Integrand size = 32, antiderivative size = 193 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=-\frac {63 c^5 \text {arctanh}(\sin (e+f x))}{2 a^3 f}+\frac {42 c^5 \tan (e+f x)}{a^3 f}-\frac {21 c^5 \sec (e+f x) \tan (e+f x)}{2 a^3 f}-\frac {6 c^2 (c-c \sec (e+f x))^3 \tan (e+f x)}{5 a f (a+a \sec (e+f x))^2}+\frac {2 c (c-c \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {42 c \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{5 f \left (a^3+a^3 \sec (e+f x)\right )} \]
-63/2*c^5*arctanh(sin(f*x+e))/a^3/f+42*c^5*tan(f*x+e)/a^3/f-21/2*c^5*sec(f *x+e)*tan(f*x+e)/a^3/f-6/5*c^2*(c-c*sec(f*x+e))^3*tan(f*x+e)/a/f/(a+a*sec( f*x+e))^2+2/5*c*(c-c*sec(f*x+e))^4*tan(f*x+e)/f/(a+a*sec(f*x+e))^3+42/5*c* (c^2-c^2*sec(f*x+e))^2*tan(f*x+e)/f/(a^3+a^3*sec(f*x+e))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.39 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.39 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=-\frac {32 c^5 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},-\frac {5}{2},-\frac {3}{2},\frac {1}{2} (1+\sec (e+f x))\right ) \sqrt {2-2 \sec (e+f x)} \tan (e+f x)}{5 a^3 f (-1+\sec (e+f x)) (1+\sec (e+f x))^3} \]
(-32*c^5*Hypergeometric2F1[-9/2, -5/2, -3/2, (1 + Sec[e + f*x])/2]*Sqrt[2 - 2*Sec[e + f*x]]*Tan[e + f*x])/(5*a^3*f*(-1 + Sec[e + f*x])*(1 + Sec[e + f*x])^3)
Time = 1.14 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3042, 4445, 3042, 4445, 3042, 4445, 3042, 4275, 3042, 4254, 24, 4534, 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.
| \(\displaystyle \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a \sec (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^5}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(\sec (e+f x) a+a)^2}dx}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}{\left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a}\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \int \frac {\sec (e+f x) (c-c \sec (e+f x))^3}{\sec (e+f x) a+a}dx}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3}{\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}dx}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \int \sec (e+f x) (c-c \sec (e+f x))^2dx}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^2dx}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 4275 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\int \sec (e+f x) \left (\sec ^2(e+f x) c^2+c^2\right )dx-2 c^2 \int \sec ^2(e+f x)dx\right )}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )^2 c^2+c^2\right )dx-2 c^2 \int \csc \left (e+f x+\frac {\pi }{2}\right )^2dx\right )}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\frac {2 c^2 \int 1d(-\tan (e+f x))}{f}+\int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )^2 c^2+c^2\right )dx\right )}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )^2 c^2+c^2\right )dx-\frac {2 c^2 \tan (e+f x)}{f}\right )}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\frac {3}{2} c^2 \int \sec (e+f x)dx-\frac {2 c^2 \tan (e+f x)}{f}+\frac {c^2 \tan (e+f x) \sec (e+f x)}{2 f}\right )}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\frac {3}{2} c^2 \int \csc \left (e+f x+\frac {\pi }{2}\right )dx-\frac {2 c^2 \tan (e+f x)}{f}+\frac {c^2 \tan (e+f x) \sec (e+f x)}{2 f}\right )}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}-\frac {9 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\frac {3 c^2 \text {arctanh}(\sin (e+f x))}{2 f}-\frac {2 c^2 \tan (e+f x)}{f}+\frac {c^2 \tan (e+f x) \sec (e+f x)}{2 f}\right )}{a}\right )}{3 a}\right )}{5 a}\) |
(2*c*(c - c*Sec[e + f*x])^4*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) - ( 9*c*((2*c*(c - c*Sec[e + f*x])^3*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2 ) - (7*c*((2*c*(c - c*Sec[e + f*x])^2*Tan[e + f*x])/(f*(a + a*Sec[e + f*x] )) - (5*c*((3*c^2*ArcTanh[Sin[e + f*x]])/(2*f) - (2*c^2*Tan[e + f*x])/f + (c^2*Sec[e + f*x]*Tan[e + f*x])/(2*f)))/a))/(3*a)))/(5*a)
3.1.53.3.1 Defintions of rubi rules used
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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[2*a*(b/d) Int[(d*Csc[e + f*x])^(n + 1), x], x] + Int[(d*Csc[e + f*x])^n*(a^2 + b^2*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^(n - 1)/(b*f*(2*m + 1))), x] - Simp[d*((2*n - 1)/(b*(2*m + 1))) Int[Csc[e + f*x]*(a + b*Csc[e + f* x])^(m + 1)*(c + d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0] && LtQ[m, -2^ (-1)] && IntegerQ[2*m]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) )), x] + Simp[(C*m + A*(m + 1))/(m + 1) Int[(b*Csc[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && !LeQ[m, -1]
