Integrand size = 31, antiderivative size = 87 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4} \, dx=-\frac {i a^3}{f (c-i c \tan (e+f x))^4}+\frac {4 i a^3}{3 c f (c-i c \tan (e+f x))^3}-\frac {i a^3}{2 f \left (c^2-i c^2 \tan (e+f x)\right )^2} \] Output:
-I*a^3/f/(c-I*c*tan(f*x+e))^4+4/3*I*a^3/c/f/(c-I*c*tan(f*x+e))^3-1/2*I*a^3 /f/(c^2-I*c^2*tan(f*x+e))^2
Time = 1.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.56 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4} \, dx=\frac {a^3 \left (-i+2 \tan (e+f x)+3 i \tan ^2(e+f x)\right )}{6 c^4 f (i+\tan (e+f x))^4} \] Input:
Integrate[(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^4,x]
Output:
(a^3*(-I + 2*Tan[e + f*x] + (3*I)*Tan[e + f*x]^2))/(6*c^4*f*(I + Tan[e + f *x])^4)
Time = 0.36 (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.
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4}dx\) |
\(\Big \downarrow \) 4005 |
\(\displaystyle a^3 c^3 \int \frac {\sec ^6(e+f x)}{(c-i c \tan (e+f x))^7}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^3 c^3 \int \frac {\sec (e+f x)^6}{(c-i c \tan (e+f x))^7}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle \frac {i a^3 \int \frac {(i \tan (e+f x) c+c)^2}{(c-i c \tan (e+f x))^5}d(-i c \tan (e+f x))}{c^2 f}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {i a^3 \int \left (\frac {4 c^2}{(c-i c \tan (e+f x))^5}-\frac {4 c}{(c-i c \tan (e+f x))^4}+\frac {1}{(c-i c \tan (e+f x))^3}\right )d(-i c \tan (e+f x))}{c^2 f}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i a^3 \left (-\frac {c^2}{(c-i c \tan (e+f x))^4}+\frac {4 c}{3 (c-i c \tan (e+f x))^3}-\frac {1}{2 (c-i c \tan (e+f x))^2}\right )}{c^2 f}\) |
Input:
Int[(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^4,x]
Output:
(I*a^3*(-(c^2/(c - I*c*Tan[e + f*x])^4) + (4*c)/(3*(c - I*c*Tan[e + f*x])^ 3) - 1/(2*(c - I*c*Tan[e + f*x])^2)))/(c^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 = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.51
method | result | size |
risch | \(-\frac {i a^{3} {\mathrm e}^{8 i \left (f x +e \right )}}{16 c^{4} f}-\frac {i a^{3} {\mathrm e}^{6 i \left (f x +e \right )}}{12 c^{4} f}\) | \(44\) |
derivativedivides | \(\frac {a^{3} \left (\frac {4}{3 \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {i}{2 \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {i}{\left (i+\tan \left (f x +e \right )\right )^{4}}\right )}{f \,c^{4}}\) | \(53\) |
default | \(\frac {a^{3} \left (\frac {4}{3 \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {i}{2 \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {i}{\left (i+\tan \left (f x +e \right )\right )^{4}}\right )}{f \,c^{4}}\) | \(53\) |
norman | \(\frac {\frac {a^{3} \tan \left (f x +e \right )}{c f}-\frac {i a^{3}}{6 c f}-\frac {14 a^{3} \tan \left (f x +e \right )^{3}}{3 c f}+\frac {7 a^{3} \tan \left (f x +e \right )^{5}}{3 c f}+\frac {i a^{3} \tan \left (f x +e \right )^{6}}{2 c f}+\frac {17 i a^{3} \tan \left (f x +e \right )^{2}}{6 c f}-\frac {9 i a^{3} \tan \left (f x +e \right )^{4}}{2 c f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{4} c^{3}}\) | \(144\) |
Input:
int((a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^4,x,method=_RETURNVERBOSE)
Output:
-1/16*I*a^3/c^4/f*exp(8*I*(f*x+e))-1/12*I*a^3/c^4/f*exp(6*I*(f*x+e))
