}\mathrm {i}-4\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - c*tan(e + f*x)*1i)^4/(a + a*tan(e + f*x)*1i)^4,x)

[Out]

-(c^4*tan(e + f*x)*(tan(e + f*x)^2*1i - 1i))/(a^4*f*(4*tan(e + f*x)^3 - tan(e + f*x)^2*6i - 4*tan(e + f*x) + t
an(e + f*x)^4*1i + 1i))

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