\(\int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx\) [41]

Optimal result

Integrand size = 26, antiderivative size = 252 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=\frac {x}{a^3 c^6}+\frac {\cot (e+f x)}{a^3 c^6 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f} \]

[Out]

x/a^3/c^6+cot(f*x+e)/a^3/c^6/f-1/3*cot(f*x+e)^3/a^3/c^6/f+1/5*cot(f*x+e)^5/a^3/c^6/f-1/7*cot(f*x+e)^7/a^3/c^6/
f+1/9*cot(f*x+e)^9/a^3/c^6/f-4/11*cot(f*x+e)^11/a^3/c^6/f+3*csc(f*x+e)/a^3/c^6/f-16/3*csc(f*x+e)^3/a^3/c^6/f+3
4/5*csc(f*x+e)^5/a^3/c^6/f-36/7*csc(f*x+e)^7/a^3/c^6/f+19/9*csc(f*x+e)^9/a^3/c^6/f-4/11*csc(f*x+e)^11/a^3/c^6/
f

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3989, 3971, 3554, 8, 2686, 200, 2687, 30, 276} \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot (e+f x)}{a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}+\frac {x}{a^3 c^6} \]

[In]

Int[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^6),x]

[Out]

x/(a^3*c^6) + Cot[e + f*x]/(a^3*c^6*f) - Cot[e + f*x]^3/(3*a^3*c^6*f) + Cot[e + f*x]^5/(5*a^3*c^6*f) - Cot[e +
f*x]^7/(7*a^3*c^6*f) + Cot[e + f*x]^9/(9*a^3*c^6*f) - (4*Cot[e + f*x]^11)/(11*a^3*c^6*f) + (3*Csc[e + f*x])/(
a^3*c^6*f) - (16*Csc[e + f*x]^3)/(3*a^3*c^6*f) + (34*Csc[e + f*x]^5)/(5*a^3*c^6*f) - (36*Csc[e + f*x]^7)/(7*a^
3*c^6*f) + (19*Csc[e + f*x]^9)/(9*a^3*c^6*f) - (4*Csc[e + f*x]^11)/(11*a^3*c^6*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 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && 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 ^{12}(e+f x) (a+a \sec (e+f x))^3 \, dx}{a^6 c^6} \\ & = \frac {\int \left (a^3 \cot ^{12}(e+f x)+3 a^3 \cot ^{11}(e+f x) \csc (e+f x)+3 a^3 \cot ^{10}(e+f x) \csc ^2(e+f x)+a^3 \cot ^9(e+f x) \csc ^3(e+f x)\right ) \, dx}{a^6 c^6} \\ & = \frac {\int \cot ^{12}(e+f x) \, dx}{a^3 c^6}+\frac {\int \cot ^9(e+f x) \csc ^3(e+f x) \, dx}{a^3 c^6}+\frac {3 \int \cot ^{11}(e+f x) \csc (e+f x) \, dx}{a^3 c^6}+\frac {3 \int \cot ^{10}(e+f x) \csc ^2(e+f x) \, dx}{a^3 c^6} \\ & = -\frac {\cot ^{11}(e+f x)}{11 a^3 c^6 f}-\frac {\int \cot ^{10}(e+f x) \, dx}{a^3 c^6}-\frac {\text {Subst}\left (\int x^2 \left (-1+x^2\right )^4 \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f}+\frac {3 \text {Subst}\left (\int x^{10} \, dx,x,-\cot (e+f x)\right )}{a^3 c^6 f}-\frac {3 \text {Subst}\left (\int \left (-1+x^2\right )^5 \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f} \\ & = \frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {\int \cot ^8(e+f x) \, dx}{a^3 c^6}-\frac {\text {Subst}\left (\int \left (x^2-4 x^4+6 x^6-4 x^8+x^{10}\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f}-\frac {3 \text {Subst}\left (\int \left (-1+5 x^2-10 x^4+10 x^6-5 x^8+x^{10}\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f} \\ & = -\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}-\frac {\int \cot ^6(e+f x) \, dx}{a^3 c^6} \\ & = \frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {\int \cot ^4(e+f x) \, dx}{a^3 c^6} \\ & = -\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}-\frac {\int \cot ^2(e+f x) \, dx}{a^3 c^6} \\ & = \frac {\cot (e+f x)}{a^3 c^6 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {\int 1 \, dx}{a^3 c^6} \\ & = \frac {x}{a^3 c^6}+\frac {\cot (e+f x)}{a^3 c^6 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 11.12 (sec) , antiderivative size = 787, normalized size of antiderivative = 3.12 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=\frac {\cot ^9(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {11}{2},1,-\frac {9}{2},-\tan ^2(e+f x)\right )}{11 a^2 c^5 f (a+a \sec (e+f x)) (c-c \sec (e+f x))}+\frac {16 \tan (e+f x)}{55 a^3 f (c-c \sec (e+f x))^6}-\frac {2 a^3 \tan (e+f x)}{11 f (a+a \sec (e+f x))^6 (c-c \sec (e+f x))^6}-\frac {3 a^3 \sec (e+f x) \tan (e+f x)}{11 f (a+a \sec (e+f x))^6 (c-c \sec (e+f x))^6}-\frac {a^2 \tan (e+f x)}{9 f (a+a \sec (e+f x))^5 (c-c \sec (e+f x))^6}-\frac {a \tan (e+f x)}{63 f (a+a \sec (e+f x))^4 (c-c \sec (e+f x))^6}-\frac {\tan (e+f x)}{35 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6}-\frac {8 \tan (e+f x)}{105 a f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^6}-\frac {8 \tan (e+f x)}{15 a^2 f (a+a \sec (e+f x)) (c-c \sec (e+f x))^6}+\frac {16 \tan (e+f x)}{99 a^3 c f (c-c \sec (e+f x))^5}-\frac {10 a^2 \sec (e+f x) \tan (e+f x)}{33 c f (a+a \sec (e+f x))^5 (c-c \sec (e+f x))^5}+\frac {64 \tan (e+f x)}{693 a^3 c^2 f (c-c \sec (e+f x))^4}-\frac {80 a \sec (e+f x) \tan (e+f x)}{231 c^2 f (a+a \sec (e+f x))^4 (c-c \sec (e+f x))^4}+\frac {64 \tan (e+f x)}{1155 a^3 c^3 f (c-c \sec (e+f x))^3}-\frac {32 \sec (e+f x) \tan (e+f x)}{77 c^3 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3}+\frac {128 \tan (e+f x)}{3465 a^3 c^4 f (c-c \sec (e+f x))^2}-\frac {128 \sec (e+f x) \tan (e+f x)}{231 a c^4 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2}+\frac {128 \tan (e+f x)}{3465 a^3 c^5 f (c-c \sec (e+f x))}-\frac {256 \sec (e+f x) \tan (e+f x)}{231 a^2 c^5 f (a+a \sec (e+f x)) (c-c \sec (e+f x))} \]

[In]

Integrate[1/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^6),x]

[Out]

