Integrand size = 31, antiderivative size = 50 \[ \int \frac {(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx=-\frac {i a^4 \left (c^2+i c^2 \tan (e+f x)\right )^4}{8 f \left (c^3-i c^3 \tan (e+f x)\right )^4} \] Output:
-1/8*I*a^4*(c^2+I*c^2*tan(f*x+e))^4/f/(c^3-I*c^3*tan(f*x+e))^4
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68 \[ \int \frac {(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx=\frac {a^4 (-i \cos (8 (e+f x))+\sin (8 (e+f x)))}{8 c^4 f} \] Input:
Integrate[(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x])^4,x]
Output:
(a^4*((-I)*Cos[8*(e + f*x)] + Sin[8*(e + f*x)]))/(8*c^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.
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4}dx\) |
\(\Big \downarrow \) 4005 |
\(\displaystyle a^4 c^4 \int \frac {\sec ^8(e+f x)}{(c-i c \tan (e+f x))^8}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^4 c^4 \int \frac {\sec (e+f x)^8}{(c-i c \tan (e+f x))^8}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle \frac {i a^4 \int \frac {(i \tan (e+f x) c+c)^3}{(c-i c \tan (e+f x))^5}d(-i c \tan (e+f x))}{c^3 f}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {i a^4 (c+i c \tan (e+f x))^4}{8 c^4 f (c-i c \tan (e+f x))^4}\) |
Input:
Int[(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x])^4,x]
Output:
((-1/8*I)*a^4*(c + I*c*Tan[e + f*x])^4)/(c^4*f*(c - I*c*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 = 0.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.44
method | result | size |
risch | \(-\frac {i a^{4} {\mathrm e}^{8 i \left (f x +e \right )}}{8 c^{4} f}\) | \(22\) |
orering | \(-\frac {i \left (a +i a \tan \left (f x +e \right )\right )^{4}}{8 f \left (c -i c \tan \left (f x +e \right )\right )^{4}}\) | \(35\) |
derivativedivides | \(\frac {a^{4} \left (\frac {4}{\left (i+\tan \left (f x +e \right )\right )^{3}}-\frac {2 i}{\left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {3 i}{\left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {1}{i+\tan \left (f x +e \right )}\right )}{f \,c^{4}}\) | \(66\) |
default | \(\frac {a^{4} \left (\frac {4}{\left (i+\tan \left (f x +e \right )\right )^{3}}-\frac {2 i}{\left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {3 i}{\left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {1}{i+\tan \left (f x +e \right )}\right )}{f \,c^{4}}\) | \(66\) |
norman | \(\frac {\frac {a^{4} \tan \left (f x +e \right )}{c f}-\frac {7 a^{4} \tan \left (f x +e \right )^{3}}{c f}+\frac {7 a^{4} \tan \left (f x +e \right )^{5}}{c f}-\frac {a^{4} \tan \left (f x +e \right )^{7}}{c f}-\frac {8 i a^{4} \tan \left (f x +e \right )^{4}}{c f}+\frac {4 i a^{4} \tan \left (f x +e \right )^{2}}{c f}+\frac {4 i a^{4} \tan \left (f x +e \right )^{6}}{c f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{4} c^{3}}\) | \(151\) |
Input:
int((a+I*a*tan(f*x+e))^4/(c-I*c*tan(f*x+e))^4,x,method=_RETURNVERBOSE)
Output:
-1/8*I*a^4/c^4/f*exp(8*I*(f*x+e))
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.40 \[ \int \frac {(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx=-\frac {i \, a^{4} e^{\left (8 i \, f x + 8 i \, e\right )}}{8 \, c^{4} f} \] Input:
integrate((a+I*a*tan(f*x+e))^4/(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")
Output:
-1/8*I*a^4*e^(8*I*f*x + 8*I*e)/(c^4*f)
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92 \[ \int \frac {(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx=\begin {cases} - \frac {i a^{4} e^{8 i e} e^{8 i f x}}{8 c^{4} f} & \text {for}\: c^{4} f \neq 0 \\\frac {a^{4} x e^{8 i e}}{c^{4}} & \text {otherwise} \end {cases} \] Input:
integrate((a+I*a*tan(f*x+e))**4/(c-I*c*tan(f*x+e))**4,x)
Output:
Piecewise((-I*a**4*exp(8*I*e)*exp(8*I*f*x)/(8*c**4*f), Ne(c**4*f, 0)), (a* *4*x*exp(8*I*e)/c**4, True))
Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+I*a*tan(f*x+e))^4/(c-I*c*tan(f*x+e))^4,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.84 \[ \int \frac {(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx=-\frac {a^{4} \tan \left (f x + e\right )^{3} - a^{4} \tan \left (f x + e\right )}{c^{4} f {\left (\tan \left (f x + e\right ) + i\right )}^{4}} \] Input:
integrate((a+I*a*tan(f*x+e))^4/(c-I*c*tan(f*x+e))^4,x, algorithm="giac")
Output:
-(a^4*tan(f*x + e)^3 - a^4*tan(f*x + e))/(c^4*f*(tan(f*x + e) + I)^4)
Time = 1.94 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.38 \[ \int \frac {(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx=-\frac {a^4\,\mathrm {tan}\left (e+f\,x\right )\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2-1\right )}{c^4\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4+{\mathrm {tan}\left (e+f\,x\right )}^3\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (e+f\,x\right )}^2-\mathrm {tan}\left (e+f\,x\right )\,4{}\mathrm {i}+1\right )} \] Input:
int((a + a*tan(e + f*x)*1i)^4/(c - c*tan(e + f*x)*1i)^4,x)
Output:
-(a^4*tan(e + f*x)*(tan(e + f*x)^2 - 1))/(c^4*f*(tan(e + f*x)^3*4i - 6*tan (e + f*x)^2 - tan(e + f*x)*4i + tan(e + f*x)^4 + 1))
\[ \int \frac {(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx=\frac {a^{4} \left (32 \left (\int \frac {\tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{4} i -4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i +4 \tan \left (f x +e \right )+i}d x \right ) f -32 \left (\int \frac {\tan \left (f x +e \right )}{\tan \left (f x +e \right )^{4} i -4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i +4 \tan \left (f x +e \right )+i}d x \right ) f +\mathrm {log}\left (\tan \left (f x +e \right )^{4}+4 \tan \left (f x +e \right )^{3} i -6 \tan \left (f x +e \right )^{2}-4 \tan \left (f x +e \right ) i +1\right ) i -2 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) i \right )}{4 c^{4} f} \] Input:
int((a+I*a*tan(f*x+e))^4/(c-I*c*tan(f*x+e))^4,x)
Output:
(a**4*(32*int(tan(e + f*x)**3/(tan(e + f*x)**4*i - 4*tan(e + f*x)**3 - 6*t an(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*c**4*f)