\(\int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx\) [40]

Optimal result

Integrand size = 26, antiderivative size = 210 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {x}{a^3 c^5}+\frac {\cot (e+f x)}{a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f} \]

[Out]

x/a^3/c^5+cot(f*x+e)/a^3/c^5/f-1/3*cot(f*x+e)^3/a^3/c^5/f+1/5*cot(f*x+e)^5/a^3/c^5/f-1/7*cot(f*x+e)^7/a^3/c^5/
f+2/9*cot(f*x+e)^9/a^3/c^5/f+2*csc(f*x+e)/a^3/c^5/f-8/3*csc(f*x+e)^3/a^3/c^5/f+12/5*csc(f*x+e)^5/a^3/c^5/f-8/7
*csc(f*x+e)^7/a^3/c^5/f+2/9*csc(f*x+e)^9/a^3/c^5/f

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3989, 3971, 3554, 8, 2686, 200, 2687, 30} \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot (e+f x)}{a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}+\frac {x}{a^3 c^5} \]

[In]

Int[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5),x]

[Out]

x/(a^3*c^5) + Cot[e + f*x]/(a^3*c^5*f) - Cot[e + f*x]^3/(3*a^3*c^5*f) + Cot[e + f*x]^5/(5*a^3*c^5*f) - Cot[e +
f*x]^7/(7*a^3*c^5*f) + (2*Cot[e + f*x]^9)/(9*a^3*c^5*f) + (2*Csc[e + f*x])/(a^3*c^5*f) - (8*Csc[e + f*x]^3)/(
3*a^3*c^5*f) + (12*Csc[e + f*x]^5)/(5*a^3*c^5*f) - (8*Csc[e + f*x]^7)/(7*a^3*c^5*f) + (2*Csc[e + f*x]^9)/(9*a^
3*c^5*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3971

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

Rule 3989

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Di
st[((-a)*c)^m, Int[Cot[e + f*x]^(2*m)*(c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
&& EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] &&  !(IntegerQ[n] && GtQ[m - n, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cot ^{10}(e+f x) (a+a \sec (e+f x))^2 \, dx}{a^5 c^5} \\ & = -\frac {\int \left (a^2 \cot ^{10}(e+f x)+2 a^2 \cot ^9(e+f x) \csc (e+f x)+a^2 \cot ^8(e+f x) \csc ^2(e+f x)\right ) \, dx}{a^5 c^5} \\ & = -\frac {\int \cot ^{10}(e+f x) \, dx}{a^3 c^5}-\frac {\int \cot ^8(e+f x) \csc ^2(e+f x) \, dx}{a^3 c^5}-\frac {2 \int \cot ^9(e+f x) \csc (e+f x) \, dx}{a^3 c^5} \\ & = \frac {\cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {\int \cot ^8(e+f x) \, dx}{a^3 c^5}-\frac {\text {Subst}\left (\int x^8 \, dx,x,-\cot (e+f x)\right )}{a^3 c^5 f}+\frac {2 \text {Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (e+f x)\right )}{a^3 c^5 f} \\ & = -\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac {\int \cot ^6(e+f x) \, dx}{a^3 c^5}+\frac {2 \text {Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c^5 f} \\ & = \frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}+\frac {\int \cot ^4(e+f x) \, dx}{a^3 c^5} \\ & = -\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac {\int \cot ^2(e+f x) \, dx}{a^3 c^5} \\ & = \frac {\cot (e+f x)}{a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}+\frac {\int 1 \, dx}{a^3 c^5} \\ & = \frac {x}{a^3 c^5}+\frac {\cot (e+f x)}{a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 6.33 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {\cot ^9(e+f x) \left (35+35 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},-\tan ^2(e+f x)\right )+630 \sec (e+f x)-1680 \sec ^3(e+f x)+2016 \sec ^5(e+f x)-1152 \sec ^7(e+f x)+256 \sec ^9(e+f x)\right )}{315 a^3 c^5 f} \]

[In]

Integrate[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5),x]

[Out]

(Cot[e + f*x]^9*(35 + 35*Hypergeometric2F1[-9/2, 1, -7/2, -Tan[e + f*x]^2] + 630*Sec[e + f*x] - 1680*Sec[e + f
*x]^3 + 2016*Sec[e + f*x]^5 - 1152*Sec[e + f*x]^7 + 256*Sec[e + f*x]^9))/(315*a^3*c^5*f)

