Integrand size = 31, antiderivative size = 87 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5} \, dx=\frac {i c^4 (1-i \tan (e+f x))^4}{10 f (a+i a \tan (e+f x))^5}+\frac {i c^4 (a-i a \tan (e+f x))^4}{80 a^5 f (a+i a \tan (e+f x))^4} \] Output:
1/10*I*c^4*(1-I*tan(f*x+e))^4/f/(a+I*a*tan(f*x+e))^5+1/80*I*c^4*(a-I*a*tan (f*x+e))^4/a^5/f/(a+I*a*tan(f*x+e))^4
Time = 5.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.56 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5} \, dx=\frac {c^4 (9+i \tan (e+f x)) (i+\tan (e+f x))^4}{80 a^5 f (-i+\tan (e+f x))^5} \] Input:
Integrate[(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^5,x]
Output:
(c^4*(9 + I*Tan[e + f*x])*(I + Tan[e + f*x])^4)/(80*a^5*f*(-I + Tan[e + f* x])^5)
Time = 0.34 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3042, 4005, 3042, 3968, 55, 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.
\(\displaystyle \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5}dx\) |
\(\Big \downarrow \) 4005 |
\(\displaystyle a^4 c^4 \int \frac {\sec ^8(e+f x)}{(i \tan (e+f x) a+a)^9}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^4 c^4 \int \frac {\sec (e+f x)^8}{(i \tan (e+f x) a+a)^9}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i c^4 \int \frac {(a-i a \tan (e+f x))^3}{(i \tan (e+f x) a+a)^6}d(i a \tan (e+f x))}{a^3 f}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {i c^4 \left (\frac {\int \frac {(a-i a \tan (e+f x))^3}{(i \tan (e+f x) a+a)^5}d(i a \tan (e+f x))}{10 a}-\frac {(a-i a \tan (e+f x))^4}{10 a (a+i a \tan (e+f x))^5}\right )}{a^3 f}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {i c^4 \left (-\frac {(a-i a \tan (e+f x))^4}{80 a^2 (a+i a \tan (e+f x))^4}-\frac {(a-i a \tan (e+f x))^4}{10 a (a+i a \tan (e+f x))^5}\right )}{a^3 f}\) |
Input:
Int[(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^5,x]
Output:
((-I)*c^4*(-1/10*(a - I*a*Tan[e + f*x])^4/(a*(a + I*a*Tan[e + f*x])^5) - ( a - I*a*Tan[e + f*x])^4/(80*a^2*(a + I*a*Tan[e + f*x])^4)))/(a^3*f)
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[((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] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 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 = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.51
method | result | size |
risch | \(\frac {i c^{4} {\mathrm e}^{-8 i \left (f x +e \right )}}{16 a^{5} f}+\frac {i c^{4} {\mathrm e}^{-10 i \left (f x +e \right )}}{20 a^{5} f}\) | \(44\) |
derivativedivides | \(\frac {c^{4} \left (\frac {i}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {3 i}{\left (-i+\tan \left (f x +e \right )\right )^{4}}+\frac {8}{5 \left (-i+\tan \left (f x +e \right )\right )^{5}}-\frac {2}{\left (-i+\tan \left (f x +e \right )\right )^{3}}\right )}{f \,a^{5}}\) | \(66\) |
default | \(\frac {c^{4} \left (\frac {i}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {3 i}{\left (-i+\tan \left (f x +e \right )\right )^{4}}+\frac {8}{5 \left (-i+\tan \left (f x +e \right )\right )^{5}}-\frac {2}{\left (-i+\tan \left (f x +e \right )\right )^{3}}\right )}{f \,a^{5}}\) | \(66\) |
norman | \(\frac {\frac {c^{4} \tan \left (f x +e \right )}{a f}-\frac {4 i c^{4} \tan \left (f x +e \right )^{2}}{a f}+\frac {13 i c^{4} \tan \left (f x +e \right )^{4}}{a f}+\frac {i c^{4}}{10 a f}-\frac {9 c^{4} \tan \left (f x +e \right )^{3}}{a f}+\frac {63 c^{4} \tan \left (f x +e \right )^{5}}{5 a f}-\frac {3 c^{4} \tan \left (f x +e \right )^{7}}{a f}+\frac {i c^{4} \tan \left (f x +e \right )^{8}}{2 a f}-\frac {8 i c^{4} \tan \left (f x +e \right )^{6}}{a f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{5} a^{4}}\) | \(183\) |
Input:
int((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^5,x,method=_RETURNVERBOSE)
Output:
1/16*I*c^4/a^5/f*exp(-8*I*(f*x+e))+1/20*I*c^4/a^5/f*exp(-10*I*(f*x+e))
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.43 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5} \, dx=\frac {{\left (5 i \, c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i \, c^{4}\right )} e^{\left (-10 i \, f x - 10 i \, e\right )}}{80 \, a^{5} f} \] Input:
integrate((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^5,x, algorithm="fricas")
Output:
