Integrand size = 32, antiderivative size = 164 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx=-\frac {7 c^4 \text {arctanh}(\sin (e+f x))}{a^3 f}+\frac {7 c^4 \tan (e+f x)}{a^3 f}+\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {14 \left (c^4-c^4 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^3+a^3 \sec (e+f x)\right )} \] Output:
-7*c^4*arctanh(sin(f*x+e))/a^3/f+7*c^4*tan(f*x+e)/a^3/f+2/5*c*(c-c*sec(f*x +e))^3*tan(f*x+e)/f/(a+a*sec(f*x+e))^3-14/15*(c^2-c^2*sec(f*x+e))^2*tan(f* x+e)/a/f/(a+a*sec(f*x+e))^2+14/3*(c^4-c^4*sec(f*x+e))*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 = 0.83 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.46 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx=-\frac {16 c^4 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {5}{2},-\frac {3}{2},\frac {1}{2} (1+\sec (e+f x))\right ) \sqrt {2-2 \sec (e+f x)} \tan (e+f x)}{5 a^3 f (-1+\sec (e+f x)) (1+\sec (e+f x))^3} \] Input:
Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^4)/(a + a*Sec[e + f*x])^3,x]
Output:
(-16*c^4*Hypergeometric2F1[-7/2, -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 = 0.98 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.01, 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, 4445, 3042, 4274, 3042, 4254, 24, 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.
| \(\displaystyle \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a \sec (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}-\frac {7 c \int \frac {\sec (e+f x) (c-c \sec (e+f x))^3}{(\sec (e+f x) a+a)^2}dx}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}-\frac {7 c \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3}{\left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a}\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2}-\frac {5 c \int \frac {\sec (e+f x) (c-c \sec (e+f x))^2}{\sec (e+f x) a+a}dx}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2}-\frac {5 c \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^2}{\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}dx}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 4445 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2}-\frac {5 c \left (\frac {2 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )}{f (a \sec (e+f x)+a)}-\frac {3 c \int \sec (e+f x) (c-c \sec (e+f x))dx}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2}-\frac {5 c \left (\frac {2 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )}{f (a \sec (e+f x)+a)}-\frac {3 c \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )dx}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2}-\frac {5 c \left (\frac {2 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )}{f (a \sec (e+f x)+a)}-\frac {3 c \left (c \int \sec (e+f x)dx-c \int \sec ^2(e+f x)dx\right )}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2}-\frac {5 c \left (\frac {2 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )}{f (a \sec (e+f x)+a)}-\frac {3 c \left (c \int \csc \left (e+f x+\frac {\pi }{2}\right )dx-c \int \csc \left (e+f x+\frac {\pi }{2}\right )^2dx\right )}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2}-\frac {5 c \left (\frac {2 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )}{f (a \sec (e+f x)+a)}-\frac {3 c \left (\frac {c \int 1d(-\tan (e+f x))}{f}+c \int \csc \left (e+f x+\frac {\pi }{2}\right )dx\right )}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2}-\frac {5 c \left (\frac {2 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )}{f (a \sec (e+f x)+a)}-\frac {3 c \left (c \int \csc \left (e+f x+\frac {\pi }{2}\right )dx-\frac {c \tan (e+f x)}{f}\right )}{a}\right )}{3 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}-\frac {7 c \left (\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2}-\frac {5 c \left (\frac {2 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )}{f (a \sec (e+f x)+a)}-\frac {3 c \left (\frac {c \text {arctanh}(\sin (e+f x))}{f}-\frac {c \tan (e+f x)}{f}\right )}{a}\right )}{3 a}\right )}{5 a}\) |
Input:
Int[(Sec[e + f*x]*(c - c*Sec[e + f*x])^4)/(a + a*Sec[e + f*x])^3,x]
Output:
(2*c*(c - c*Sec[e + f*x])^3*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) - ( 7*c*((2*c*(c - c*Sec[e + f*x])^2*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2 ) - (5*c*((2*(c^2 - c^2*Sec[e + f*x])*Tan[e + f*x])/(f*(a + a*Sec[e + f*x] )) - (3*c*((c*ArcTanh[Sin[e + f*x]])/f - (c*Tan[e + f*x])/f))/a))/(3*a)))/ (5*a)
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_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], 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]
Time = 0.36 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.66
| method | result | size |
| derivativedivides | \(\frac {4 c^{4} \left (\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {7 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{4}-\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {7 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{4}\right )}{f \,a^{3}}\) | \(108\) |