(Cot[e + f*x]^9*Hypergeometric2F1[-11/2, 1, -9/2, -Tan[e + f*x]^2])/(11*a^2*c^5*f*(a + a*Sec[e + f*x])*(c - c*
Sec[e + f*x])) + (16*Tan[e + f*x])/(55*a^3*f*(c - c*Sec[e + f*x])^6) - (2*a^3*Tan[e + f*x])/(11*f*(a + a*Sec[e
+ f*x])^6*(c - c*Sec[e + f*x])^6) - (3*a^3*Sec[e + f*x]*Tan[e + f*x])/(11*f*(a + a*Sec[e + f*x])^6*(c - c*Sec
[e + f*x])^6) - (a^2*Tan[e + f*x])/(9*f*(a + a*Sec[e + f*x])^5*(c - c*Sec[e + f*x])^6) - (a*Tan[e + f*x])/(63*
f*(a + a*Sec[e + f*x])^4*(c - c*Sec[e + f*x])^6) - Tan[e + f*x]/(35*f*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*
x])^6) - (8*Tan[e + f*x])/(105*a*f*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^6) - (8*Tan[e + f*x])/(15*a^2*f
*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^6) + (16*Tan[e + f*x])/(99*a^3*c*f*(c - c*Sec[e + f*x])^5) - (10*a^
2*Sec[e + f*x]*Tan[e + f*x])/(33*c*f*(a + a*Sec[e + f*x])^5*(c - c*Sec[e + f*x])^5) + (64*Tan[e + f*x])/(693*a
^3*c^2*f*(c - c*Sec[e + f*x])^4) - (80*a*Sec[e + f*x]*Tan[e + f*x])/(231*c^2*f*(a + a*Sec[e + f*x])^4*(c - c*S
ec[e + f*x])^4) + (64*Tan[e + f*x])/(1155*a^3*c^3*f*(c - c*Sec[e + f*x])^3) - (32*Sec[e + f*x]*Tan[e + f*x])/(
77*c^3*f*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3) + (128*Tan[e + f*x])/(3465*a^3*c^4*f*(c - c*Sec[e + f*
x])^2) - (128*Sec[e + f*x]*Tan[e + f*x])/(231*a*c^4*f*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2) + (128*Ta
n[e + f*x])/(3465*a^3*c^5*f*(c - c*Sec[e + f*x])) - (256*Sec[e + f*x]*Tan[e + f*x])/(231*a^2*c^5*f*(a + a*Sec[
e + f*x])*(c - c*Sec[e + f*x]))

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.52

method	result	size
parallelrisch	\(\frac {-315 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}+3850 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-22770 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-693 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+90090 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+11550 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-295680 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+887040 f x -159390 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1323630 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{887040 f \,a^{3} c^{6}}\)	\(130\)
derivativedivides	\(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\frac {10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-46 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+512 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {1}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}+\frac {10}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {46}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {26}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {256}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {382}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{256 f \,a^{3} c^{6}}\)	\(140\)
default	\(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\frac {10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-46 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+512 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {1}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}+\frac {10}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {46}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {26}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {256}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {382}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{256 f \,a^{3} c^{6}}\)	\(140\)
risch	\(\frac {x}{a^{3} c^{6}}+\frac {2 i \left (10395 \,{\mathrm e}^{15 i \left (f x +e \right )}-31185 \,{\mathrm e}^{14 i \left (f x +e \right )}+1155 \,{\mathrm e}^{13 i \left (f x +e \right )}+148995 \,{\mathrm e}^{12 i \left (f x +e \right )}-190113 \,{\mathrm e}^{11 i \left (f x +e \right )}-117117 \,{\mathrm e}^{10 i \left (f x +e \right )}+434775 \,{\mathrm e}^{9 i \left (f x +e \right )}-138105 \,{\mathrm e}^{8 i \left (f x +e \right )}-385055 \,{\mathrm e}^{7 i \left (f x +e \right )}+374781 \,{\mathrm e}^{6 i \left (f x +e \right )}+63289 \,{\mathrm e}^{5 i \left (f x +e \right )}-223655 \,{\mathrm e}^{4 i \left (f x +e \right )}+75685 \,{\mathrm e}^{3 i \left (f x +e \right )}+43345 \,{\mathrm e}^{2 i \left (f x +e \right )}-34323 \,{\mathrm e}^{i \left (f x +e \right )}+7453\right )}{3465 f \,a^{3} c^{6} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{11} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5}}\)	\(215\)
norman	\(\frac {\frac {x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{c a}-\frac {1}{2816 a c f}+\frac {5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{1152 a c f}-\frac {23 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{896 a c f}+\frac {13 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{128 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{3 a c f}+\frac {191 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{128 a c f}-\frac {23 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{128 a c f}+\frac {5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{384 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{16}}{1280 a c f}}{c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}\)	\(226\)

