Integrand size = 31, antiderivative size = 90 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5} \, dx=\frac {4 i c^3}{5 f (a+i a \tan (e+f x))^5}+\frac {i c^3}{3 a^2 f (a+i a \tan (e+f x))^3}-\frac {i a^3 c^3}{f \left (a^2+i a^2 \tan (e+f x)\right )^4} \] Output:
4/5*I*c^3/f/(a+I*a*tan(f*x+e))^5+1/3*I*c^3/a^2/f/(a+I*a*tan(f*x+e))^3-I*a^ 3*c^3/f/(a^2+I*a^2*tan(f*x+e))^4
Time = 3.69 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.52 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5} \, dx=-\frac {c^3 \left (-2+5 i \tan (e+f x)+5 \tan ^2(e+f x)\right )}{15 a^5 f (-i+\tan (e+f x))^5} \] Input:
Integrate[(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^5,x]
Output:
-1/15*(c^3*(-2 + (5*I)*Tan[e + f*x] + 5*Tan[e + f*x]^2))/(a^5*f*(-I + Tan[ e + f*x])^5)
Time = 0.36 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.80, 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, 53, 2009}
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.
\(\displaystyle \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5}dx\) |
\(\Big \downarrow \) 4005 |
\(\displaystyle a^3 c^3 \int \frac {\sec ^6(e+f x)}{(i \tan (e+f x) a+a)^8}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^3 c^3 \int \frac {\sec (e+f x)^6}{(i \tan (e+f x) a+a)^8}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i c^3 \int \frac {(a-i a \tan (e+f x))^2}{(i \tan (e+f x) a+a)^6}d(i a \tan (e+f x))}{a^2 f}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {i c^3 \int \left (\frac {4 a^2}{(i \tan (e+f x) a+a)^6}-\frac {4 a}{(i \tan (e+f x) a+a)^5}+\frac {1}{(i \tan (e+f x) a+a)^4}\right )d(i a \tan (e+f x))}{a^2 f}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i c^3 \left (-\frac {4 a^2}{5 (a+i a \tan (e+f x))^5}+\frac {a}{(a+i a \tan (e+f x))^4}-\frac {1}{3 (a+i a \tan (e+f x))^3}\right )}{a^2 f}\) |
Input:
Int[(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^5,x]
Output:
((-I)*c^3*((-4*a^2)/(5*(a + I*a*Tan[e + f*x])^5) + a/(a + I*a*Tan[e + f*x] )^4 - 1/(3*(a + I*a*Tan[e + f*x])^3)))/(a^2*f)
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] && 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])
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 = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.58
method | result | size |
derivativedivides | \(\frac {c^{3} \left (-\frac {i}{\left (-i+\tan \left (f x +e \right )\right )^{4}}-\frac {1}{3 \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {4}{5 \left (-i+\tan \left (f x +e \right )\right )^{5}}\right )}{f \,a^{5}}\) | \(52\) |
default | \(\frac {c^{3} \left (-\frac {i}{\left (-i+\tan \left (f x +e \right )\right )^{4}}-\frac {1}{3 \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {4}{5 \left (-i+\tan \left (f x +e \right )\right )^{5}}\right )}{f \,a^{5}}\) | \(52\) |
risch | \(\frac {i c^{3} {\mathrm e}^{-6 i \left (f x +e \right )}}{24 a^{5} f}+\frac {i c^{3} {\mathrm e}^{-8 i \left (f x +e \right )}}{16 a^{5} f}+\frac {i c^{3} {\mathrm e}^{-10 i \left (f x +e \right )}}{40 a^{5} f}\) | \(65\) |
norman | \(\frac {\frac {c^{3} \tan \left (f x +e \right )}{a f}+\frac {2 i c^{3}}{15 a f}-\frac {19 c^{3} \tan \left (f x +e \right )^{3}}{3 a f}+\frac {77 c^{3} \tan \left (f x +e \right )^{5}}{15 a f}-\frac {c^{3} \tan \left (f x +e \right )^{7}}{3 a f}-\frac {10 i c^{3} \tan \left (f x +e \right )^{2}}{3 a f}-\frac {2 i c^{3} \tan \left (f x +e \right )^{6}}{a f}+\frac {22 i c^{3} \tan \left (f x +e \right )^{4}}{3 a f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{5} a^{4}}\) | \(163\) |
Input:
int((c-I*c*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^5,x,method=_RETURNVERBOSE)
Output:
1/f*c^3/a^5*(-I/(-I+tan(f*x+e))^4-1/3/(-I+tan(f*x+e))^3+4/5/(-I+tan(f*x+e) )^5)
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.57 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5} \, dx=\frac {{\left (10 i \, c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 15 i \, c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i \, c^{3}\right )} e^{\left (-10 i \, f x - 10 i \, e\right )}}{240 \, a^{5} f} \] Input:
integrate((c-I*c*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^5,x, algorithm="fricas")
Output:
1/240*(10*I*c^3*e^(4*I*f*x + 4*I*e) + 15*I*c^3*e^(2*I*f*x + 2*I*e) + 6*I*c ^3)*e^(-10*I*f*x - 10*I*e)/(a^5*f)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (75) = 150\).
