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

Optimal. Leaf size=252 \[ -\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot (e+f x)}{a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}+\frac {x}{a^3 c^6} \]

[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

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Rubi [A]  time = 0.30, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3904, 3886, 3473, 8, 2606, 194, 2607, 30, 270} \[ -\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot (e+f x)}{a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}+\frac {x}{a^3 c^6} \]

Antiderivative was successfully verified.

[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 194

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 270

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 2606

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 2607

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 3473

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 3886

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 3904

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*} \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx &=\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 {\operatorname {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 \operatorname {Subst}\left (\int x^{10} \, dx,x,-\cot (e+f x)\right )}{a^3 c^6 f}-\frac {3 \operatorname {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 {\operatorname {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 \operatorname {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*}

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Mathematica [A]  time = 2.36, size = 499, normalized size = 1.98 \[ \frac {\csc \left (\frac {e}{2}\right ) \sec \left (\frac {e}{2}\right ) \tan (e+f x) \sec ^8(e+f x) (-86058610 \sin (e+f x)[phone] \sin (2 (e+f x))[phone] \sin (3 (e+f x))-39117550 \sin (4 (e+f x))+7823510 \sin (5 (e+f x))+7823510 \sin (6 (e+f x))-4694106 \sin (7 (e+f x))+782351 \sin (8 (e+f x))-55651200 \sin (2 e+f x)[phone] \sin (e+2 f x)[phone] \sin (3 e+2 f x)+8100992 \sin (2 e+3 f x)[phone] \sin (4 e+3 f x)-28627840 \sin (3 e+4 f x)-19071360 \sin (5 e+4 f x)+9687680 \sin (4 e+5 f x)-147840 \sin (6 e+5 f x)+5548160 \sin (5 e+6 f x)+3991680 \sin (7 e+6 f x)-4393344 \sin (6 e+7 f x)-1330560 \sin (8 e+7 f x)+953984 \sin (7 e+8 f x)-24393600 f x \cos (2 e+f x)-14636160 f x \cos (e+2 f x)[phone] f x \cos (3 e+2 f x)-7539840 f x \cos (2 e+3 f x)+7539840 f x \cos (4 e+3 f x)[phone] f x \cos (3 e+4 f x)-11088000 f x \cos (5 e+4 f x)-2217600 f x \cos (4 e+5 f x)+2217600 f x \cos (6 e+5 f x)-2217600 f x \cos (5 e+6 f x)+2217600 f x \cos (7 e+6 f x)+1330560 f x \cos (6 e+7 f x)-1330560 f x \cos (8 e+7 f x)-221760 f x \cos (7 e+8 f x)+221760 f x \cos (9 e+8 f x)[phone] \sin (e)-49287040 \sin (f x)[phone] f x \cos (f x))}{113541120 a^3 c^6 f (\sec (e+f x)-1)^6 (\sec (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[e/2]*Sec[e/2]*Sec[e + f*x]^8*(24393600*f*x*Cos[f*x] - 24393600*f*x*Cos[2*e + f*x] - 14636160*f*x*Cos[e +
2*f*x] + 14636160*f*x*Cos[3*e + 2*f*x] - 7539840*f*x*Cos[2*e + 3*f*x] + 7539840*f*x*Cos[4*e + 3*f*x] + 1108800
0*f*x*Cos[3*e + 4*f*x] - 11088000*f*x*Cos[5*e + 4*f*x] - 2217600*f*x*Cos[4*e + 5*f*x] + 2217600*f*x*Cos[6*e +
5*f*x] - 2217600*f*x*Cos[5*e + 6*f*x] + 2217600*f*x*Cos[7*e + 6*f*x] + 1330560*f*x*Cos[6*e + 7*f*x] - 1330560*
f*x*Cos[8*e + 7*f*x] - 221760*f*x*Cos[7*e + 8*f*x] + 221760*f*x*Cos[9*e + 8*f*x] + 17677440*Sin[e] - 49287040*
Sin[f*x] - 86058610*Sin[e + f*x] + 51635166*Sin[2*(e + f*x)] + 26599934*Sin[3*(e + f*x)] - 39117550*Sin[4*(e +
f*x)] + 7823510*Sin[5*(e + f*x)] + 7823510*Sin[6*(e + f*x)] - 4694106*Sin[7*(e + f*x)] + 782351*Sin[8*(e + f*
x)] - 55651200*Sin[2*e + f*x] + 47971968*Sin[e + 2*f*x] + 14990976*Sin[3*e + 2*f*x] + 8100992*Sin[2*e + 3*f*x]
+ 24334464*Sin[4*e + 3*f*x] - 28627840*Sin[3*e + 4*f*x] - 19071360*Sin[5*e + 4*f*x] + 9687680*Sin[4*e + 5*f*x
] - 147840*Sin[6*e + 5*f*x] + 5548160*Sin[5*e + 6*f*x] + 3991680*Sin[7*e + 6*f*x] - 4393344*Sin[6*e + 7*f*x] -
1330560*Sin[8*e + 7*f*x] + 953984*Sin[7*e + 8*f*x])*Tan[e + f*x])/(113541120*a^3*c^6*f*(-1 + Sec[e + f*x])^6*
(1 + Sec[e + f*x])^3)

