Integrand size = 31, antiderivative size = 100 \[ \int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx=\frac {i a^5 c^4 \sec ^8(e+f x)}{8 f}+\frac {a^5 c^4 \tan (e+f x)}{f}+\frac {a^5 c^4 \tan ^3(e+f x)}{f}+\frac {3 a^5 c^4 \tan ^5(e+f x)}{5 f}+\frac {a^5 c^4 \tan ^7(e+f x)}{7 f} \] Output:
1/8*I*a^5*c^4*sec(f*x+e)^8/f+a^5*c^4*tan(f*x+e)/f+a^5*c^4*tan(f*x+e)^3/f+3 /5*a^5*c^4*tan(f*x+e)^5/f+1/7*a^5*c^4*tan(f*x+e)^7/f
Time = 0.66 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.84 \[ \int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx=\frac {a^5 c^4 \sec ^8(e+f x) (36 \cos (e+f x)+57 \cos (3 (e+f x))-5 i (4 \sin (e+f x)+11 \sin (3 (e+f x)))) (-i \cos (5 (e+f x))+\sin (5 (e+f x)))}{280 f} \] Input:
Integrate[(a + I*a*Tan[e + f*x])^5*(c - I*c*Tan[e + f*x])^4,x]
Output:
(a^5*c^4*Sec[e + f*x]^8*(36*Cos[e + f*x] + 57*Cos[3*(e + f*x)] - (5*I)*(4* Sin[e + f*x] + 11*Sin[3*(e + f*x)]))*((-I)*Cos[5*(e + f*x)] + Sin[5*(e + f *x)]))/(280*f)
Time = 0.40 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.75, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3042, 4005, 3042, 3967, 3042, 4254, 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.
| \(\displaystyle \int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4dx\) |
| \(\Big \downarrow \) 4005 |
| \(\displaystyle a^4 c^4 \int \sec ^8(e+f x) (i \tan (e+f x) a+a)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \int \sec (e+f x)^8 (i \tan (e+f x) a+a)dx\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle a^4 c^4 \left (a \int \sec ^8(e+f x)dx+\frac {i a \sec ^8(e+f x)}{8 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (a \int \csc \left (e+f x+\frac {\pi }{2}\right )^8dx+\frac {i a \sec ^8(e+f x)}{8 f}\right )\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle a^4 c^4 \left (-\frac {a \int \left (\tan ^6(e+f x)+3 \tan ^4(e+f x)+3 \tan ^2(e+f x)+1\right )d(-\tan (e+f x))}{f}+\frac {i a \sec ^8(e+f x)}{8 f}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 c^4 \left (-\frac {a \left (-\frac {1}{7} \tan ^7(e+f x)-\frac {3}{5} \tan ^5(e+f x)-\tan ^3(e+f x)-\tan (e+f x)\right )}{f}+\frac {i a \sec ^8(e+f x)}{8 f}\right )\) |
Input:
Int[(a + I*a*Tan[e + f*x])^5*(c - I*c*Tan[e + f*x])^4,x]
Output:
a^4*c^4*(((I/8)*a*Sec[e + f*x]^8)/f - (a*(-Tan[e + f*x] - Tan[e + f*x]^3 - (3*Tan[e + f*x]^5)/5 - Tan[e + f*x]^7/7))/f)
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
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]))
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Time = 0.43 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {32 i a^{5} c^{4} \left (70 \,{\mathrm e}^{8 i \left (f x +e \right )}+56 \,{\mathrm e}^{6 i \left (f x +e \right )}+28 \,{\mathrm e}^{4 i \left (f x +e \right )}+8 \,{\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{35 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{8}}\) | \(72\) |
| derivativedivides | \(\frac {i a^{5} c^{4} \left (\frac {\tan \left (f x +e \right )^{8}}{8}+\frac {\tan \left (f x +e \right )^{6}}{2}-\frac {i \tan \left (f x +e \right )^{7}}{7}+\frac {3 \tan \left (f x +e \right )^{4}}{4}-\frac {3 i \tan \left (f x +e \right )^{5}}{5}+\frac {\tan \left (f x +e \right )^{2}}{2}-i \tan \left (f x +e \right )^{3}-i \tan \left (f x +e \right )\right )}{f}\) | \(96\) |
| default | \(\frac {i a^{5} c^{4} \left (\frac {\tan \left (f x +e \right )^{8}}{8}+\frac {\tan \left (f x +e \right )^{6}}{2}-\frac {i \tan \left (f x +e \right )^{7}}{7}+\frac {3 \tan \left (f x +e \right )^{4}}{4}-\frac {3 i \tan \left (f x +e \right )^{5}}{5}+\frac {\tan \left (f x +e \right )^{2}}{2}-i \tan \left (f x +e \right )^{3}-i \tan \left (f x +e \right )\right )}{f}\) | \(96\) |
