Integrand size = 31, antiderivative size = 87 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4} \, dx=\frac {i c^3}{f (a+i a \tan (e+f x))^4}-\frac {4 i c^3}{3 a f (a+i a \tan (e+f x))^3}+\frac {i c^3}{2 f \left (a^2+i a^2 \tan (e+f x)\right )^2} \] Output:
I*c^3/f/(a+I*a*tan(f*x+e))^4-4/3*I*c^3/a/f/(a+I*a*tan(f*x+e))^3+1/2*I*c^3/ f/(a^2+I*a^2*tan(f*x+e))^2
Time = 5.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.56 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4} \, dx=\frac {c^3 \left (i+2 \tan (e+f x)-3 i \tan ^2(e+f x)\right )}{6 a^4 f (-i+\tan (e+f x))^4} \] Input:
Integrate[(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^4,x]
Output:
(c^3*(I + 2*Tan[e + f*x] - (3*I)*Tan[e + f*x]^2))/(6*a^4*f*(-I + Tan[e + f *x])^4)
Time = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84, 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.
\(\displaystyle \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4}dx\) |
\(\Big \downarrow \) 4005 |
\(\displaystyle a^3 c^3 \int \frac {\sec ^6(e+f x)}{(i \tan (e+f x) a+a)^7}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^3 c^3 \int \frac {\sec (e+f x)^6}{(i \tan (e+f x) a+a)^7}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i c^3 \int \frac {(a-i a \tan (e+f x))^2}{(i \tan (e+f x) a+a)^5}d(i a \tan (e+f x))}{a^2 f}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {i c^3 \int \left (\frac {4 a^2}{(i \tan (e+f x) a+a)^5}-\frac {4 a}{(i \tan (e+f x) a+a)^4}+\frac {1}{(i \tan (e+f x) a+a)^3}\right )d(i a \tan (e+f x))}{a^2 f}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i c^3 \left (-\frac {a^2}{(a+i a \tan (e+f x))^4}+\frac {4 a}{3 (a+i a \tan (e+f x))^3}-\frac {1}{2 (a+i a \tan (e+f x))^2}\right )}{a^2 f}\) |
Input:
Int[(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^4,x]
Output:
((-I)*c^3*(-(a^2/(a + I*a*Tan[e + f*x])^4) + (4*a)/(3*(a + I*a*Tan[e + f*x ])^3) - 1/(2*(a + I*a*Tan[e + f*x])^2)))/(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 = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.51
method | result | size |
risch | \(\frac {i c^{3} {\mathrm e}^{-6 i \left (f x +e \right )}}{12 a^{4} f}+\frac {i c^{3} {\mathrm e}^{-8 i \left (f x +e \right )}}{16 a^{4} f}\) | \(44\) |
derivativedivides | \(\frac {c^{3} \left (\frac {4}{3 \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {i}{\left (-i+\tan \left (f x +e \right )\right )^{4}}-\frac {i}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}\right )}{f \,a^{4}}\) | \(53\) |
default | \(\frac {c^{3} \left (\frac {4}{3 \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {i}{\left (-i+\tan \left (f x +e \right )\right )^{4}}-\frac {i}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}\right )}{f \,a^{4}}\) | \(53\) |
norman | \(\frac {\frac {c^{3} \tan \left (f x +e \right )}{a f}+\frac {i c^{3}}{6 a f}-\frac {14 c^{3} \tan \left (f x +e \right )^{3}}{3 a f}+\frac {7 c^{3} \tan \left (f x +e \right )^{5}}{3 a f}-\frac {17 i c^{3} \tan \left (f x +e \right )^{2}}{6 a f}-\frac {i c^{3} \tan \left (f x +e \right )^{6}}{2 a f}+\frac {9 i c^{3} \tan \left (f x +e \right )^{4}}{2 a f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{4} a^{3}}\) | \(144\) |
Input:
int((c-I*c*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^4,x,method=_RETURNVERBOSE)
Output:
1/12*I*c^3/a^4/f*exp(-6*I*(f*x+e))+1/16*I*c^3/a^4/f*exp(-8*I*(f*x+e))
