Integrand size = 31, antiderivative size = 77 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx=\frac {a^4 c^4 \tan (e+f x)}{f}+\frac {a^4 c^4 \tan ^3(e+f x)}{f}+\frac {3 a^4 c^4 \tan ^5(e+f x)}{5 f}+\frac {a^4 c^4 \tan ^7(e+f x)}{7 f} \] Output:
a^4*c^4*tan(f*x+e)/f+a^4*c^4*tan(f*x+e)^3/f+3/5*a^4*c^4*tan(f*x+e)^5/f+1/7 *a^4*c^4*tan(f*x+e)^7/f
Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.64 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx=\frac {a^4 c^4 \left (\tan (e+f x)+\tan ^3(e+f x)+\frac {3}{5} \tan ^5(e+f x)+\frac {1}{7} \tan ^7(e+f x)\right )}{f} \] Input:
Integrate[(a + I*a*Tan[e + f*x])^4*(c - I*c*Tan[e + f*x])^4,x]
Output:
(a^4*c^4*(Tan[e + f*x] + Tan[e + f*x]^3 + (3*Tan[e + f*x]^5)/5 + Tan[e + f *x]^7/7))/f
Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.70, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3042, 4005, 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.
\(\displaystyle \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4dx\) |
\(\Big \downarrow \) 4005 |
\(\displaystyle a^4 c^4 \int \sec ^8(e+f x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^4 c^4 \int \csc \left (e+f x+\frac {\pi }{2}\right )^8dx\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {a^4 c^4 \int \left (\tan ^6(e+f x)+3 \tan ^4(e+f x)+3 \tan ^2(e+f x)+1\right )d(-\tan (e+f x))}{f}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^4 c^4 \left (-\frac {1}{7} \tan ^7(e+f x)-\frac {3}{5} \tan ^5(e+f x)-\tan ^3(e+f x)-\tan (e+f x)\right )}{f}\) |
Input:
Int[(a + I*a*Tan[e + f*x])^4*(c - I*c*Tan[e + f*x])^4,x]
Output:
-((a^4*c^4*(-Tan[e + f*x] - Tan[e + f*x]^3 - (3*Tan[e + f*x]^5)/5 - Tan[e + f*x]^7/7))/f)
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 = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.60
method | result | size |
derivativedivides | \(\frac {a^{4} c^{4} \left (\frac {\tan \left (f x +e \right )^{7}}{7}+\frac {3 \tan \left (f x +e \right )^{5}}{5}+\tan \left (f x +e \right )^{3}+\tan \left (f x +e \right )\right )}{f}\) | \(46\) |
default | \(\frac {a^{4} c^{4} \left (\frac {\tan \left (f x +e \right )^{7}}{7}+\frac {3 \tan \left (f x +e \right )^{5}}{5}+\tan \left (f x +e \right )^{3}+\tan \left (f x +e \right )\right )}{f}\) | \(46\) |
risch | \(\frac {32 i a^{4} c^{4} \left (35 \,{\mathrm e}^{6 i \left (f x +e \right )}+21 \,{\mathrm e}^{4 i \left (f x +e \right )}+7 \,{\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{35 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{7}}\) | \(61\) |
parallelrisch | \(\frac {5 a^{4} c^{4} \tan \left (f x +e \right )^{7}+21 a^{4} c^{4} \tan \left (f x +e \right )^{5}+35 a^{4} c^{4} \tan \left (f x +e \right )^{3}+35 a^{4} c^{4} \tan \left (f x +e \right )}{35 f}\) | \(69\) |
norman | \(\frac {a^{4} c^{4} \tan \left (f x +e \right )}{f}+\frac {a^{4} c^{4} \tan \left (f x +e \right )^{3}}{f}+\frac {3 a^{4} c^{4} \tan \left (f x +e \right )^{5}}{5 f}+\frac {a^{4} c^{4} \tan \left (f x +e \right )^{7}}{7 f}\) | \(74\) |
parts | \(a^{4} c^{4} x +\frac {a^{4} c^{4} \left (\frac {\tan \left (f x +e \right )^{7}}{7}-\frac {\tan \left (f x +e \right )^{5}}{5}+\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {4 a^{4} c^{4} \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {6 a^{4} c^{4} \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {4 a^{4} c^{4} \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(177\) |
Input:
int((a+I*a*tan(f*x+e))^4*(c-I*c*tan(f*x+e))^4,x,method=_RETURNVERBOSE)
Output:
