Integrand size = 32, antiderivative size = 100 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=\frac {5 a^3 c^3 \text {arctanh}(\sin (e+f x))}{16 f}-\frac {5 a^3 c^3 \sec (e+f x) \tan (e+f x)}{16 f}+\frac {5 a^3 c^3 \sec (e+f x) \tan ^3(e+f x)}{24 f}-\frac {a^3 c^3 \sec (e+f x) \tan ^5(e+f x)}{6 f} \]
5/16*a^3*c^3*arctanh(sin(f*x+e))/f-5/16*a^3*c^3*sec(f*x+e)*tan(f*x+e)/f+5/ 24*a^3*c^3*sec(f*x+e)*tan(f*x+e)^3/f-1/6*a^3*c^3*sec(f*x+e)*tan(f*x+e)^5/f
Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.25 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=-a^3 c^3 \left (-\frac {5 \text {arctanh}(\sin (e+f x))}{16 f}-\frac {5 \sec (e+f x) \tan (e+f x)}{16 f}-\frac {5 \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac {5 \sec ^5(e+f x) \tan (e+f x)}{6 f}-\frac {5 \sec ^3(e+f x) \tan ^3(e+f x)}{3 f}+\frac {\sec (e+f x) \tan ^5(e+f x)}{f}\right ) \]
-(a^3*c^3*((-5*ArcTanh[Sin[e + f*x]])/(16*f) - (5*Sec[e + f*x]*Tan[e + f*x ])/(16*f) - (5*Sec[e + f*x]^3*Tan[e + f*x])/(24*f) + (5*Sec[e + f*x]^5*Tan [e + f*x])/(6*f) - (5*Sec[e + f*x]^3*Tan[e + f*x]^3)/(3*f) + (Sec[e + f*x] *Tan[e + f*x]^5)/f))
Time = 0.54 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3042, 4446, 3042, 3091, 3042, 3091, 3042, 3091, 3042, 4257}
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.
| \(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4446 |
| \(\displaystyle -a^3 c^3 \int \sec (e+f x) \tan ^6(e+f x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^3 c^3 \int \sec (e+f x) \tan (e+f x)^6dx\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle -a^3 c^3 \left (\frac {\tan ^5(e+f x) \sec (e+f x)}{6 f}-\frac {5}{6} \int \sec (e+f x) \tan ^4(e+f x)dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^3 c^3 \left (\frac {\tan ^5(e+f x) \sec (e+f x)}{6 f}-\frac {5}{6} \int \sec (e+f x) \tan (e+f x)^4dx\right )\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle -a^3 c^3 \left (\frac {\tan ^5(e+f x) \sec (e+f x)}{6 f}-\frac {5}{6} \left (\frac {\tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac {3}{4} \int \sec (e+f x) \tan ^2(e+f x)dx\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^3 c^3 \left (\frac {\tan ^5(e+f x) \sec (e+f x)}{6 f}-\frac {5}{6} \left (\frac {\tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac {3}{4} \int \sec (e+f x) \tan (e+f x)^2dx\right )\right )\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle -a^3 c^3 \left (\frac {\tan ^5(e+f x) \sec (e+f x)}{6 f}-\frac {5}{6} \left (\frac {\tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac {3}{4} \left (\frac {\tan (e+f x) \sec (e+f x)}{2 f}-\frac {1}{2} \int \sec (e+f x)dx\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^3 c^3 \left (\frac {\tan ^5(e+f x) \sec (e+f x)}{6 f}-\frac {5}{6} \left (\frac {\tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac {3}{4} \left (\frac {\tan (e+f x) \sec (e+f x)}{2 f}-\frac {1}{2} \int \csc \left (e+f x+\frac {\pi }{2}\right )dx\right )\right )\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -a^3 c^3 \left (\frac {\tan ^5(e+f x) \sec (e+f x)}{6 f}-\frac {5}{6} \left (\frac {\tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac {3}{4} \left (\frac {\tan (e+f x) \sec (e+f x)}{2 f}-\frac {\text {arctanh}(\sin (e+f x))}{2 f}\right )\right )\right )\) |
-(a^3*c^3*((Sec[e + f*x]*Tan[e + f*x]^5)/(6*f) - (5*((Sec[e + f*x]*Tan[e + f*x]^3)/(4*f) - (3*(-1/2*ArcTanh[Sin[e + f*x]]/f + (Sec[e + f*x]*Tan[e + f*x])/(2*f)))/4))/6))
3.1.24.3.1 Defintions of rubi rules used
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(c sc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[((-a)*c)^m Int[ExpandTrig[csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n - m ), x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && Eq Q[a^2 - b^2, 0] && IntegersQ[m, n] && GeQ[n - m, 0] && GtQ[m*n, 0]
Result contains complex when optimal does not.