| parallelrisch | \(\frac {35 i a^{5} c^{4} \tan \left (f x +e \right )^{8}+140 i a^{5} c^{4} \tan \left (f x +e \right )^{6}+40 \tan \left (f x +e \right )^{7} a^{5} c^{4}+210 i a^{5} c^{4} \tan \left (f x +e \right )^{4}+168 \tan \left (f x +e \right )^{5} a^{5} c^{4}+140 i a^{5} c^{4} \tan \left (f x +e \right )^{2}+280 \tan \left (f x +e \right )^{3} a^{5} c^{4}+280 \tan \left (f x +e \right ) a^{5} c^{4}}{280 f}\) | \(137\) |
| norman | \(\frac {a^{5} c^{4} \tan \left (f x +e \right )}{f}+\frac {a^{5} c^{4} \tan \left (f x +e \right )^{3}}{f}+\frac {3 a^{5} c^{4} \tan \left (f x +e \right )^{5}}{5 f}+\frac {a^{5} c^{4} \tan \left (f x +e \right )^{7}}{7 f}+\frac {i a^{5} c^{4} \tan \left (f x +e \right )^{2}}{2 f}+\frac {3 i a^{5} c^{4} \tan \left (f x +e \right )^{4}}{4 f}+\frac {i a^{5} c^{4} \tan \left (f x +e \right )^{6}}{2 f}+\frac {i a^{5} c^{4} \tan \left (f x +e \right )^{8}}{8 f}\) | \(154\) |
| parts | \(a^{5} c^{4} x +\frac {a^{5} c^{4} \left (\frac {\tan \left (f x +e \right )^{7}}{7}-\frac {\tan \left (f x +e \right )^{5}}{5}+\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {i a^{5} c^{4} \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}+\frac {i a^{5} c^{4} \left (\frac {\tan \left (f x +e \right )^{8}}{8}-\frac {\tan \left (f x +e \right )^{6}}{6}+\frac {\tan \left (f x +e \right )^{4}}{4}-\frac {\tan \left (f x +e \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {4 i a^{5} c^{4} \left (\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {4 i a^{5} c^{4} \left (\frac {\tan \left (f x +e \right )^{6}}{6}-\frac {\tan \left (f x +e \right )^{4}}{4}+\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {6 i a^{5} c^{4} \left (\frac {\tan \left (f x +e \right )^{4}}{4}-\frac {\tan \left (f x +e \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {4 a^{5} c^{4} \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {6 a^{5} c^{4} \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {4 a^{5} c^{4} \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(404\) |
Input:
int((a+I*a*tan(f*x+e))^5*(c-I*c*tan(f*x+e))^4,x,method=_RETURNVERBOSE)
Output:
32/35*I*a^5*c^4*(70*exp(8*I*(f*x+e))+56*exp(6*I*(f*x+e))+28*exp(4*I*(f*x+e ))+8*exp(2*I*(f*x+e))+1)/f/(exp(2*I*(f*x+e))+1)^8
Time = 0.07 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.78 \[ \int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx=-\frac {32 \, {\left (-70 i \, a^{5} c^{4} e^{\left (8 i \, f x + 8 i \, e\right )} - 56 i \, a^{5} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} - 28 i \, a^{5} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} - 8 i \, a^{5} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{5} c^{4}\right )}}{35 \, {\left (f e^{\left (16 i \, f x + 16 i \, e\right )} + 8 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 28 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 56 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 70 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 56 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 28 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \] Input:
integrate((a+I*a*tan(f*x+e))^5*(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")
Output:
-32/35*(-70*I*a^5*c^4*e^(8*I*f*x + 8*I*e) - 56*I*a^5*c^4*e^(6*I*f*x + 6*I* e) - 28*I*a^5*c^4*e^(4*I*f*x + 4*I*e) - 8*I*a^5*c^4*e^(2*I*f*x + 2*I*e) - I*a^5*c^4)/(f*e^(16*I*f*x + 16*I*e) + 8*f*e^(14*I*f*x + 14*I*e) + 28*f*e^( 12*I*f*x + 12*I*e) + 56*f*e^(10*I*f*x + 10*I*e) + 70*f*e^(8*I*f*x + 8*I*e) + 56*f*e^(6*I*f*x + 6*I*e) + 28*f*e^(4*I*f*x + 4*I*e) + 8*f*e^(2*I*f*x + 2*I*e) + f)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (90) = 180\).