1/f*a^4*c^4*(1/7*tan(f*x+e)^7+3/5*tan(f*x+e)^5+tan(f*x+e)^3+tan(f*x+e))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.94 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx=-\frac {32 \, {\left (-35 i \, a^{4} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} - 21 i \, a^{4} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} - 7 i \, a^{4} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{4} c^{4}\right )}}{35 \, {\left (f e^{\left (14 i \, f x + 14 i \, e\right )} + 7 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 21 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 35 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 35 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 21 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 7 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \] Input:
integrate((a+I*a*tan(f*x+e))^4*(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")
Output:
-32/35*(-35*I*a^4*c^4*e^(6*I*f*x + 6*I*e) - 21*I*a^4*c^4*e^(4*I*f*x + 4*I* e) - 7*I*a^4*c^4*e^(2*I*f*x + 2*I*e) - I*a^4*c^4)/(f*e^(14*I*f*x + 14*I*e) + 7*f*e^(12*I*f*x + 12*I*e) + 21*f*e^(10*I*f*x + 10*I*e) + 35*f*e^(8*I*f* x + 8*I*e) + 35*f*e^(6*I*f*x + 6*I*e) + 21*f*e^(4*I*f*x + 4*I*e) + 7*f*e^( 2*I*f*x + 2*I*e) + f)
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.84 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx=\frac {1120 i a^{4} c^{4} e^{6 i e} e^{6 i f x} + 672 i a^{4} c^{4} e^{4 i e} e^{4 i f x} + 224 i a^{4} c^{4} e^{2 i e} e^{2 i f x} + 32 i a^{4} c^{4}}{35 f e^{14 i e} e^{14 i f x} + 245 f e^{12 i e} e^{12 i f x} + 735 f e^{10 i e} e^{10 i f x} + 1225 f e^{8 i e} e^{8 i f x} + 1225 f e^{6 i e} e^{6 i f x} + 735 f e^{4 i e} e^{4 i f x} + 245 f e^{2 i e} e^{2 i f x} + 35 f} \] Input:
integrate((a+I*a*tan(f*x+e))**4*(c-I*c*tan(f*x+e))**4,x)
Output:
(1120*I*a**4*c**4*exp(6*I*e)*exp(6*I*f*x) + 672*I*a**4*c**4*exp(4*I*e)*exp (4*I*f*x) + 224*I*a**4*c**4*exp(2*I*e)*exp(2*I*f*x) + 32*I*a**4*c**4)/(35* f*exp(14*I*e)*exp(14*I*f*x) + 245*f*exp(12*I*e)*exp(12*I*f*x) + 735*f*exp( 10*I*e)*exp(10*I*f*x) + 1225*f*exp(8*I*e)*exp(8*I*f*x) + 1225*f*exp(6*I*e) *exp(6*I*f*x) + 735*f*exp(4*I*e)*exp(4*I*f*x) + 245*f*exp(2*I*e)*exp(2*I*f *x) + 35*f)
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx=\frac {5 \, a^{4} c^{4} \tan \left (f x + e\right )^{7} + 21 \, a^{4} c^{4} \tan \left (f x + e\right )^{5} + 35 \, a^{4} c^{4} \tan \left (f x + e\right )^{3} + 35 \, a^{4} c^{4} \tan \left (f x + e\right )}{35 \, f} \] Input:
integrate((a+I*a*tan(f*x+e))^4*(c-I*c*tan(f*x+e))^4,x, algorithm="maxima")
Output:
1/35*(5*a^4*c^4*tan(f*x + e)^7 + 21*a^4*c^4*tan(f*x + e)^5 + 35*a^4*c^4*ta n(f*x + e)^3 + 35*a^4*c^4*tan(f*x + e))/f
Time = 0.64 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx=\frac {5 \, a^{4} c^{4} \tan \left (f x + e\right )^{7} + 21 \, a^{4} c^{4} \tan \left (f x + e\right )^{5} + 35 \, a^{4} c^{4} \tan \left (f x + e\right )^{3} + 35 \, a^{4} c^{4} \tan \left (f x + e\right )}{35 \, f} \] Input:
integrate((a+I*a*tan(f*x+e))^4*(c-I*c*tan(f*x+e))^4,x, algorithm="giac")
Output:
1/35*(5*a^4*c^4*tan(f*x + e)^7 + 21*a^4*c^4*tan(f*x + e)^5 + 35*a^4*c^4*ta n(f*x + e)^3 + 35*a^4*c^4*tan(f*x + e))/f
Time = 1.77 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.06 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx=\frac {a^4\,c^4\,\sin \left (e+f\,x\right )\,\left (35\,{\cos \left (e+f\,x\right )}^6+35\,{\cos \left (e+f\,x\right )}^4\,{\sin \left (e+f\,x\right )}^2+21\,{\cos \left (e+f\,x\right )}^2\,{\sin \left (e+f\,x\right )}^4+5\,{\sin \left (e+f\,x\right )}^6\right )}{35\,f\,{\cos \left (e+f\,x\right )}^7} \] Input:
int((a + a*tan(e + f*x)*1i)^4*(c - c*tan(e + f*x)*1i)^4,x)
Output:
(a^4*c^4*sin(e + f*x)*(35*cos(e + f*x)^6 + 5*sin(e + f*x)^6 + 21*cos(e + f *x)^2*sin(e + f*x)^4 + 35*cos(e + f*x)^4*sin(e + f*x)^2))/(35*f*cos(e + f* x)^7)
Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.64 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx=\frac {\tan \left (f x +e \right ) a^{4} c^{4} \left (5 \tan \left (f x +e \right )^{6}+21 \tan \left (f x +e \right )^{4}+35 \tan \left (f x +e \right )^{2}+35\right )}{35 f} \] Input:
int((a+I*a*tan(f*x+e))^4*(c-I*c*tan(f*x+e))^4,x)
Output:
(tan(e + f*x)*a**4*c**4*(5*tan(e + f*x)**6 + 21*tan(e + f*x)**4 + 35*tan(e + f*x)**2 + 35))/(35*f)