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.56

method	result	size
parallelrisch	\(\frac {35 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-405 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-63 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+2331 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+945 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-9765 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+40320 f x -11655 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+51345 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{40320 f \,a^{3} c^{5}}\)	\(117\)
derivativedivides	\(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-37 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+256 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {9}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {37}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {31}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {163}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{128 f \,c^{5} a^{3}}\)	\(127\)
default	\(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-37 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+256 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {9}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {37}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {31}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {163}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{128 f \,c^{5} a^{3}}\)	\(127\)
risch	\(\frac {x}{a^{3} c^{5}}+\frac {4 i \left (315 \,{\mathrm e}^{13 i \left (f x +e \right )}-315 \,{\mathrm e}^{12 i \left (f x +e \right )}-1470 \,{\mathrm e}^{11 i \left (f x +e \right )}+3360 \,{\mathrm e}^{10 i \left (f x +e \right )}+1113 \,{\mathrm e}^{9 i \left (f x +e \right )}-6447 \,{\mathrm e}^{8 i \left (f x +e \right )}+2028 \,{\mathrm e}^{7 i \left (f x +e \right )}+7008 \,{\mathrm e}^{6 i \left (f x +e \right )}-4867 \,{\mathrm e}^{5 i \left (f x +e \right )}-2321 \,{\mathrm e}^{4 i \left (f x +e \right )}+3314 \,{\mathrm e}^{3 i \left (f x +e \right )}-16 \,{\mathrm e}^{2 i \left (f x +e \right )}-881 \,{\mathrm e}^{i \left (f x +e \right )}+299\right )}{315 f \,c^{5} a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5}}\)	\(193\)
norman	\(\frac {\frac {x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{c a}+\frac {1}{1152 a c f}-\frac {9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{896 a c f}+\frac {37 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{640 a c f}-\frac {31 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{128 a c f}+\frac {163 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{128 a c f}-\frac {37 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{128 a c f}+\frac {3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{128 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{640 a c f}}{c^{4} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}\)	\(204\)

[In]

int(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^5,x,method=_RETURNVERBOSE)

[Out]

1/40320*(35*cot(1/2*f*x+1/2*e)^9-405*cot(1/2*f*x+1/2*e)^7-63*tan(1/2*f*x+1/2*e)^5+2331*cot(1/2*f*x+1/2*e)^5+94
5*tan(1/2*f*x+1/2*e)^3-9765*cot(1/2*f*x+1/2*e)^3+40320*f*x-11655*tan(1/2*f*x+1/2*e)+51345*cot(1/2*f*x+1/2*e))/
f/a^3/c^5

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.29 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {598 \, \cos \left (f x + e\right )^{7} - 566 \, \cos \left (f x + e\right )^{6} - 1212 \, \cos \left (f x + e\right )^{5} + 1310 \, \cos \left (f x + e\right )^{4} + 860 \, \cos \left (f x + e\right )^{3} - 1014 \, \cos \left (f x + e\right )^{2} + 315 \, {\left (f x \cos \left (f x + e\right )^{6} - 2 \, f x \cos \left (f x + e\right )^{5} - f x \cos \left (f x + e\right )^{4} + 4 \, f x \cos \left (f x + e\right )^{3} - f x \cos \left (f x + e\right )^{2} - 2 \, f x \cos \left (f x + e\right ) + f x\right )} \sin \left (f x + e\right ) - 197 \, \cos \left (f x + e\right ) + 256}{315 \, {\left (a^{3} c^{5} f \cos \left (f x + e\right )^{6} - 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5} - a^{3} c^{5} f \cos \left (f x + e\right )^{4} + 4 \, a^{3} c^{5} f \cos \left (f x + e\right )^{3} - a^{3} c^{5} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} c^{5} f \cos \left (f x + e\right ) + a^{3} c^{5} f\right )} \sin \left (f x + e\right )} \]

[In]

integrate(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^5,x, algorithm="fricas")

[Out]

1/315*(598*cos(f*x + e)^7 - 566*cos(f*x + e)^6 - 1212*cos(f*x + e)^5 + 1310*cos(f*x + e)^4 + 860*cos(f*x + e)^
3 - 1014*cos(f*x + e)^2 + 315*(f*x*cos(f*x + e)^6 - 2*f*x*cos(f*x + e)^5 - f*x*cos(f*x + e)^4 + 4*f*x*cos(f*x
+ e)^3 - f*x*cos(f*x + e)^2 - 2*f*x*cos(f*x + e) + f*x)*sin(f*x + e) - 197*cos(f*x + e) + 256)/((a^3*c^5*f*cos
(f*x + e)^6 - 2*a^3*c^5*f*cos(f*x + e)^5 - a^3*c^5*f*cos(f*x + e)^4 + 4*a^3*c^5*f*cos(f*x + e)^3 - a^3*c^5*f*c
os(f*x + e)^2 - 2*a^3*c^5*f*cos(f*x + e) + a^3*c^5*f)*sin(f*x + e))