Time = 0.53 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.64 \[ \int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx=\frac {2240 i a^{5} c^{4} e^{8 i e} e^{8 i f x} + 1792 i a^{5} c^{4} e^{6 i e} e^{6 i f x} + 896 i a^{5} c^{4} e^{4 i e} e^{4 i f x} + 256 i a^{5} c^{4} e^{2 i e} e^{2 i f x} + 32 i a^{5} c^{4}}{35 f e^{16 i e} e^{16 i f x} + 280 f e^{14 i e} e^{14 i f x} + 980 f e^{12 i e} e^{12 i f x} + 1960 f e^{10 i e} e^{10 i f x} + 2450 f e^{8 i e} e^{8 i f x} + 1960 f e^{6 i e} e^{6 i f x} + 980 f e^{4 i e} e^{4 i f x} + 280 f e^{2 i e} e^{2 i f x} + 35 f} \] Input:
integrate((a+I*a*tan(f*x+e))**5*(c-I*c*tan(f*x+e))**4,x)
Output:
(2240*I*a**5*c**4*exp(8*I*e)*exp(8*I*f*x) + 1792*I*a**5*c**4*exp(6*I*e)*ex p(6*I*f*x) + 896*I*a**5*c**4*exp(4*I*e)*exp(4*I*f*x) + 256*I*a**5*c**4*exp (2*I*e)*exp(2*I*f*x) + 32*I*a**5*c**4)/(35*f*exp(16*I*e)*exp(16*I*f*x) + 2 80*f*exp(14*I*e)*exp(14*I*f*x) + 980*f*exp(12*I*e)*exp(12*I*f*x) + 1960*f* exp(10*I*e)*exp(10*I*f*x) + 2450*f*exp(8*I*e)*exp(8*I*f*x) + 1960*f*exp(6* I*e)*exp(6*I*f*x) + 980*f*exp(4*I*e)*exp(4*I*f*x) + 280*f*exp(2*I*e)*exp(2 *I*f*x) + 35*f)
Time = 0.15 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.32 \[ \int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx=-\frac {-35 i \, a^{5} c^{4} \tan \left (f x + e\right )^{8} - 40 \, a^{5} c^{4} \tan \left (f x + e\right )^{7} - 140 i \, a^{5} c^{4} \tan \left (f x + e\right )^{6} - 168 \, a^{5} c^{4} \tan \left (f x + e\right )^{5} - 210 i \, a^{5} c^{4} \tan \left (f x + e\right )^{4} - 280 \, a^{5} c^{4} \tan \left (f x + e\right )^{3} - 140 i \, a^{5} c^{4} \tan \left (f x + e\right )^{2} - 280 \, a^{5} c^{4} \tan \left (f x + e\right )}{280 \, f} \] Input:
integrate((a+I*a*tan(f*x+e))^5*(c-I*c*tan(f*x+e))^4,x, algorithm="maxima")
Output:
-1/280*(-35*I*a^5*c^4*tan(f*x + e)^8 - 40*a^5*c^4*tan(f*x + e)^7 - 140*I*a ^5*c^4*tan(f*x + e)^6 - 168*a^5*c^4*tan(f*x + e)^5 - 210*I*a^5*c^4*tan(f*x + e)^4 - 280*a^5*c^4*tan(f*x + e)^3 - 140*I*a^5*c^4*tan(f*x + e)^2 - 280* a^5*c^4*tan(f*x + e))/f
