Integrand size = 26, antiderivative size = 96 \[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx=\frac {c^3 x}{a^3}-\frac {8 c^3 \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}+\frac {4 c^3 \tan (e+f x)}{15 a^3 f (1+\sec (e+f x))^2}-\frac {26 c^3 \tan (e+f x)}{15 a^3 f (1+\sec (e+f x))} \]
c^3*x/a^3-8/5*c^3*tan(f*x+e)/a^3/f/(1+sec(f*x+e))^3+4/15*c^3*tan(f*x+e)/a^ 3/f/(1+sec(f*x+e))^2-26/15*c^3*tan(f*x+e)/a^3/f/(1+sec(f*x+e))
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx=-\frac {c^3 \left (-\frac {2 \arctan \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f}+\frac {2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {2 \tan ^3\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}+\frac {2 \tan ^5\left (\frac {e}{2}+\frac {f x}{2}\right )}{5 f}\right )}{a^3} \]
-((c^3*((-2*ArcTan[Tan[e/2 + (f*x)/2]])/f + (2*Tan[e/2 + (f*x)/2])/f - (2* Tan[e/2 + (f*x)/2]^3)/(3*f) + (2*Tan[e/2 + (f*x)/2]^5)/(5*f)))/a^3)
Time = 0.54 (sec) , antiderivative size = 88, 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.
\(\displaystyle \int \frac {(c-c \sec (e+f x))^3}{(a \sec (e+f x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
\(\Big \downarrow \) 4391 |
\(\displaystyle \frac {\int \left (-\frac {\sec ^3(e+f x) c^3}{(\sec (e+f x)+1)^3}+\frac {3 \sec ^2(e+f x) c^3}{(\sec (e+f x)+1)^3}-\frac {3 \sec (e+f x) c^3}{(\sec (e+f x)+1)^3}+\frac {c^3}{(\sec (e+f x)+1)^3}\right )dx}{a^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {26 c^3 \tan (e+f x)}{15 f (\sec (e+f x)+1)}+\frac {4 c^3 \tan (e+f x)}{15 f (\sec (e+f x)+1)^2}-\frac {8 c^3 \tan (e+f x)}{5 f (\sec (e+f x)+1)^3}+c^3 x}{a^3}\) |
(c^3*x - (8*c^3*Tan[e + f*x])/(5*f*(1 + Sec[e + f*x])^3) + (4*c^3*Tan[e + f*x])/(15*f*(1 + Sec[e + f*x])^2) - (26*c^3*Tan[e + f*x])/(15*f*(1 + Sec[e + f*x])))/a^3
3.1.33.3.1 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 = 0.68 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.56
method | result | size |
parallelrisch | \(-\frac {c^{3} \left (6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-15 f x +30 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a^{3} f}\) | \(54\) |
derivativedivides | \(\frac {2 c^{3} \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,a^{3}}\) | \(60\) |
default | \(\frac {2 c^{3} \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,a^{3}}\) | \(60\) |
risch | \(\frac {c^{3} x}{a^{3}}-\frac {4 i c^{3} \left (45 \,{\mathrm e}^{4 i \left (f x +e \right )}+90 \,{\mathrm e}^{3 i \left (f x +e \right )}+140 \,{\mathrm e}^{2 i \left (f x +e \right )}+70 \,{\mathrm e}^{i \left (f x +e \right )}+23\right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5}}\) | \(81\) |
norman | \(\frac {\frac {c^{3} x}{a}+\frac {c^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{a}-\frac {2 c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {14 c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 a f}-\frac {56 c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{15 a f}+\frac {22 c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{15 a f}-\frac {2 c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{5 a f}-\frac {2 c^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{a}}{a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{2}}\) | \(176\) |
-1/15*c^3*(6*tan(1/2*f*x+1/2*e)^5-10*tan(1/2*f*x+1/2*e)^3-15*f*x+30*tan(1/ 2*f*x+1/2*e))/a^3/f
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.44 \[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx=\frac {15 \, c^{3} f x \cos \left (f x + e\right )^{3} + 45 \, c^{3} f x \cos \left (f x + e\right )^{2} + 45 \, c^{3} f x \cos \left (f x + e\right ) + 15 \, c^{3} f x - 2 \, {\left (23 \, c^{3} \cos \left (f x + e\right )^{2} + 24 \, c^{3} \cos \left (f x + e\right ) + 13 \, c^{3}\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}} \]
1/15*(15*c^3*f*x*cos(f*x + e)^3 + 45*c^3*f*x*cos(f*x + e)^2 + 45*c^3*f*x*c os(f*x + e) + 15*c^3*f*x - 2*(23*c^3*cos(f*x + e)^2 + 24*c^3*cos(f*x + e) + 13*c^3)*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 + 3 *a^3*f*cos(f*x + e) + a^3*f)
\[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx=- \frac {c^{3} \left (\int \frac {3 \sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {1}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx\right )}{a^{3}} \]
-c**3*(Integral(3*sec(e + f*x)/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*se c(e + f*x) + 1), x) + Integral(-3*sec(e + f*x)**2/(sec(e + f*x)**3 + 3*sec (e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(sec(e + f*x)**3/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(-1/(sec( e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x))/a**3
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (90) = 180\).
Time = 0.29 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.89 \[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx=-\frac {c^{3} {\left (\frac {\frac {105 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3}}\right )} + \frac {c^{3} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac {3 \, c^{3} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac {9 \, c^{3} {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \]
-1/60*(c^3*((105*sin(f*x + e)/(cos(f*x + e) + 1) - 20*sin(f*x + e)^3/(cos( f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3 - 120*arctan( sin(f*x + e)/(cos(f*x + e) + 1))/a^3) + c^3*(15*sin(f*x + e)/(cos(f*x + e) + 1) + 10*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3 + 3*c^3*(15*sin(f*x + e)/(cos(f*x + e) + 1) - 10*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3 - 9*c^3*(5*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3)/f
Time = 0.33 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.83 \[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx=\frac {\frac {15 \, {\left (f x + e\right )} c^{3}}{a^{3}} - \frac {2 \, {\left (3 \, a^{12} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 5 \, a^{12} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{12} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15}}}{15 \, f} \]
1/15*(15*(f*x + e)*c^3/a^3 - 2*(3*a^12*c^3*tan(1/2*f*x + 1/2*e)^5 - 5*a^12 *c^3*tan(1/2*f*x + 1/2*e)^3 + 15*a^12*c^3*tan(1/2*f*x + 1/2*e))/a^15)/f
Time = 14.42 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.97 \[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^3} \, dx=\frac {c^3\,x}{a^3}-\frac {\frac {46\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{15}-\frac {22\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{15}+\frac {2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3}{5}}{a^3\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5} \]