| default | \(\frac {4 c^{4} \left (\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {7 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{4}-\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {7 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{4}\right )}{f \,a^{3}}\) | \(108\) |
| parallelrisch | \(\frac {1609 \left (\frac {1680 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \cos \left (f x +e \right )}{1609}-\frac {1680 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \cos \left (f x +e \right )}{1609}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \left (\cos \left (f x +e \right )+\frac {762 \cos \left (2 f x +2 e \right )}{1609}+\frac {167 \cos \left (3 f x +3 e \right )}{1609}+\frac {822}{1609}\right )\right ) c^{4}}{240 f \,a^{3} \cos \left (f x +e \right )}\) | \(112\) |
| risch | \(\frac {2 i c^{4} \left (120 \,{\mathrm e}^{6 i \left (f x +e \right )}+495 \,{\mathrm e}^{5 i \left (f x +e \right )}+1235 \,{\mathrm e}^{4 i \left (f x +e \right )}+1270 \,{\mathrm e}^{3 i \left (f x +e \right )}+1342 \,{\mathrm e}^{2 i \left (f x +e \right )}+715 \,{\mathrm e}^{i \left (f x +e \right )}+167\right )}{15 f \,a^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5}}+\frac {7 c^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{a^{3} f}-\frac {7 c^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{a^{3} f}\) | \(156\) |
| norman | \(\frac {\frac {14 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {154 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 a f}+\frac {1022 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{15 a f}-\frac {186 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{5 a f}+\frac {92 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{15 a f}-\frac {8 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{15 a f}+\frac {4 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{5 a f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4} a^{2}}+\frac {7 c^{4} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{a^{3} f}-\frac {7 c^{4} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{a^{3} f}\) | \(220\) |
Input:
int(sec(f*x+e)*(c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^3,x,method=_RETURNVERBO SE)
Output:
4/f*c^4/a^3*(1/5*tan(1/2*f*x+1/2*e)^5+2/3*tan(1/2*f*x+1/2*e)^3+3*tan(1/2*f *x+1/2*e)-1/4/(tan(1/2*f*x+1/2*e)+1)-7/4*ln(tan(1/2*f*x+1/2*e)+1)-1/4/(tan (1/2*f*x+1/2*e)-1)+7/4*ln(tan(1/2*f*x+1/2*e)-1))
Time = 0.13 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.41 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx=-\frac {105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + 3 \, c^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} \cos \left (f x + e\right )^{2} + c^{4} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + 3 \, c^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} \cos \left (f x + e\right )^{2} + c^{4} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (167 \, c^{4} \cos \left (f x + e\right )^{3} + 381 \, c^{4} \cos \left (f x + e\right )^{2} + 277 \, c^{4} \cos \left (f x + e\right ) + 15 \, c^{4}\right )} \sin \left (f x + e\right )}{30 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} + 3 \, a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f \cos \left (f x + e\right )\right )}} \] Input:
integrate(sec(f*x+e)*(c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^3,x, algorithm="f ricas")
Output:
-1/30*(105*(c^4*cos(f*x + e)^4 + 3*c^4*cos(f*x + e)^3 + 3*c^4*cos(f*x + e) ^2 + c^4*cos(f*x + e))*log(sin(f*x + e) + 1) - 105*(c^4*cos(f*x + e)^4 + 3 *c^4*cos(f*x + e)^3 + 3*c^4*cos(f*x + e)^2 + c^4*cos(f*x + e))*log(-sin(f* x + e) + 1) - 2*(167*c^4*cos(f*x + e)^3 + 381*c^4*cos(f*x + e)^2 + 277*c^4 *cos(f*x + e) + 15*c^4)*sin(f*x + e))/(a^3*f*cos(f*x + e)^4 + 3*a^3*f*cos( f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 + a^3*f*cos(f*x + e))
\[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx=\frac {c^{4} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {4 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {6 \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {4 \sec ^{4}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{5}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx\right )}{a^{3}} \] Input:
integrate(sec(f*x+e)*(c-c*sec(f*x+e))**4/(a+a*sec(f*x+e))**3,x)
Output:
c**4*(Integral(sec(e + f*x)/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(-4*sec(e + f*x)**2/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(6*sec(e + f*x)**3/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(-4*sec(e + f*x)**4/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(sec(e + f*x)**5/(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. 470 vs. \(2 (161) = 322\).