Time = 0.68 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.32 \[ \int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx=-\frac {-35 i \, a^{5} c^{4} \tan \left (f x + e\right )^{8} - 40 \, a^{5} c^{4} \tan \left (f x + e\right )^{7} - 140 i \, a^{5} c^{4} \tan \left (f x + e\right )^{6} - 168 \, a^{5} c^{4} \tan \left (f x + e\right )^{5} - 210 i \, a^{5} c^{4} \tan \left (f x + e\right )^{4} - 280 \, a^{5} c^{4} \tan \left (f x + e\right )^{3} - 140 i \, a^{5} c^{4} \tan \left (f x + e\right )^{2} - 280 \, a^{5} c^{4} \tan \left (f x + e\right )}{280 \, f} \] Input:
integrate((a+I*a*tan(f*x+e))^5*(c-I*c*tan(f*x+e))^4,x, algorithm="giac")
Output:
-1/280*(-35*I*a^5*c^4*tan(f*x + e)^8 - 40*a^5*c^4*tan(f*x + e)^7 - 140*I*a ^5*c^4*tan(f*x + e)^6 - 168*a^5*c^4*tan(f*x + e)^5 - 210*I*a^5*c^4*tan(f*x + e)^4 - 280*a^5*c^4*tan(f*x + e)^3 - 140*I*a^5*c^4*tan(f*x + e)^2 - 280* a^5*c^4*tan(f*x + e))/f
Time = 1.92 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95 \[ \int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx=\frac {a^5\,c^4\,\left (-{\cos \left (e+f\,x\right )}^8\,35{}\mathrm {i}+128\,\sin \left (e+f\,x\right )\,{\cos \left (e+f\,x\right )}^7+64\,\sin \left (e+f\,x\right )\,{\cos \left (e+f\,x\right )}^5+48\,\sin \left (e+f\,x\right )\,{\cos \left (e+f\,x\right )}^3+40\,\sin \left (e+f\,x\right )\,\cos \left (e+f\,x\right )+35{}\mathrm {i}\right )}{280\,f\,{\cos \left (e+f\,x\right )}^8} \] Input:
int((a + a*tan(e + f*x)*1i)^5*(c - c*tan(e + f*x)*1i)^4,x)
Output:
(a^5*c^4*(40*cos(e + f*x)*sin(e + f*x) + 48*cos(e + f*x)^3*sin(e + f*x) + 64*cos(e + f*x)^5*sin(e + f*x) + 128*cos(e + f*x)^7*sin(e + f*x) - cos(e + f*x)^8*35i + 35i))/(280*f*cos(e + f*x)^8)
Time = 0.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.91 \[ \int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx=\frac {\tan \left (f x +e \right ) a^{5} c^{4} \left (35 \tan \left (f x +e \right )^{7} i +40 \tan \left (f x +e \right )^{6}+140 \tan \left (f x +e \right )^{5} i +168 \tan \left (f x +e \right )^{4}+210 \tan \left (f x +e \right )^{3} i +280 \tan \left (f x +e \right )^{2}+140 \tan \left (f x +e \right ) i +280\right )}{280 f} \] Input:
int((a+I*a*tan(f*x+e))^5*(c-I*c*tan(f*x+e))^4,x)
Output:
(tan(e + f*x)*a**5*c**4*(35*tan(e + f*x)**7*i + 40*tan(e + f*x)**6 + 140*t an(e + f*x)**5*i + 168*tan(e + f*x)**4 + 210*tan(e + f*x)**3*i + 280*tan(e + f*x)**2 + 140*tan(e + f*x)*i + 280))/(280*f)