Time = 0.98 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {8 c^{5} \left (\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {17}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {63 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16}-\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {17}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {63 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16}\right )}{f \,a^{3}}\) | \(136\) |
| default | \(\frac {8 c^{5} \left (\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {17}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {63 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16}-\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {17}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {63 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16}\right )}{f \,a^{3}}\) | \(136\) |
| parallelrisch | \(\frac {3749 \left (\frac {2520 \left (1+\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{3749}+\frac {2520 \left (-1-\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{3749}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \left (\cos \left (f x +e \right )+\frac {2594 \cos \left (2 f x +2 e \right )}{3749}+\frac {1163 \cos \left (3 f x +3 e \right )}{3749}+\frac {248 \cos \left (4 f x +4 e \right )}{3749}+\frac {2326}{3749}\right )\right ) c^{5}}{80 f \,a^{3} \left (1+\cos \left (2 f x +2 e \right )\right )}\) | \(140\) |
| risch | \(\frac {i c^{5} \left (325 \,{\mathrm e}^{8 i \left (f x +e \right )}+1545 \,{\mathrm e}^{7 i \left (f x +e \right )}+3805 \,{\mathrm e}^{6 i \left (f x +e \right )}+5545 \,{\mathrm e}^{5 i \left (f x +e \right )}+7351 \,{\mathrm e}^{4 i \left (f x +e \right )}+6115 \,{\mathrm e}^{3 i \left (f x +e \right )}+4407 \,{\mathrm e}^{2 i \left (f x +e \right )}+2155 \,{\mathrm e}^{i \left (f x +e \right )}+496\right )}{5 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5} \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2}}+\frac {63 c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 a^{3} f}-\frac {63 c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 a^{3} f}\) | \(178\) |
| norman | \(\frac {-\frac {63 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {294 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{a f}-\frac {2688 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5 a f}+\frac {474 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{a f}-\frac {193 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{a f}+\frac {24 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{a f}+\frac {8 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{15}}{5 a f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{5} a^{2}}+\frac {63 c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 a^{3} f}-\frac {63 c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 a^{3} f}\) | \(220\) |
8/f*c^5/a^3*(1/5*tan(1/2*f*x+1/2*e)^5+tan(1/2*f*x+1/2*e)^3+6*tan(1/2*f*x+1 /2*e)+1/16/(tan(1/2*f*x+1/2*e)+1)^2-17/16/(tan(1/2*f*x+1/2*e)+1)-63/16*ln( tan(1/2*f*x+1/2*e)+1)-1/16/(tan(1/2*f*x+1/2*e)-1)^2-17/16/(tan(1/2*f*x+1/2 *e)-1)+63/16*ln(tan(1/2*f*x+1/2*e)-1))
Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.30 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=-\frac {315 \, {\left (c^{5} \cos \left (f x + e\right )^{5} + 3 \, c^{5} \cos \left (f x + e\right )^{4} + 3 \, c^{5} \cos \left (f x + e\right )^{3} + c^{5} \cos \left (f x + e\right )^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 315 \, {\left (c^{5} \cos \left (f x + e\right )^{5} + 3 \, c^{5} \cos \left (f x + e\right )^{4} + 3 \, c^{5} \cos \left (f x + e\right )^{3} + c^{5} \cos \left (f x + e\right )^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (496 \, c^{5} \cos \left (f x + e\right )^{4} + 1163 \, c^{5} \cos \left (f x + e\right )^{3} + 801 \, c^{5} \cos \left (f x + e\right )^{2} + 65 \, c^{5} \cos \left (f x + e\right ) - 5 \, c^{5}\right )} \sin \left (f x + e\right )}{20 \, {\left (a^{3} f \cos \left (f x + e\right )^{5} + 3 \, a^{3} f \cos \left (f x + e\right )^{4} + 3 \, a^{3} f \cos \left (f x + e\right )^{3} + a^{3} f \cos \left (f x + e\right )^{2}\right )}} \]
-1/20*(315*(c^5*cos(f*x + e)^5 + 3*c^5*cos(f*x + e)^4 + 3*c^5*cos(f*x + e) ^3 + c^5*cos(f*x + e)^2)*log(sin(f*x + e) + 1) - 315*(c^5*cos(f*x + e)^5 + 3*c^5*cos(f*x + e)^4 + 3*c^5*cos(f*x + e)^3 + c^5*cos(f*x + e)^2)*log(-si n(f*x + e) + 1) - 2*(496*c^5*cos(f*x + e)^4 + 1163*c^5*cos(f*x + e)^3 + 80 1*c^5*cos(f*x + e)^2 + 65*c^5*cos(f*x + e) - 5*c^5)*sin(f*x + e))/(a^3*f*c os(f*x + e)^5 + 3*a^3*f*cos(f*x + e)^4 + 3*a^3*f*cos(f*x + e)^3 + a^3*f*co s(f*x + e)^2)
