Integrand size = 32, antiderivative size = 150 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {35 c^4 \text {arctanh}(\sin (e+f x))}{2 a^2 f}-\frac {70 c^4 \tan (e+f x)}{3 a^2 f}+\frac {35 c^4 \sec (e+f x) \tan (e+f x)}{6 a^2 f}+\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )} \]
35/2*c^4*arctanh(sin(f*x+e))/a^2/f-70/3*c^4*tan(f*x+e)/a^2/f+35/6*c^4*sec( f*x+e)*tan(f*x+e)/a^2/f+2/3*c*(c-c*sec(f*x+e))^3*tan(f*x+e)/f/(a+a*sec(f*x +e))^2-14/3*(c^2-c^2*sec(f*x+e))^2*tan(f*x+e)/f/(a^2+a^2*sec(f*x+e))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.96 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.50 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=-\frac {16 c^4 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1+\sec (e+f x))\right ) \sqrt {2-2 \sec (e+f x)} \tan (e+f x)}{3 a^2 f (-1+\sec (e+f x)) (1+\sec (e+f x))^2} \]
(-16*c^4*Hypergeometric2F1[-7/2, -3/2, -1/2, (1 + Sec[e + f*x])/2]*Sqrt[2 - 2*Sec[e + f*x]]*Tan[e + f*x])/(3*a^2*f*(-1 + Sec[e + f*x])*(1 + Sec[e + f*x])^2)
Time = 0.88 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {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.
| \(\displaystyle \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a \sec (e+f x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \int \frac {\sec (e+f x) (c-c \sec (e+f x))^3}{\sec (e+f x) a+a}dx}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3}{\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}dx}{3 a}\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \int \sec (e+f x) (c-c \sec (e+f x))^2dx}{a}\right )}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^2dx}{a}\right )}{3 a}\) |
| \(\Big \downarrow \) 4275 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\int \sec (e+f x) \left (\sec ^2(e+f x) c^2+c^2\right )dx-2 c^2 \int \sec ^2(e+f x)dx\right )}{a}\right )}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )^2 c^2+c^2\right )dx-2 c^2 \int \csc \left (e+f x+\frac {\pi }{2}\right )^2dx\right )}{a}\right )}{3 a}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\frac {2 c^2 \int 1d(-\tan (e+f x))}{f}+\int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )^2 c^2+c^2\right )dx\right )}{a}\right )}{3 a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )^2 c^2+c^2\right )dx-\frac {2 c^2 \tan (e+f x)}{f}\right )}{a}\right )}{3 a}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\frac {3}{2} c^2 \int \sec (e+f x)dx-\frac {2 c^2 \tan (e+f x)}{f}+\frac {c^2 \tan (e+f x) \sec (e+f x)}{2 f}\right )}{a}\right )}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\frac {3}{2} c^2 \int \csc \left (e+f x+\frac {\pi }{2}\right )dx-\frac {2 c^2 \tan (e+f x)}{f}+\frac {c^2 \tan (e+f x) \sec (e+f x)}{2 f}\right )}{a}\right )}{3 a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)}-\frac {5 c \left (\frac {3 c^2 \text {arctanh}(\sin (e+f x))}{2 f}-\frac {2 c^2 \tan (e+f x)}{f}+\frac {c^2 \tan (e+f x) \sec (e+f x)}{2 f}\right )}{a}\right )}{3 a}\) |
(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)
3.1.43.3.1 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 = 1.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {8 c^{4} \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}+\frac {13}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {35 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16}-\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {13}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {35 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16}\right )}{f \,a^{2}}\) | \(125\) |
| default | \(\frac {8 c^{4} \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}+\frac {13}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {35 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16}-\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {13}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {35 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16}\right )}{f \,a^{2}}\) | \(125\) |
| parallelrisch | \(-\frac {51 \left (\frac {35 \left (1+\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{51}+\frac {35 \left (-1-\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{51}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \left (\cos \left (f x +e \right )+\frac {229 \cos \left (2 f x +2 e \right )}{306}+\frac {41 \cos \left (3 f x +3 e \right )}{153}+\frac {223}{306}\right )\right ) c^{4}}{2 f \,a^{2} \left (1+\cos \left (2 f x +2 e \right )\right )}\) | \(129\) |
| risch | \(-\frac {i c^{4} \left (99 \,{\mathrm e}^{6 i \left (f x +e \right )}+333 \,{\mathrm e}^{5 i \left (f x +e \right )}+434 \,{\mathrm e}^{4 i \left (f x +e \right )}+714 \,{\mathrm e}^{3 i \left (f x +e \right )}+487 \,{\mathrm e}^{2 i \left (f x +e \right )}+393 \,{\mathrm e}^{i \left (f x +e \right )}+164\right )}{3 f \,a^{2} \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}+\frac {35 c^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 a^{2} f}-\frac {35 c^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 a^{2} f}\) | \(156\) |
| norman | \(\frac {-\frac {35 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {385 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 a f}-\frac {511 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3 a f}+\frac {93 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{a f}-\frac {40 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{3 a f}-\frac {8 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{3 a f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4} a}-\frac {35 c^{4} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 a^{2} f}+\frac {35 c^{4} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 a^{2} f}\) | \(198\) |