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fricas [A]  time = 0.45, size = 310, normalized size = 1.23 \[ \frac {7453 \, \cos \left (f x + e\right )^{8} - 11964 \, \cos \left (f x + e\right )^{7} - 11866 \, \cos \left (f x + e\right )^{6} + 30542 \, \cos \left (f x + e\right )^{5} + 90 \, \cos \left (f x + e\right )^{4} - 26438 \, \cos \left (f x + e\right )^{3} + 8539 \, \cos \left (f x + e\right )^{2} + 3465 \, {\left (f x \cos \left (f x + e\right )^{7} - 3 \, f x \cos \left (f x + e\right )^{6} + f x \cos \left (f x + e\right )^{5} + 5 \, f x \cos \left (f x + e\right )^{4} - 5 \, f x \cos \left (f x + e\right )^{3} - f x \cos \left (f x + e\right )^{2} + 3 \, f x \cos \left (f x + e\right ) - f x\right )} \sin \left (f x + e\right ) + 7671 \, \cos \left (f x + e\right ) - 3712}{3465 \, {\left (a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 3 \, a^{3} c^{6} f \cos \left (f x + e\right )^{6} + a^{3} c^{6} f \cos \left (f x + e\right )^{5} + 5 \, a^{3} c^{6} f \cos \left (f x + e\right )^{4} - 5 \, a^{3} c^{6} f \cos \left (f x + e\right )^{3} - a^{3} c^{6} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} c^{6} f \cos \left (f x + e\right ) - a^{3} c^{6} f\right )} \sin \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.98, size = 179, normalized size = 0.71 \[ \frac {\frac {887040 \, {\left (f x + e\right )}}{a^{3} c^{6}} + \frac {5 \, {\left (264726 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 59136 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 18018 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 4554 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 770 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 63\right )}}{a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11}} - \frac {231 \, {\left (3 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 50 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 690 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15} c^{30}}}{887040 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 1.00, size = 219, normalized size = 0.87 \[ -\frac {\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )}{1280 f \,a^{3} c^{6}}+\frac {5 \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{384 f \,a^{3} c^{6}}-\frac {23 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{128 f \,a^{3} c^{6}}-\frac {1}{2816 f \,a^{3} c^{6} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{11}}+\frac {5}{1152 f \,a^{3} c^{6} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{9}}-\frac {23}{896 f \,a^{3} c^{6} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{7}}+\frac {13}{128 f \,a^{3} c^{6} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{5}}-\frac {1}{3 f \,a^{3} c^{6} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{3}}+\frac {191}{128 f \,a^{3} c^{6} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}+\frac {2 \arctan \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f \,a^{3} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1280/f/a^3/c^6*tan(1/2*e+1/2*f*x)^5+5/384/f/a^3/c^6*tan(1/2*e+1/2*f*x)^3-23/128/f/a^3/c^6*tan(1/2*e+1/2*f*x
)-1/2816/f/a^3/c^6/tan(1/2*e+1/2*f*x)^11+5/1152/f/a^3/c^6/tan(1/2*e+1/2*f*x)^9-23/896/f/a^3/c^6/tan(1/2*e+1/2*
f*x)^7+13/128/f/a^3/c^6/tan(1/2*e+1/2*f*x)^5-1/3/f/a^3/c^6/tan(1/2*e+1/2*f*x)^3+191/128/f/a^3/c^6/tan(1/2*e+1/
2*f*x)+2/f/a^3/c^6*arctan(tan(1/2*e+1/2*f*x))

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maxima [A]  time = 0.45, size = 227, normalized size = 0.90 \[ -\frac {\frac {231 \, {\left (\frac {690 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {50 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{6}} - \frac {1774080 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3} c^{6}} - \frac {5 \, {\left (\frac {770 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {4554 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {18018 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {59136 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {264726 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 63\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{a^{3} c^{6} \sin \left (f x + e\right )^{11}}}{887040 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 2.33, size = 257, normalized size = 1.02 \[ -\frac {315\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}+693\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}-11550\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+159390\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-1323630\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+295680\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-90090\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+22770\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-3850\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-887040\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (e+f\,x\right )}{887040\,a^3\,c^6\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{\sec ^{9}{\left (e + f x \right )} - 3 \sec ^{8}{\left (e + f x \right )} + 8 \sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} - 6 \sec ^{4}{\left (e + f x \right )} + 8 \sec ^{3}{\left (e + f x \right )} - 3 \sec {\left (e + f x \right )} + 1}\, dx}{a^{3} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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