\[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=- \frac {c^{5} \left (\int \left (- \frac {\sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {5 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {10 \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {10 \sec ^{4}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {5 \sec ^{5}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{6}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx\right )}{a^{3}} \]
-c**5*(Integral(-sec(e + f*x)/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec (e + f*x) + 1), x) + Integral(5*sec(e + f*x)**2/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(-10*sec(e + f*x)**3/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(10*sec (e + f*x)**4/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x ) + Integral(-5*sec(e + f*x)**5/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*s ec(e + f*x) + 1), x) + Integral(sec(e + f*x)**6/(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. 680 vs. \(2 (186) = 372\).
Time = 0.21 (sec) , antiderivative size = 680, normalized size of antiderivative = 3.52 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {c^{5} {\left (\frac {60 \, {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {7 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{3} - \frac {2 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \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 {390 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {390 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} + 15 \, c^{5} {\left (\frac {40 \, \sin \left (f x + e\right )}{{\left (a^{3} - \frac {a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}} + \frac {\frac {85 \, \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 {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} + 10 \, c^{5} {\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 {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} + \frac {10 \, c^{5} {\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 {c^{5} {\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 {15 \, c^{5} {\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^5*(60*(5*sin(f*x + e)/(cos(f*x + e) + 1) - 7*sin(f*x + e)^3/(cos(f *x + e) + 1)^3)/(a^3 - 2*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^3*sin (f*x + e)^4/(cos(f*x + e) + 1)^4) + (465*sin(f*x + e)/(cos(f*x + e) + 1) + 40*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 - 390*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a^3 + 390*log(si n(f*x + e)/(cos(f*x + e) + 1) - 1)/a^3) + 15*c^5*(40*sin(f*x + e)/((a^3 - a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)*(cos(f*x + e) + 1)) + (85*sin(f*x + e)/(cos(f*x + e) + 1) + 10*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + sin(f* x + e)^5/(cos(f*x + e) + 1)^5)/a^3 - 60*log(sin(f*x + e)/(cos(f*x + e) + 1 ) + 1)/a^3 + 60*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^3) + 10*c^5*((1 05*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 - 60*log(sin(f*x + e)/(cos( f*x + e) + 1) + 1)/a^3 + 60*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^3) + 10*c^5*(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 + c^5*(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 - 15*c^5*(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 = 0.41 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.82 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=-\frac {\frac {315 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {315 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} + \frac {10 \, {\left (17 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{3}} - \frac {16 \, {\left (a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 5 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 30 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15}}}{10 \, f} \]
-1/10*(315*c^5*log(abs(tan(1/2*f*x + 1/2*e) + 1))/a^3 - 315*c^5*log(abs(ta n(1/2*f*x + 1/2*e) - 1))/a^3 + 10*(17*c^5*tan(1/2*f*x + 1/2*e)^3 - 15*c^5* tan(1/2*f*x + 1/2*e))/((tan(1/2*f*x + 1/2*e)^2 - 1)^2*a^3) - 16*(a^12*c^5* tan(1/2*f*x + 1/2*e)^5 + 5*a^12*c^5*tan(1/2*f*x + 1/2*e)^3 + 30*a^12*c^5*t an(1/2*f*x + 1/2*e))/a^15)/f
Time = 12.96 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.82 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {48\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^3\,f}-\frac {17\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-15\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a^3\right )}+\frac {8\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{a^3\,f}+\frac {8\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{5\,a^3\,f}-\frac {63\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^3\,f} \]