Time = 3.91 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.42
| method | result | size |
| risch | \(\frac {i a^{3} c^{3} \left (33 \,{\mathrm e}^{11 i \left (f x +e \right )}-5 \,{\mathrm e}^{9 i \left (f x +e \right )}+90 \,{\mathrm e}^{7 i \left (f x +e \right )}-90 \,{\mathrm e}^{5 i \left (f x +e \right )}+5 \,{\mathrm e}^{3 i \left (f x +e \right )}-33 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{24 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{6}}-\frac {5 c^{3} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{16 f}+\frac {5 c^{3} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{16 f}\) | \(142\) |
| parallelrisch | \(\frac {5 a^{3} c^{3} \left (\left (-\frac {45 \cos \left (2 f x +2 e \right )}{2}-9 \cos \left (4 f x +4 e \right )-\frac {3 \cos \left (6 f x +6 e \right )}{2}-15\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+\left (\frac {3 \cos \left (6 f x +6 e \right )}{2}+9 \cos \left (4 f x +4 e \right )+\frac {45 \cos \left (2 f x +2 e \right )}{2}+15\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\sin \left (3 f x +3 e \right )-\frac {33 \sin \left (5 f x +5 e \right )}{5}-18 \sin \left (f x +e \right )\right )}{24 f \left (6 \cos \left (4 f x +4 e \right )+10+15 \cos \left (2 f x +2 e \right )+\cos \left (6 f x +6 e \right )\right )}\) | \(172\) |
| derivativedivides | \(\frac {-c^{3} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+3 c^{3} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-3 c^{3} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+c^{3} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}\) | \(180\) |
| default | \(\frac {-c^{3} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+3 c^{3} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-3 c^{3} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+c^{3} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}\) | \(180\) |
| parts | \(\frac {c^{3} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}-\frac {3 c^{3} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}+\frac {3 c^{3} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}-\frac {c^{3} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}\) | \(188\) |
| norman | \(\frac {-\frac {5 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {85 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{24 f}-\frac {33 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{4 f}-\frac {33 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{4 f}+\frac {85 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{24 f}-\frac {5 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{8 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{6}}-\frac {5 c^{3} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 f}+\frac {5 c^{3} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 f}\) | \(195\) |
1/24*I*a^3*c^3/f/(1+exp(2*I*(f*x+e)))^6*(33*exp(11*I*(f*x+e))-5*exp(9*I*(f *x+e))+90*exp(7*I*(f*x+e))-90*exp(5*I*(f*x+e))+5*exp(3*I*(f*x+e))-33*exp(I *(f*x+e)))-5/16*c^3*a^3/f*ln(exp(I*(f*x+e))-I)+5/16*c^3*a^3/f*ln(exp(I*(f* x+e))+I)
Time = 0.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.15 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=\frac {15 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (33 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} - 26 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 8 \, a^{3} c^{3}\right )} \sin \left (f x + e\right )}{96 \, f \cos \left (f x + e\right )^{6}} \]
1/96*(15*a^3*c^3*cos(f*x + e)^6*log(sin(f*x + e) + 1) - 15*a^3*c^3*cos(f*x + e)^6*log(-sin(f*x + e) + 1) - 2*(33*a^3*c^3*cos(f*x + e)^4 - 26*a^3*c^3 *cos(f*x + e)^2 + 8*a^3*c^3)*sin(f*x + e))/(f*cos(f*x + e)^6)
\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=- a^{3} c^{3} \left (\int \left (- \sec {\left (e + f x \right )}\right )\, dx + \int 3 \sec ^{3}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{5}{\left (e + f x \right )}\right )\, dx + \int \sec ^{7}{\left (e + f x \right )}\, dx\right ) \]
-a**3*c**3*(Integral(-sec(e + f*x), x) + Integral(3*sec(e + f*x)**3, x) + Integral(-3*sec(e + f*x)**5, x) + Integral(sec(e + f*x)**7, x))
Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (92) = 184\).