[In]

int(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^6,x,method=_RETURNVERBOSE)

[Out]

1/887040*(-315*cot(1/2*f*x+1/2*e)^11+3850*cot(1/2*f*x+1/2*e)^9-22770*cot(1/2*f*x+1/2*e)^7-693*tan(1/2*f*x+1/2*
e)^5+90090*cot(1/2*f*x+1/2*e)^5+11550*tan(1/2*f*x+1/2*e)^3-295680*cot(1/2*f*x+1/2*e)^3+887040*f*x-159390*tan(1
/2*f*x+1/2*e)+1323630*cot(1/2*f*x+1/2*e))/f/a^3/c^6

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=\frac {7453 \, \cos \left (f x + e\right )^{8} - 11964 \, \cos \left (f x + e\right )^{7} - 11866 \, \cos \left (f x + e\right )^{6} + 30542 \, \cos \left (f x + e\right )^{5} + 90 \, \cos \left (f x + e\right )^{4} - 26438 \, \cos \left (f x + e\right )^{3} + 8539 \, \cos \left (f x + e\right )^{2} + 3465 \, {\left (f x \cos \left (f x + e\right )^{7} - 3 \, f x \cos \left (f x + e\right )^{6} + f x \cos \left (f x + e\right )^{5} + 5 \, f x \cos \left (f x + e\right )^{4} - 5 \, f x \cos \left (f x + e\right )^{3} - f x \cos \left (f x + e\right )^{2} + 3 \, f x \cos \left (f x + e\right ) - f x\right )} \sin \left (f x + e\right ) + 7671 \, \cos \left (f x + e\right ) - 3712}{3465 \, {\left (a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 3 \, a^{3} c^{6} f \cos \left (f x + e\right )^{6} + a^{3} c^{6} f \cos \left (f x + e\right )^{5} + 5 \, a^{3} c^{6} f \cos \left (f x + e\right )^{4} - 5 \, a^{3} c^{6} f \cos \left (f x + e\right )^{3} - a^{3} c^{6} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} c^{6} f \cos \left (f x + e\right ) - a^{3} c^{6} f\right )} \sin \left (f x + e\right )} \]

[In]

integrate(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^6,x, algorithm="fricas")

[Out]

1/3465*(7453*cos(f*x + e)^8 - 11964*cos(f*x + e)^7 - 11866*cos(f*x + e)^6 + 30542*cos(f*x + e)^5 + 90*cos(f*x
+ e)^4 - 26438*cos(f*x + e)^3 + 8539*cos(f*x + e)^2 + 3465*(f*x*cos(f*x + e)^7 - 3*f*x*cos(f*x + e)^6 + f*x*co
s(f*x + e)^5 + 5*f*x*cos(f*x + e)^4 - 5*f*x*cos(f*x + e)^3 - f*x*cos(f*x + e)^2 + 3*f*x*cos(f*x + e) - f*x)*si
n(f*x + e) + 7671*cos(f*x + e) - 3712)/((a^3*c^6*f*cos(f*x + e)^7 - 3*a^3*c^6*f*cos(f*x + e)^6 + a^3*c^6*f*cos
(f*x + e)^5 + 5*a^3*c^6*f*cos(f*x + e)^4 - 5*a^3*c^6*f*cos(f*x + e)^3 - a^3*c^6*f*cos(f*x + e)^2 + 3*a^3*c^6*f
*cos(f*x + e) - a^3*c^6*f)*sin(f*x + e))

Sympy [F]

\[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=\frac {\int \frac {1}{\sec ^{9}{\left (e + f x \right )} - 3 \sec ^{8}{\left (e + f x \right )} + 8 \sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} - 6 \sec ^{4}{\left (e + f x \right )} + 8 \sec ^{3}{\left (e + f x \right )} - 3 \sec {\left (e + f x \right )} + 1}\, dx}{a^{3} c^{6}} \]