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.43 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4} \, dx=\frac {{\left (4 i \, c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, c^{3}\right )} e^{\left (-8 i \, f x - 8 i \, e\right )}}{48 \, a^{4} f} \] Input:
integrate((c-I*c*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^4,x, algorithm="fricas")
Output:
1/48*(4*I*c^3*e^(2*I*f*x + 2*I*e) + 3*I*c^3)*e^(-8*I*f*x - 8*I*e)/(a^4*f)
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.23 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4} \, dx=\begin {cases} \frac {\left (16 i a^{4} c^{3} f e^{8 i e} e^{- 6 i f x} + 12 i a^{4} c^{3} f e^{6 i e} e^{- 8 i f x}\right ) e^{- 14 i e}}{192 a^{8} f^{2}} & \text {for}\: a^{8} f^{2} e^{14 i e} \neq 0 \\\frac {x \left (c^{3} e^{2 i e} + c^{3}\right ) e^{- 8 i e}}{2 a^{4}} & \text {otherwise} \end {cases} \] Input:
integrate((c-I*c*tan(f*x+e))**3/(a+I*a*tan(f*x+e))**4,x)
Output:
Piecewise(((16*I*a**4*c**3*f*exp(8*I*e)*exp(-6*I*f*x) + 12*I*a**4*c**3*f*e xp(6*I*e)*exp(-8*I*f*x))*exp(-14*I*e)/(192*a**8*f**2), Ne(a**8*f**2*exp(14 *I*e), 0)), (x*(c**3*exp(2*I*e) + c**3)*exp(-8*I*e)/(2*a**4), True))
Exception generated. \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((c-I*c*tan(f*x+e))^3/(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 = 0.65 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.55 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4} \, dx=-\frac {3 i \, c^{3} \tan \left (f x + e\right )^{2} - 2 \, c^{3} \tan \left (f x + e\right ) - i \, c^{3}}{6 \, a^{4} f {\left (\tan \left (f x + e\right ) - i\right )}^{4}} \] Input:
integrate((c-I*c*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^4,x, algorithm="giac")
Output:
-1/6*(3*I*c^3*tan(f*x + e)^2 - 2*c^3*tan(f*x + e) - I*c^3)/(a^4*f*(tan(f*x + e) - I)^4)
Time = 1.86 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4} \, dx=\frac {c^3\,\left (3\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}-1\right )}{6\,a^4\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4\,1{}\mathrm {i}+4\,{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,6{}\mathrm {i}-4\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )} \] Input:
int((c - c*tan(e + f*x)*1i)^3/(a + a*tan(e + f*x)*1i)^4,x)
Output:
(c^3*(tan(e + f*x)*2i + 3*tan(e + f*x)^2 - 1))/(6*a^4*f*(4*tan(e + f*x)^3 - tan(e + f*x)^2*6i - 4*tan(e + f*x) + tan(e + f*x)^4*1i + 1i))
\[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4} \, dx=\frac {c^{3} \left (-\left (\int \frac {\tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{4} i +4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i -4 \tan \left (f x +e \right )+i}d x \right )-3 \left (\int \frac {\tan \left (f x +e \right )^{2}}{\tan \left (f x +e \right )^{4} i +4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i -4 \tan \left (f x +e \right )+i}d x \right ) i +3 \left (\int \frac {\tan \left (f x +e \right )}{\tan \left (f x +e \right )^{4} i +4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i -4 \tan \left (f x +e \right )+i}d x \right )+\left (\int \frac {1}{\tan \left (f x +e \right )^{4} i +4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i -4 \tan \left (f x +e \right )+i}d x \right ) i \right )}{a^{4}} \] Input:
int((c-I*c*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^4,x)
Output:
(c**3*( - 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) - 3*int(tan(e + f*x)**2/(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)*i + 3*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) + int(1/(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)*i))/a**4