Time = 0.23 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.44 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=\frac {a^{3} c^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 18 \, a^{3} c^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 72 \, a^{3} c^{3} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 96 \, a^{3} c^{3} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right )}{96 \, f} \]
1/96*(a^3*c^3*(2*(15*sin(f*x + e)^5 - 40*sin(f*x + e)^3 + 33*sin(f*x + e)) /(sin(f*x + e)^6 - 3*sin(f*x + e)^4 + 3*sin(f*x + e)^2 - 1) - 15*log(sin(f *x + e) + 1) + 15*log(sin(f*x + e) - 1)) - 18*a^3*c^3*(2*(3*sin(f*x + e)^3 - 5*sin(f*x + e))/(sin(f*x + e)^4 - 2*sin(f*x + e)^2 + 1) - 3*log(sin(f*x + e) + 1) + 3*log(sin(f*x + e) - 1)) + 72*a^3*c^3*(2*sin(f*x + e)/(sin(f* x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) + 96*a^3*c^ 3*log(sec(f*x + e) + tan(f*x + e)))/f
Time = 0.37 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.03 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=\frac {15 \, a^{3} c^{3} \log \left ({\left | \sin \left (f x + e\right ) + 1 \right |}\right ) - 15 \, a^{3} c^{3} \log \left ({\left | \sin \left (f x + e\right ) - 1 \right |}\right ) + \frac {2 \, {\left (33 \, a^{3} c^{3} \sin \left (f x + e\right )^{5} - 40 \, a^{3} c^{3} \sin \left (f x + e\right )^{3} + 15 \, a^{3} c^{3} \sin \left (f x + e\right )\right )}}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{3}}}{96 \, f} \]
1/96*(15*a^3*c^3*log(abs(sin(f*x + e) + 1)) - 15*a^3*c^3*log(abs(sin(f*x + e) - 1)) + 2*(33*a^3*c^3*sin(f*x + e)^5 - 40*a^3*c^3*sin(f*x + e)^3 + 15* a^3*c^3*sin(f*x + e))/(sin(f*x + e)^2 - 1)^3)/f
Time = 16.50 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.20 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx=\frac {5\,a^3\,c^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{8\,f}-\frac {\frac {5\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{8}-\frac {85\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{24}+\frac {33\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{4}+\frac {33\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{4}-\frac {85\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24}+\frac {5\,a^3\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
(5*a^3*c^3*atanh(tan(e/2 + (f*x)/2)))/(8*f) - ((33*a^3*c^3*tan(e/2 + (f*x) /2)^5)/4 - (85*a^3*c^3*tan(e/2 + (f*x)/2)^3)/24 + (33*a^3*c^3*tan(e/2 + (f *x)/2)^7)/4 - (85*a^3*c^3*tan(e/2 + (f*x)/2)^9)/24 + (5*a^3*c^3*tan(e/2 + (f*x)/2)^11)/8 + (5*a^3*c^3*tan(e/2 + (f*x)/2))/8)/(f*(15*tan(e/2 + (f*x)/ 2)^4 - 6*tan(e/2 + (f*x)/2)^2 - 20*tan(e/2 + (f*x)/2)^6 + 15*tan(e/2 + (f* x)/2)^8 - 6*tan(e/2 + (f*x)/2)^10 + tan(e/2 + (f*x)/2)^12 + 1))