Integrand size = 31, antiderivative size = 50 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx=\frac {i c^4 \left (a^2-i a^2 \tan (e+f x)\right )^4}{8 f \left (a^3+i a^3 \tan (e+f x)\right )^4} \] Output:
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
Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx=\frac {c^4 (i \cos (8 (e+f x))+\sin (8 (e+f x)))}{8 a^4 f} \] Input:
Integrate[(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^4,x]
Output:
(c^4*(I*Cos[8*(e + f*x)] + Sin[8*(e + f*x)]))/(8*a^4*f)
Time = 0.33 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3042, 4005, 3042, 3968, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4}dx\) |
\(\Big \downarrow \) 4005 |
\(\displaystyle a^4 c^4 \int \frac {\sec ^8(e+f x)}{(i \tan (e+f x) a+a)^8}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^4 c^4 \int \frac {\sec (e+f x)^8}{(i \tan (e+f x) a+a)^8}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i c^4 \int \frac {(a-i a \tan (e+f x))^3}{(i \tan (e+f x) a+a)^5}d(i a \tan (e+f x))}{a^3 f}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {i c^4 (a-i a \tan (e+f x))^4}{8 a^4 f (a+i a \tan (e+f x))^4}\) |
Input:
Int[(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^4,x]
Output:
((I/8)*c^4*(a - I*a*Tan[e + f*x])^4)/(a^4*f*(a + I*a*Tan[e + f*x])^4)
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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[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]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[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]))
Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.44
method | result | size |
risch | \(\frac {i c^{4} {\mathrm e}^{-8 i \left (f x +e \right )}}{8 a^{4} f}\) | \(22\) |
orering | \(\frac {i \left (c -i c \tan \left (f x +e \right )\right )^{4}}{8 f \left (a +i a \tan \left (f x +e \right )\right )^{4}}\) | \(35\) |
derivativedivides | \(\frac {c^{4} \left (-\frac {1}{-i+\tan \left (f x +e \right )}-\frac {3 i}{\left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {4}{\left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {2 i}{\left (-i+\tan \left (f x +e \right )\right )^{4}}\right )}{f \,a^{4}}\) | \(66\) |
default | \(\frac {c^{4} \left (-\frac {1}{-i+\tan \left (f x +e \right )}-\frac {3 i}{\left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {4}{\left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {2 i}{\left (-i+\tan \left (f x +e \right )\right )^{4}}\right )}{f \,a^{4}}\) | \(66\) |
norman | \(\frac {\frac {c^{4} \tan \left (f x +e \right )}{a f}-\frac {7 c^{4} \tan \left (f x +e \right )^{3}}{a f}+\frac {7 c^{4} \tan \left (f x +e \right )^{5}}{a f}-\frac {c^{4} \tan \left (f x +e \right )^{7}}{a f}-\frac {4 i c^{4} \tan \left (f x +e \right )^{2}}{a f}-\frac {4 i c^{4} \tan \left (f x +e \right )^{6}}{a f}+\frac {8 i c^{4} \tan \left (f x +e \right )^{4}}{a f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{4} a^{3}}\) | \(151\) |
Input:
int((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^4,x,method=_RETURNVERBOSE)
Output:
1/8*I*c^4/a^4/f*exp(-8*I*(f*x+e))
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.40 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx=\frac {i \, c^{4} e^{\left (-8 i \, f x - 8 i \, e\right )}}{8 \, a^{4} f} \] Input:
integrate((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^4,x, algorithm="fricas")
Output:
1/8*I*c^4*e^(-8*I*f*x - 8*I*e)/(a^4*f)
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.02 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx=\begin {cases} \frac {i c^{4} e^{- 8 i e} e^{- 8 i f x}}{8 a^{4} f} & \text {for}\: a^{4} f e^{8 i e} \neq 0 \\\frac {c^{4} x e^{- 8 i e}}{a^{4}} & \text {otherwise} \end {cases} \] Input:
integrate((c-I*c*tan(f*x+e))**4/(a+I*a*tan(f*x+e))**4,x)
Output:
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, True))
Exception generated. \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^4,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.67 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.84 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx=-\frac {c^{4} \tan \left (f x + e\right )^{3} - c^{4} \tan \left (f x + e\right )}{a^{4} f {\left (\tan \left (f x + e\right ) - i\right )}^{4}} \] Input:
integrate((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^4,x, algorithm="giac")
Output:
-(c^4*tan(f*x + e)^3 - c^4*tan(f*x + e))/(a^4*f*(tan(f*x + e) - I)^4)
Time = 1.87 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.52 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx=-\frac {c^4\,\mathrm {tan}\left (e+f\,x\right )\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}-\mathrm {i}\right )}{a^4\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4\,1{}\mathrm {i}+4\,{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,6{}\mathrm {i}-4\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )} \] Input:
int((c - c*tan(e + f*x)*1i)^4/(a + a*tan(e + f*x)*1i)^4,x)
Output:
-(c^4*tan(e + f*x)*(tan(e + f*x)^2*1i - 1i))/(a^4*f*(4*tan(e + f*x)^3 - ta n(e + f*x)^2*6i - 4*tan(e + f*x) + tan(e + f*x)^4*1i + 1i))
\[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx=\frac {c^{4} \left (-32 \left (\int \frac {\tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{4} i +4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i -4 \tan \left (f x +e \right )+i}d x \right ) f +32 \left (\int \frac {\tan \left (f x +e \right )}{\tan \left (f x +e \right )^{4} i +4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i -4 \tan \left (f x +e \right )+i}d x \right ) f -\mathrm {log}\left (\tan \left (f x +e \right )^{4}-4 \tan \left (f x +e \right )^{3} i -6 \tan \left (f x +e \right )^{2}+4 \tan \left (f x +e \right ) i +1\right ) i +2 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) i \right )}{4 a^{4} f} \] Input:
int((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^4,x)
Output:
(c**4*( - 32*int(tan(e + f*x)**3/(tan(e + f*x)**4*i + 4*tan(e + f*x)**3 - 6*tan(e + f*x)**2*i - 4*tan(e + f*x) + i),x)*f + 32*int(tan(e + f*x)/(tan( e + f*x)**4*i + 4*tan(e + f*x)**3 - 6*tan(e + f*x)**2*i - 4*tan(e + f*x) + i),x)*f - log(tan(e + f*x)**4 - 4*tan(e + f*x)**3*i - 6*tan(e + f*x)**2 + 4*tan(e + f*x)*i + 1)*i + 2*log(tan(e + f*x)**2 + 1)*i))/(4*a**4*f)