1/80*(5*I*c^4*e^(2*I*f*x + 2*I*e) + 4*I*c^4)*e^(-10*I*f*x - 10*I*e)/(a^5*f )
Time = 0.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.23 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5} \, dx=\begin {cases} \frac {\left (20 i a^{5} c^{4} f e^{10 i e} e^{- 8 i f x} + 16 i a^{5} c^{4} f e^{8 i e} e^{- 10 i f x}\right ) e^{- 18 i e}}{320 a^{10} f^{2}} & \text {for}\: a^{10} f^{2} e^{18 i e} \neq 0 \\\frac {x \left (c^{4} e^{2 i e} + c^{4}\right ) e^{- 10 i e}}{2 a^{5}} & \text {otherwise} \end {cases} \] Input:
integrate((c-I*c*tan(f*x+e))**4/(a+I*a*tan(f*x+e))**5,x)
Output:
Piecewise(((20*I*a**5*c**4*f*exp(10*I*e)*exp(-8*I*f*x) + 16*I*a**5*c**4*f* exp(8*I*e)*exp(-10*I*f*x))*exp(-18*I*e)/(320*a**10*f**2), Ne(a**10*f**2*ex p(18*I*e), 0)), (x*(c**4*exp(2*I*e) + c**4)*exp(-10*I*e)/(2*a**5), True))
Exception generated. \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^5,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.74 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.70 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5} \, dx=-\frac {-5 i \, c^{4} \tan \left (f x + e\right )^{3} + 5 \, c^{4} \tan \left (f x + e\right )^{2} + 5 i \, c^{4} \tan \left (f x + e\right ) - c^{4}}{10 \, a^{5} f {\left (\tan \left (f x + e\right ) - i\right )}^{5}} \] Input:
integrate((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^5,x, algorithm="giac")
Output:
-1/10*(-5*I*c^4*tan(f*x + e)^3 + 5*c^4*tan(f*x + e)^2 + 5*I*c^4*tan(f*x + e) - c^4)/(a^5*f*(tan(f*x + e) - I)^5)
Time = 1.93 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5} \, dx=\frac {c^4\,\left (-5\,{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,5{}\mathrm {i}+5\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}{10\,a^5\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^5\,1{}\mathrm {i}+5\,{\mathrm {tan}\left (e+f\,x\right )}^4-{\mathrm {tan}\left (e+f\,x\right )}^3\,10{}\mathrm {i}-10\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,5{}\mathrm {i}+1\right )} \] Input:
int((c - c*tan(e + f*x)*1i)^4/(a + a*tan(e + f*x)*1i)^5,x)
Output:
(c^4*(5*tan(e + f*x) - tan(e + f*x)^2*5i - 5*tan(e + f*x)^3 + 1i))/(10*a^5 *f*(tan(e + f*x)*5i - 10*tan(e + f*x)^2 - tan(e + f*x)^3*10i + 5*tan(e + f *x)^4 + tan(e + f*x)^5*1i + 1))
\[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5} \, dx=\frac {c^{4} \left (\int \frac {\tan \left (f x +e \right )^{4}}{\tan \left (f x +e \right )^{5} i +5 \tan \left (f x +e \right )^{4}-10 \tan \left (f x +e \right )^{3} i -10 \tan \left (f x +e \right )^{2}+5 \tan \left (f x +e \right ) i +1}d x +4 \left (\int \frac {\tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{5} i +5 \tan \left (f x +e \right )^{4}-10 \tan \left (f x +e \right )^{3} i -10 \tan \left (f x +e \right )^{2}+5 \tan \left (f x +e \right ) i +1}d x \right ) i -6 \left (\int \frac {\tan \left (f x +e \right )^{2}}{\tan \left (f x +e \right )^{5} i +5 \tan \left (f x +e \right )^{4}-10 \tan \left (f x +e \right )^{3} i -10 \tan \left (f x +e \right )^{2}+5 \tan \left (f x +e \right ) i +1}d x \right )-4 \left (\int \frac {\tan \left (f x +e \right )}{\tan \left (f x +e \right )^{5} i +5 \tan \left (f x +e \right )^{4}-10 \tan \left (f x +e \right )^{3} i -10 \tan \left (f x +e \right )^{2}+5 \tan \left (f x +e \right ) i +1}d x \right ) i +\int \frac {1}{\tan \left (f x +e \right )^{5} i +5 \tan \left (f x +e \right )^{4}-10 \tan \left (f x +e \right )^{3} i -10 \tan \left (f x +e \right )^{2}+5 \tan \left (f x +e \right ) i +1}d x \right )}{a^{5}} \] Input:
int((c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^5,x)
Output:
(c**4*(int(tan(e + f*x)**4/(tan(e + f*x)**5*i + 5*tan(e + f*x)**4 - 10*tan (e + f*x)**3*i - 10*tan(e + f*x)**2 + 5*tan(e + f*x)*i + 1),x) + 4*int(tan (e + f*x)**3/(tan(e + f*x)**5*i + 5*tan(e + f*x)**4 - 10*tan(e + f*x)**3*i - 10*tan(e + f*x)**2 + 5*tan(e + f*x)*i + 1),x)*i - 6*int(tan(e + f*x)**2 /(tan(e + f*x)**5*i + 5*tan(e + f*x)**4 - 10*tan(e + f*x)**3*i - 10*tan(e + f*x)**2 + 5*tan(e + f*x)*i + 1),x) - 4*int(tan(e + f*x)/(tan(e + f*x)**5 *i + 5*tan(e + f*x)**4 - 10*tan(e + f*x)**3*i - 10*tan(e + f*x)**2 + 5*tan (e + f*x)*i + 1),x)*i + int(1/(tan(e + f*x)**5*i + 5*tan(e + f*x)**4 - 10* tan(e + f*x)**3*i - 10*tan(e + f*x)**2 + 5*tan(e + f*x)*i + 1),x)))/a**5