8/f*c^4/a^2*(-1/3*tan(1/2*f*x+1/2*e)^3-3*tan(1/2*f*x+1/2*e)+1/16/(tan(1/2* f*x+1/2*e)-1)^2+13/16/(tan(1/2*f*x+1/2*e)-1)-35/16*ln(tan(1/2*f*x+1/2*e)-1 )-1/16/(tan(1/2*f*x+1/2*e)+1)^2+13/16/(tan(1/2*f*x+1/2*e)+1)+35/16*ln(tan( 1/2*f*x+1/2*e)+1))
Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.31 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + 2 \, c^{4} \cos \left (f x + e\right )^{3} + c^{4} \cos \left (f x + e\right )^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + 2 \, c^{4} \cos \left (f x + e\right )^{3} + c^{4} \cos \left (f x + e\right )^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (164 \, c^{4} \cos \left (f x + e\right )^{3} + 229 \, c^{4} \cos \left (f x + e\right )^{2} + 30 \, c^{4} \cos \left (f x + e\right ) - 3 \, c^{4}\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{2} f \cos \left (f x + e\right )^{4} + 2 \, a^{2} f \cos \left (f x + e\right )^{3} + a^{2} f \cos \left (f x + e\right )^{2}\right )}} \]
1/12*(105*(c^4*cos(f*x + e)^4 + 2*c^4*cos(f*x + e)^3 + c^4*cos(f*x + e)^2) *log(sin(f*x + e) + 1) - 105*(c^4*cos(f*x + e)^4 + 2*c^4*cos(f*x + e)^3 + c^4*cos(f*x + e)^2)*log(-sin(f*x + e) + 1) - 2*(164*c^4*cos(f*x + e)^3 + 2 29*c^4*cos(f*x + e)^2 + 30*c^4*cos(f*x + e) - 3*c^4)*sin(f*x + e))/(a^2*f* cos(f*x + e)^4 + 2*a^2*f*cos(f*x + e)^3 + a^2*f*cos(f*x + e)^2)
\[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {c^{4} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {4 \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {6 \sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {4 \sec ^{4}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{5}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx\right )}{a^{2}} \]
c**4*(Integral(sec(e + f*x)/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x) + I ntegral(-4*sec(e + f*x)**2/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x) + In tegral(6*sec(e + f*x)**3/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x) + Inte gral(-4*sec(e + f*x)**4/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x) + Integ ral(sec(e + f*x)**5/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x))/a**2
Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (142) = 284\).
Time = 0.21 (sec) , antiderivative size = 531, normalized size of antiderivative = 3.54 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=-\frac {c^{4} {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} - \frac {2 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {21 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {21 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} + 4 \, c^{4} {\left (\frac {\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (f x + e\right )}{{\left (a^{2} - \frac {a^{2} \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 )}}\right )} + 6 \, c^{4} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} + \frac {4 \, c^{4} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} - \frac {c^{4} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, f} \]
-1/6*(c^4*(6*(3*sin(f*x + e)/(cos(f*x + e) + 1) - 5*sin(f*x + e)^3/(cos(f* x + e) + 1)^3)/(a^2 - 2*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin( f*x + e)^4/(cos(f*x + e) + 1)^4) + (21*sin(f*x + e)/(cos(f*x + e) + 1) + s in(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2 - 21*log(sin(f*x + e)/(cos(f*x + e ) + 1) + 1)/a^2 + 21*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^2) + 4*c^4 *((15*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(cos(f*x + e) + 1)^ 3)/a^2 - 12*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a^2 + 12*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^2 + 12*sin(f*x + e)/((a^2 - a^2*sin(f*x + e )^2/(cos(f*x + e) + 1)^2)*(cos(f*x + e) + 1))) + 6*c^4*((9*sin(f*x + e)/(c os(f*x + e) + 1) + sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2 - 6*log(sin(f* x + e)/(cos(f*x + e) + 1) + 1)/a^2 + 6*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^2) + 4*c^4*(3*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(co s(f*x + e) + 1)^3)/a^2 - c^4*(3*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2)/f
Time = 0.35 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {\frac {105 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} - \frac {105 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} + \frac {6 \, {\left (13 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 11 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{2}} - \frac {16 \, {\left (a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 9 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6}}}{6 \, f} \]
1/6*(105*c^4*log(abs(tan(1/2*f*x + 1/2*e) + 1))/a^2 - 105*c^4*log(abs(tan( 1/2*f*x + 1/2*e) - 1))/a^2 + 6*(13*c^4*tan(1/2*f*x + 1/2*e)^3 - 11*c^4*tan (1/2*f*x + 1/2*e))/((tan(1/2*f*x + 1/2*e)^2 - 1)^2*a^2) - 16*(a^4*c^4*tan( 1/2*f*x + 1/2*e)^3 + 9*a^4*c^4*tan(1/2*f*x + 1/2*e))/a^6)/f
Time = 12.99 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.91 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {13\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-11\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a^2\right )}-\frac {24\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^2\,f}-\frac {8\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^2\,f}+\frac {35\,c^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^2\,f} \]