[In]

integrate(1/(a+a*sec(f*x+e))**3/(c-c*sec(f*x+e))**6,x)

[Out]

Integral(1/(sec(e + f*x)**9 - 3*sec(e + f*x)**8 + 8*sec(e + f*x)**6 - 6*sec(e + f*x)**5 - 6*sec(e + f*x)**4 +
8*sec(e + f*x)**3 - 3*sec(e + f*x) + 1), x)/(a**3*c**6)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=-\frac {\frac {231 \, {\left (\frac {690 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {50 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{6}} - \frac {1774080 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3} c^{6}} - \frac {5 \, {\left (\frac {770 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {4554 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {18018 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {59136 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {264726 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 63\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{a^{3} c^{6} \sin \left (f x + e\right )^{11}}}{887040 \, f} \]

[In]

integrate(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^6,x, algorithm="maxima")

[Out]

-1/887040*(231*(690*sin(f*x + e)/(cos(f*x + e) + 1) - 50*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^6) - 1774080*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/(a^3*c^6) - 5*(770*sin(f*x
+ e)^2/(cos(f*x + e) + 1)^2 - 4554*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 18018*sin(f*x + e)^6/(cos(f*x + e) +
1)^6 - 59136*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 264726*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 63)*(cos(f*
x + e) + 1)^11/(a^3*c^6*sin(f*x + e)^11))/f

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=\frac {\frac {887040 \, {\left (f x + e\right )}}{a^{3} c^{6}} + \frac {5 \, {\left (264726 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 59136 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 18018 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 4554 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 770 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 63\right )}}{a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11}} - \frac {231 \, {\left (3 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 50 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 690 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15} c^{30}}}{887040 \, f} \]

[In]

integrate(1/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^6,x, algorithm="giac")

[Out]

1/887040*(887040*(f*x + e)/(a^3*c^6) + 5*(264726*tan(1/2*f*x + 1/2*e)^10 - 59136*tan(1/2*f*x + 1/2*e)^8 + 1801
8*tan(1/2*f*x + 1/2*e)^6 - 4554*tan(1/2*f*x + 1/2*e)^4 + 770*tan(1/2*f*x + 1/2*e)^2 - 63)/(a^3*c^6*tan(1/2*f*x
+ 1/2*e)^11) - 231*(3*a^12*c^24*tan(1/2*f*x + 1/2*e)^5 - 50*a^12*c^24*tan(1/2*f*x + 1/2*e)^3 + 690*a^12*c^24*
tan(1/2*f*x + 1/2*e))/(a^15*c^30))/f

Mupad [B] (verification not implemented)

Time = 14.79 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=-\frac {315\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}+693\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}-11550\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+159390\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-1323630\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+295680\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-90090\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+22770\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-3850\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-887040\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (e+f\,x\right )}{887040\,a^3\,c^6\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}} \]

[In]

int(1/((a + a/cos(e + f*x))^3*(c - c/cos(e + f*x))^6),x)

[Out]

-(315*cos(e/2 + (f*x)/2)^16 + 693*sin(e/2 + (f*x)/2)^16 - 11550*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2)^14 + 1
59390*cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2)^12 - 1323630*cos(e/2 + (f*x)/2)^6*sin(e/2 + (f*x)/2)^10 + 295680
*cos(e/2 + (f*x)/2)^8*sin(e/2 + (f*x)/2)^8 - 90090*cos(e/2 + (f*x)/2)^10*sin(e/2 + (f*x)/2)^6 + 22770*cos(e/2
+ (f*x)/2)^12*sin(e/2 + (f*x)/2)^4 - 3850*cos(e/2 + (f*x)/2)^14*sin(e/2 + (f*x)/2)^2 - 887040*cos(e/2 + (f*x)/
2)^5*sin(e/2 + (f*x)/2)^11*(e + f*x))/(887040*a^3*c^6*f*cos(e/2 + (f*x)/2)^5*sin(e/2 + (f*x)/2)^11)