Time = 0.36 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.68 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5} \, dx=\begin {cases} \frac {\left (640 i a^{10} c^{3} f^{2} e^{18 i e} e^{- 6 i f x} + 960 i a^{10} c^{3} f^{2} e^{16 i e} e^{- 8 i f x} + 384 i a^{10} c^{3} f^{2} e^{14 i e} e^{- 10 i f x}\right ) e^{- 24 i e}}{15360 a^{15} f^{3}} & \text {for}\: a^{15} f^{3} e^{24 i e} \neq 0 \\\frac {x \left (c^{3} e^{4 i e} + 2 c^{3} e^{2 i e} + c^{3}\right ) e^{- 10 i e}}{4 a^{5}} & \text {otherwise} \end {cases} \] Input:
integrate((c-I*c*tan(f*x+e))**3/(a+I*a*tan(f*x+e))**5,x)
Output:
Piecewise(((640*I*a**10*c**3*f**2*exp(18*I*e)*exp(-6*I*f*x) + 960*I*a**10* c**3*f**2*exp(16*I*e)*exp(-8*I*f*x) + 384*I*a**10*c**3*f**2*exp(14*I*e)*ex p(-10*I*f*x))*exp(-24*I*e)/(15360*a**15*f**3), Ne(a**15*f**3*exp(24*I*e), 0)), (x*(c**3*exp(4*I*e) + 2*c**3*exp(2*I*e) + c**3)*exp(-10*I*e)/(4*a**5) , True))
Exception generated. \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((c-I*c*tan(f*x+e))^3/(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 = 0.60 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.53 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5} \, dx=-\frac {5 \, c^{3} \tan \left (f x + e\right )^{2} + 5 i \, c^{3} \tan \left (f x + e\right ) - 2 \, c^{3}}{15 \, a^{5} f {\left (\tan \left (f x + e\right ) - i\right )}^{5}} \] Input:
integrate((c-I*c*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^5,x, algorithm="giac")
Output:
-1/15*(5*c^3*tan(f*x + e)^2 + 5*I*c^3*tan(f*x + e) - 2*c^3)/(a^5*f*(tan(f* x + e) - I)^5)
Time = 1.94 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5} \, dx=\frac {c^3\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^2\,5{}\mathrm {i}+5\,\mathrm {tan}\left (e+f\,x\right )+2{}\mathrm {i}\right )}{15\,a^5\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^5\,1{}\mathrm {i}+5\,{\mathrm {tan}\left (e+f\,x\right )}^4-{\mathrm {tan}\left (e+f\,x\right )}^3\,10{}\mathrm {i}-10\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,5{}\mathrm {i}+1\right )} \] Input:
int((c - c*tan(e + f*x)*1i)^3/(a + a*tan(e + f*x)*1i)^5,x)
Output:
(c^3*(5*tan(e + f*x) - tan(e + f*x)^2*5i + 2i))/(15*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))
\[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5} \, dx=\frac {c^{3} \left (\left (\int \frac {\tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{5} i +5 \tan \left (f x +e \right )^{4}-10 \tan \left (f x +e \right )^{3} i -10 \tan \left (f x +e \right )^{2}+5 \tan \left (f x +e \right ) i +1}d x \right ) i -3 \left (\int \frac {\tan \left (f x +e \right )^{2}}{\tan \left (f x +e \right )^{5} i +5 \tan \left (f x +e \right )^{4}-10 \tan \left (f x +e \right )^{3} i -10 \tan \left (f x +e \right )^{2}+5 \tan \left (f x +e \right ) i +1}d x \right )-3 \left (\int \frac {\tan \left (f x +e \right )}{\tan \left (f x +e \right )^{5} i +5 \tan \left (f x +e \right )^{4}-10 \tan \left (f x +e \right )^{3} i -10 \tan \left (f x +e \right )^{2}+5 \tan \left (f x +e \right ) i +1}d x \right ) i +\int \frac {1}{\tan \left (f x +e \right )^{5} i +5 \tan \left (f x +e \right )^{4}-10 \tan \left (f x +e \right )^{3} i -10 \tan \left (f x +e \right )^{2}+5 \tan \left (f x +e \right ) i +1}d x \right )}{a^{5}} \] Input:
int((c-I*c*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^5,x)
Output:
(c**3*(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 - 3*int(t an(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) - 3*int(tan(e + f*x)/(t an(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