Integrand size = 26, antiderivative size = 133 \[ \int \frac {(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx=\frac {a^2 x}{c^4}-\frac {4 a^2 \tan (e+f x)}{7 c^4 f (1-\sec (e+f x))^4}-\frac {12 a^2 \tan (e+f x)}{35 c^4 f (1-\sec (e+f x))^3}-\frac {59 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))^2}-\frac {164 a^2 \tan (e+f x)}{105 c^4 f (1-\sec (e+f x))} \]
a^2*x/c^4-4/7*a^2*tan(f*x+e)/c^4/f/(1-sec(f*x+e))^4-12/35*a^2*tan(f*x+e)/c ^4/f/(1-sec(f*x+e))^3-59/105*a^2*tan(f*x+e)/c^4/f/(1-sec(f*x+e))^2-164/105 *a^2*tan(f*x+e)/c^4/f/(1-sec(f*x+e))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.83 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.89 \[ \int \frac {(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx=-\frac {a^2 \csc ^7(e+f x) \left (7032+18165 \cos (e+f x)+19348 \cos (2 (e+f x))+9303 \cos (3 (e+f x))+3080 \cos (4 (e+f x))+2149 \cos (5 (e+f x))+1260 \cos (6 (e+f x))+143 \cos (7 (e+f x))+960 \cos ^7(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},-\tan ^2(e+f x)\right )\right )}{6720 c^4 f} \]
-1/6720*(a^2*Csc[e + f*x]^7*(7032 + 18165*Cos[e + f*x] + 19348*Cos[2*(e + f*x)] + 9303*Cos[3*(e + f*x)] + 3080*Cos[4*(e + f*x)] + 2149*Cos[5*(e + f* x)] + 1260*Cos[6*(e + f*x)] + 143*Cos[7*(e + f*x)] + 960*Cos[e + f*x]^7*Hy pergeometric2F1[-7/2, 1, -5/2, -Tan[e + f*x]^2]))/(c^4*f)
Time = 0.58 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3042, 4391, 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.
| \(\displaystyle \int \frac {(a \sec (e+f x)+a)^2}{(c-c \sec (e+f x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 4391 |
| \(\displaystyle \frac {\int \left (\frac {\sec ^2(e+f x) a^2}{(1-\sec (e+f x))^4}+\frac {2 \sec (e+f x) a^2}{(1-\sec (e+f x))^4}+\frac {a^2}{(1-\sec (e+f x))^4}\right )dx}{c^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {164 a^2 \tan (e+f x)}{105 f (1-\sec (e+f x))}-\frac {59 a^2 \tan (e+f x)}{105 f (1-\sec (e+f x))^2}-\frac {12 a^2 \tan (e+f x)}{35 f (1-\sec (e+f x))^3}-\frac {4 a^2 \tan (e+f x)}{7 f (1-\sec (e+f x))^4}+a^2 x}{c^4}\) |
(a^2*x - (4*a^2*Tan[e + f*x])/(7*f*(1 - Sec[e + f*x])^4) - (12*a^2*Tan[e + f*x])/(35*f*(1 - Sec[e + f*x])^3) - (59*a^2*Tan[e + f*x])/(105*f*(1 - Sec [e + f*x])^2) - (164*a^2*Tan[e + f*x])/(105*f*(1 - Sec[e + f*x])))/c^4
3.1.9.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_), x_Symbol] :> Simp[c^n Int[ExpandTrig[(1 + (d/c)*csc[e + f*x])^n, (a + b*csc[e + f*x])^m, x], x], x] /; FreeQ[{a, b, c, d, e, f, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && ILtQ[n, 0] && LtQ[m + n, 2]
Time = 0.55 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.50
| method | result | size |
| parallelrisch | \(-\frac {a^{2} \left (15 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-63 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+140 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-210 f x -420 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{210 c^{4} f}\) | \(67\) |
| derivativedivides | \(\frac {a^{2} \left (4 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {3}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {4}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}\right )}{2 f \,c^{4}}\) | \(77\) |
| default | \(\frac {a^{2} \left (4 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {3}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {4}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}\right )}{2 f \,c^{4}}\) | \(77\) |
| risch | \(\frac {a^{2} x}{c^{4}}+\frac {2 i a^{2} \left (630 \,{\mathrm e}^{6 i \left (f x +e \right )}-2415 \,{\mathrm e}^{5 i \left (f x +e \right )}+5215 \,{\mathrm e}^{4 i \left (f x +e \right )}-5950 \,{\mathrm e}^{3 i \left (f x +e \right )}+4284 \,{\mathrm e}^{2 i \left (f x +e \right )}-1603 \,{\mathrm e}^{i \left (f x +e \right )}+319\right )}{105 f \,c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{7}}\) | \(103\) |
| norman | \(\frac {\frac {a^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{c}+\frac {a^{2}}{14 c f}-\frac {13 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{35 c f}+\frac {29 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{30 c f}-\frac {8 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3 c f}+\frac {2 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{c f}-\frac {a^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{c}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right ) c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\) | \(170\) |