Sympy [F]

\[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=- \frac {\int \frac {1}{\sec ^{8}{\left (e + f x \right )} - 2 \sec ^{7}{\left (e + f x \right )} - 2 \sec ^{6}{\left (e + f x \right )} + 6 \sec ^{5}{\left (e + f x \right )} - 6 \sec ^{3}{\left (e + f x \right )} + 2 \sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} - 1}\, dx}{a^{3} c^{5}} \]

[In]

integrate(1/(a+a*sec(f*x+e))**3/(c-c*sec(f*x+e))**5,x)

[Out]

-Integral(1/(sec(e + f*x)**8 - 2*sec(e + f*x)**7 - 2*sec(e + f*x)**6 + 6*sec(e + f*x)**5 - 6*sec(e + f*x)**3 +
2*sec(e + f*x)**2 + 2*sec(e + f*x) - 1), x)/(a**3*c**5)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=-\frac {\frac {63 \, {\left (\frac {185 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {15 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{5}} - \frac {80640 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3} c^{5}} + \frac {{\left (\frac {405 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2331 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {9765 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {51345 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{a^{3} c^{5} \sin \left (f x + e\right )^{9}}}{40320 \, f} \]

[In]

integrate(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^5,x, algorithm="maxima")

[Out]

-1/40320*(63*(185*sin(f*x + e)/(cos(f*x + e) + 1) - 15*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + sin(f*x + e)^5/(c
os(f*x + e) + 1)^5)/(a^3*c^5) - 80640*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/(a^3*c^5) + (405*sin(f*x + e)^2/
(cos(f*x + e) + 1)^2 - 2331*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9765*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 5
1345*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 35)*(cos(f*x + e) + 1)^9/(a^3*c^5*sin(f*x + e)^9))/f

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {\frac {40320 \, {\left (f x + e\right )}}{a^{3} c^{5}} + \frac {51345 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 9765 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 2331 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 405 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 35}{a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}} - \frac {63 \, {\left (a^{12} c^{20} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 15 \, a^{12} c^{20} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 185 \, a^{12} c^{20} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15} c^{25}}}{40320 \, f} \]

[In]

integrate(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^5,x, algorithm="giac")

[Out]

1/40320*(40320*(f*x + e)/(a^3*c^5) + (51345*tan(1/2*f*x + 1/2*e)^8 - 9765*tan(1/2*f*x + 1/2*e)^6 + 2331*tan(1/
2*f*x + 1/2*e)^4 - 405*tan(1/2*f*x + 1/2*e)^2 + 35)/(a^3*c^5*tan(1/2*f*x + 1/2*e)^9) - 63*(a^12*c^20*tan(1/2*f
*x + 1/2*e)^5 - 15*a^12*c^20*tan(1/2*f*x + 1/2*e)^3 + 185*a^12*c^20*tan(1/2*f*x + 1/2*e))/(a^15*c^25))/f

Mupad [B] (verification not implemented)

Time = 14.61 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {35\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-63\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+945\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-11655\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+51345\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-9765\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+2331\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-405\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+40320\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (e+f\,x\right )}{40320\,a^3\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9} \]

[In]

int(1/((a + a/cos(e + f*x))^3*(c - c/cos(e + f*x))^5),x)

[Out]

(35*cos(e/2 + (f*x)/2)^14 - 63*sin(e/2 + (f*x)/2)^14 + 945*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2)^12 - 11655*
cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2)^10 + 51345*cos(e/2 + (f*x)/2)^6*sin(e/2 + (f*x)/2)^8 - 9765*cos(e/2 +
(f*x)/2)^8*sin(e/2 + (f*x)/2)^6 + 2331*cos(e/2 + (f*x)/2)^10*sin(e/2 + (f*x)/2)^4 - 405*cos(e/2 + (f*x)/2)^12*
sin(e/2 + (f*x)/2)^2 + 40320*cos(e/2 + (f*x)/2)^5*sin(e/2 + (f*x)/2)^9*(e + f*x))/(40320*a^3*c^5*f*cos(e/2 + (
f*x)/2)^5*sin(e/2 + (f*x)/2)^9)