Integrand size = 26, antiderivative size = 68 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=a^3 c^3 x-\frac {a^3 c^3 \tan (e+f x)}{f}+\frac {a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac {a^3 c^3 \tan ^5(e+f x)}{5 f} \]
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=-a^3 c^3 \left (-\frac {\arctan (\tan (e+f x))}{f}+\frac {\tan (e+f x)}{f}-\frac {\tan ^3(e+f x)}{3 f}+\frac {\tan ^5(e+f x)}{5 f}\right ) \]
-(a^3*c^3*(-(ArcTan[Tan[e + f*x]]/f) + Tan[e + f*x]/f - Tan[e + f*x]^3/(3* f) + Tan[e + f*x]^5/(5*f)))
Time = 0.40 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3042, 4392, 3042, 3954, 3042, 3954, 3042, 3954, 24}
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.
\(\displaystyle \int (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3dx\) |
\(\Big \downarrow \) 4392 |
\(\displaystyle -a^3 c^3 \int \tan ^6(e+f x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -a^3 c^3 \int \tan (e+f x)^6dx\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle -a^3 c^3 \left (\frac {\tan ^5(e+f x)}{5 f}-\int \tan ^4(e+f x)dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -a^3 c^3 \left (\frac {\tan ^5(e+f x)}{5 f}-\int \tan (e+f x)^4dx\right )\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle -a^3 c^3 \left (\int \tan ^2(e+f x)dx+\frac {\tan ^5(e+f x)}{5 f}-\frac {\tan ^3(e+f x)}{3 f}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -a^3 c^3 \left (\int \tan (e+f x)^2dx+\frac {\tan ^5(e+f x)}{5 f}-\frac {\tan ^3(e+f x)}{3 f}\right )\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle -a^3 c^3 \left (-\int 1dx+\frac {\tan ^5(e+f x)}{5 f}-\frac {\tan ^3(e+f x)}{3 f}+\frac {\tan (e+f x)}{f}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -a^3 c^3 \left (\frac {\tan ^5(e+f x)}{5 f}-\frac {\tan ^3(e+f x)}{3 f}+\frac {\tan (e+f x)}{f}-x\right )\) |
3.1.13.3.1 Defintions of rubi rules used
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_.), x_Symbol] :> Simp[((-a)*c)^m Int[Cot[e + f*x]^(2*m)*( c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && E qQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] && !( IntegerQ[n] && GtQ[m - n, 0])
Result contains complex when optimal does not.
Time = 2.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.19
method | result | size |
risch | \(a^{3} c^{3} x -\frac {2 i c^{3} a^{3} \left (45 \,{\mathrm e}^{8 i \left (f x +e \right )}+90 \,{\mathrm e}^{6 i \left (f x +e \right )}+140 \,{\mathrm e}^{4 i \left (f x +e \right )}+70 \,{\mathrm e}^{2 i \left (f x +e \right )}+23\right )}{15 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{5}}\) | \(81\) |
derivativedivides | \(\frac {c^{3} a^{3} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-3 c^{3} a^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-3 c^{3} a^{3} \tan \left (f x +e \right )+c^{3} a^{3} \left (f x +e \right )}{f}\) | \(93\) |
default | \(\frac {c^{3} a^{3} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-3 c^{3} a^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-3 c^{3} a^{3} \tan \left (f x +e \right )+c^{3} a^{3} \left (f x +e \right )}{f}\) | \(93\) |
parts | \(a^{3} c^{3} x -\frac {3 a^{3} c^{3} \tan \left (f x +e \right )}{f}-\frac {3 c^{3} a^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}+\frac {c^{3} a^{3} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )}{f}\) | \(94\) |
parallelrisch | \(\frac {a^{3} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10} x f -5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8} x f +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6} x f -\frac {32 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{3}-10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} x f +\frac {356 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{15}+5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} x f -\frac {32 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-f x +2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(179\) |
norman | \(\frac {a^{3} c^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}-a^{3} c^{3} x +5 a^{3} c^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-10 a^{3} c^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+10 a^{3} c^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-5 a^{3} c^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+\frac {2 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {32 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 f}+\frac {356 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{15 f}-\frac {32 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{3 f}+\frac {2 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{5}}\) | \(234\) |
a^3*c^3*x-2/15*I*c^3*a^3*(45*exp(8*I*(f*x+e))+90*exp(6*I*(f*x+e))+140*exp( 4*I*(f*x+e))+70*exp(2*I*(f*x+e))+23)/f/(1+exp(2*I*(f*x+e)))^5
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.19 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=\frac {15 \, a^{3} c^{3} f x \cos \left (f x + e\right )^{5} - {\left (23 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} - 11 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 3 \, a^{3} c^{3}\right )} \sin \left (f x + e\right )}{15 \, f \cos \left (f x + e\right )^{5}} \]
1/15*(15*a^3*c^3*f*x*cos(f*x + e)^5 - (23*a^3*c^3*cos(f*x + e)^4 - 11*a^3* c^3*cos(f*x + e)^2 + 3*a^3*c^3)*sin(f*x + e))/(f*cos(f*x + e)^5)
\[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=- a^{3} c^{3} \left (\int \left (-1\right )\, dx + \int 3 \sec ^{2}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{4}{\left (e + f x \right )}\right )\, dx + \int \sec ^{6}{\left (e + f x \right )}\, dx\right ) \]
-a**3*c**3*(Integral(-1, x) + Integral(3*sec(e + f*x)**2, x) + Integral(-3 *sec(e + f*x)**4, x) + Integral(sec(e + f*x)**6, x))
Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.38 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=-\frac {{\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{3} c^{3} - 15 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{3} - 15 \, {\left (f x + e\right )} a^{3} c^{3} + 45 \, a^{3} c^{3} \tan \left (f x + e\right )}{15 \, f} \]
-1/15*((3*tan(f*x + e)^5 + 10*tan(f*x + e)^3 + 15*tan(f*x + e))*a^3*c^3 - 15*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^3*c^3 - 15*(f*x + e)*a^3*c^3 + 45*a ^3*c^3*tan(f*x + e))/f
Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.96 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=-\frac {3 \, a^{3} c^{3} \tan \left (f x + e\right )^{5} - 5 \, a^{3} c^{3} \tan \left (f x + e\right )^{3} - 15 \, {\left (f x + e\right )} a^{3} c^{3} + 15 \, a^{3} c^{3} \tan \left (f x + e\right )}{15 \, f} \]
-1/15*(3*a^3*c^3*tan(f*x + e)^5 - 5*a^3*c^3*tan(f*x + e)^3 - 15*(f*x + e)* a^3*c^3 + 15*a^3*c^3*tan(f*x + e))/f
Time = 17.86 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.79 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=a^3\,c^3\,x+\frac {2\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9-\frac {32\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{3}+\frac {356\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{15}-\frac {32\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}+2\,a^3\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}^5} \]