-1/210*a^2*(15*cot(1/2*f*x+1/2*e)^7-63*cot(1/2*f*x+1/2*e)^5+140*cot(1/2*f* x+1/2*e)^3-210*f*x-420*cot(1/2*f*x+1/2*e))/c^4/f
Time = 0.25 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.29 \[ \int \frac {(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx=\frac {319 \, a^{2} \cos \left (f x + e\right )^{4} - 327 \, a^{2} \cos \left (f x + e\right )^{3} - 95 \, a^{2} \cos \left (f x + e\right )^{2} + 387 \, a^{2} \cos \left (f x + e\right ) - 164 \, a^{2} + 105 \, {\left (a^{2} f x \cos \left (f x + e\right )^{3} - 3 \, a^{2} f x \cos \left (f x + e\right )^{2} + 3 \, a^{2} f x \cos \left (f x + e\right ) - a^{2} f x\right )} \sin \left (f x + e\right )}{105 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )} \]
1/105*(319*a^2*cos(f*x + e)^4 - 327*a^2*cos(f*x + e)^3 - 95*a^2*cos(f*x + e)^2 + 387*a^2*cos(f*x + e) - 164*a^2 + 105*(a^2*f*x*cos(f*x + e)^3 - 3*a^ 2*f*x*cos(f*x + e)^2 + 3*a^2*f*x*cos(f*x + e) - a^2*f*x)*sin(f*x + e))/((c ^4*f*cos(f*x + e)^3 - 3*c^4*f*cos(f*x + e)^2 + 3*c^4*f*cos(f*x + e) - c^4* f)*sin(f*x + e))
\[ \int \frac {(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx=\frac {a^{2} \left (\int \frac {2 \sec {\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {1}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx\right )}{c^{4}} \]
a**2*(Integral(2*sec(e + f*x)/(sec(e + f*x)**4 - 4*sec(e + f*x)**3 + 6*sec (e + f*x)**2 - 4*sec(e + f*x) + 1), x) + Integral(sec(e + f*x)**2/(sec(e + f*x)**4 - 4*sec(e + f*x)**3 + 6*sec(e + f*x)**2 - 4*sec(e + f*x) + 1), x) + Integral(1/(sec(e + f*x)**4 - 4*sec(e + f*x)**3 + 6*sec(e + f*x)**2 - 4 *sec(e + f*x) + 1), x))/c**4
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (117) = 234\).
Time = 0.30 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.21 \[ \int \frac {(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx=\frac {5 \, a^{2} {\left (\frac {336 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{4}} + \frac {{\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {77 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {315 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 3\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}}\right )} + \frac {a^{2} {\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {105 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 15\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}} + \frac {6 \, a^{2} {\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {35 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 5\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}}}{840 \, f} \]
1/840*(5*a^2*(336*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/c^4 + (21*sin(f* x + e)^2/(cos(f*x + e) + 1)^2 - 77*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3 15*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 3)*(cos(f*x + e) + 1)^7/(c^4*sin( f*x + e)^7)) + a^2*(21*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 35*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 105*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 15)* (cos(f*x + e) + 1)^7/(c^4*sin(f*x + e)^7) + 6*a^2*(21*sin(f*x + e)^2/(cos( f*x + e) + 1)^2 - 35*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 35*sin(f*x + e) ^6/(cos(f*x + e) + 1)^6 - 5)*(cos(f*x + e) + 1)^7/(c^4*sin(f*x + e)^7))/f
Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.66 \[ \int \frac {(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx=\frac {\frac {210 \, {\left (f x + e\right )} a^{2}}{c^{4}} + \frac {420 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 140 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 63 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, a^{2}}{c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7}}}{210 \, f} \]
1/210*(210*(f*x + e)*a^2/c^4 + (420*a^2*tan(1/2*f*x + 1/2*e)^6 - 140*a^2*t an(1/2*f*x + 1/2*e)^4 + 63*a^2*tan(1/2*f*x + 1/2*e)^2 - 15*a^2)/(c^4*tan(1 /2*f*x + 1/2*e)^7))/f
Time = 14.30 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.93 \[ \int \frac {(a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx=\frac {a^2\,x}{c^4}-\frac {\frac {a^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{14}-\frac {3\,a^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{10}+\frac {2\,a^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{3}-2\,a^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6}{c^4\,f\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7} \]