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.43 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4} \, dx=\frac {-3 i \, a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} - 4 i \, a^{3} e^{\left (6 i \, f x + 6 i \, e\right )}}{48 \, c^{4} f} \] Input:
integrate((a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")
Output:
1/48*(-3*I*a^3*e^(8*I*f*x + 8*I*e) - 4*I*a^3*e^(6*I*f*x + 6*I*e))/(c^4*f)
Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.09 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4} \, dx=\begin {cases} \frac {- 12 i a^{3} c^{4} f e^{8 i e} e^{8 i f x} - 16 i a^{3} c^{4} f e^{6 i e} e^{6 i f x}}{192 c^{8} f^{2}} & \text {for}\: c^{8} f^{2} \neq 0 \\\frac {x \left (a^{3} e^{8 i e} + a^{3} e^{6 i e}\right )}{2 c^{4}} & \text {otherwise} \end {cases} \] Input:
integrate((a+I*a*tan(f*x+e))**3/(c-I*c*tan(f*x+e))**4,x)
Output:
Piecewise(((-12*I*a**3*c**4*f*exp(8*I*e)*exp(8*I*f*x) - 16*I*a**3*c**4*f*e xp(6*I*e)*exp(6*I*f*x))/(192*c**8*f**2), Ne(c**8*f**2, 0)), (x*(a**3*exp(8 *I*e) + a**3*exp(6*I*e))/(2*c**4), True))
Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+I*a*tan(f*x+e))^3/(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 = 0.55 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.55 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4} \, dx=-\frac {-3 i \, a^{3} \tan \left (f x + e\right )^{2} - 2 \, a^{3} \tan \left (f x + e\right ) + i \, a^{3}}{6 \, c^{4} f {\left (\tan \left (f x + e\right ) + i\right )}^{4}} \] Input:
integrate((a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^4,x, algorithm="giac")
Output:
-1/6*(-3*I*a^3*tan(f*x + e)^2 - 2*a^3*tan(f*x + e) + I*a^3)/(c^4*f*(tan(f* x + e) + I)^4)
Time = 1.87 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4} \, dx=\frac {a^3\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2\,3{}\mathrm {i}+2\,\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}{6\,c^4\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4+{\mathrm {tan}\left (e+f\,x\right )}^3\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (e+f\,x\right )}^2-\mathrm {tan}\left (e+f\,x\right )\,4{}\mathrm {i}+1\right )} \] Input:
int((a + a*tan(e + f*x)*1i)^3/(c - c*tan(e + f*x)*1i)^4,x)
Output:
(a^3*(2*tan(e + f*x) + tan(e + f*x)^2*3i - 1i))/(6*c^4*f*(tan(e + f*x)^3*4 i - 6*tan(e + f*x)^2 - tan(e + f*x)*4i + tan(e + f*x)^4 + 1))
\[ \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4} \, dx=\frac {a^{3} \left (\int \frac {\tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{4} i -4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i +4 \tan \left (f x +e \right )+i}d x -3 \left (\int \frac {\tan \left (f x +e \right )^{2}}{\tan \left (f x +e \right )^{4} i -4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i +4 \tan \left (f x +e \right )+i}d x \right ) i -3 \left (\int \frac {\tan \left (f x +e \right )}{\tan \left (f x +e \right )^{4} i -4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i +4 \tan \left (f x +e \right )+i}d x \right )+\left (\int \frac {1}{\tan \left (f x +e \right )^{4} i -4 \tan \left (f x +e \right )^{3}-6 \tan \left (f x +e \right )^{2} i +4 \tan \left (f x +e \right )+i}d x \right ) i \right )}{c^{4}} \] Input:
int((a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^4,x)
Output:
(a**3*(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) - 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))/c**4