Time = 0.05 (sec) , antiderivative size = 470, normalized size of antiderivative = 2.87 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx=\frac {3 \, c^{4} {\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 )} + 4 \, c^{4} {\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 {6 \, c^{4} {\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^{4} {\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 {12 \, c^{4} {\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} \] Input:
integrate(sec(f*x+e)*(c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^3,x, algorithm="m axima")
Output:
1/60*(3*c^4*(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) + 4*c^4*((105*sin(f*x + e)/(cos(f*x + e) + 1) + 2 0*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) + 6*c^4*(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^4*(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 - 12*c^4*(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 = 0.21 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.86 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx=-\frac {\frac {105 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {105 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} + \frac {30 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{3}} - \frac {4 \, {\left (3 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 10 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15}}}{15 \, f} \] Input:
integrate(sec(f*x+e)*(c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^3,x, algorithm="g iac")
Output:
-1/15*(105*c^4*log(abs(tan(1/2*f*x + 1/2*e) + 1))/a^3 - 105*c^4*log(abs(ta n(1/2*f*x + 1/2*e) - 1))/a^3 + 30*c^4*tan(1/2*f*x + 1/2*e)/((tan(1/2*f*x + 1/2*e)^2 - 1)*a^3) - 4*(3*a^12*c^4*tan(1/2*f*x + 1/2*e)^5 + 10*a^12*c^4*t an(1/2*f*x + 1/2*e)^3 + 45*a^12*c^4*tan(1/2*f*x + 1/2*e))/a^15)/f
Time = 10.87 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.77 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx=\frac {12\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^3\,f}+\frac {8\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^3\,f}+\frac {4\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{5\,a^3\,f}-\frac {14\,c^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^3\,f}-\frac {2\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-a^3\right )} \] Input:
int((c - c/cos(e + f*x))^4/(cos(e + f*x)*(a + a/cos(e + f*x))^3),x)
Output:
(12*c^4*tan(e/2 + (f*x)/2))/(a^3*f) + (8*c^4*tan(e/2 + (f*x)/2)^3)/(3*a^3* f) + (4*c^4*tan(e/2 + (f*x)/2)^5)/(5*a^3*f) - (14*c^4*atanh(tan(e/2 + (f*x )/2)))/(a^3*f) - (2*c^4*tan(e/2 + (f*x)/2))/(f*(a^3*tan(e/2 + (f*x)/2)^2 - a^3))
Time = 0.17 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.95 \[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx=\frac {c^{4} \left (105 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-105 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )-105 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+105 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+12 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+28 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+140 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-210 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a^{3} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )} \] Input:
int(sec(f*x+e)*(c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^3,x)
Output:
(c**4*(105*log(tan((e + f*x)/2) - 1)*tan((e + f*x)/2)**2 - 105*log(tan((e + f*x)/2) - 1) - 105*log(tan((e + f*x)/2) + 1)*tan((e + f*x)/2)**2 + 105*l og(tan((e + f*x)/2) + 1) + 12*tan((e + f*x)/2)**7 + 28*tan((e + f*x)/2)**5 + 140*tan((e + f*x)/2)**3 - 210*tan((e + f*x)/2)))/(15*a**3*f*(tan((e + f *x)/2)**2 - 1))