3.30 \(\int \frac {1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5} \, dx\)

Optimal. Leaf size=210 \[ \frac {4 \cot ^9(e+f x)}{9 a^2 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac {\cot (e+f x)}{a^2 c^5 f}+\frac {4 \csc ^9(e+f x)}{9 a^2 c^5 f}-\frac {15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac {21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac {13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac {3 \csc (e+f x)}{a^2 c^5 f}+\frac {x}{a^2 c^5} \]

[Out]

x/a^2/c^5+cot(f*x+e)/a^2/c^5/f-1/3*cot(f*x+e)^3/a^2/c^5/f+1/5*cot(f*x+e)^5/a^2/c^5/f-1/7*cot(f*x+e)^7/a^2/c^5/
f+4/9*cot(f*x+e)^9/a^2/c^5/f+3*csc(f*x+e)/a^2/c^5/f-13/3*csc(f*x+e)^3/a^2/c^5/f+21/5*csc(f*x+e)^5/a^2/c^5/f-15
/7*csc(f*x+e)^7/a^2/c^5/f+4/9*csc(f*x+e)^9/a^2/c^5/f

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Rubi [A]  time = 0.29, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 17, 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} \[ \frac {4 \cot ^9(e+f x)}{9 a^2 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac {\cot (e+f x)}{a^2 c^5 f}+\frac {4 \csc ^9(e+f x)}{9 a^2 c^5 f}-\frac {15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac {21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac {13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac {3 \csc (e+f x)}{a^2 c^5 f}+\frac {x}{a^2 c^5} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5),x]

[Out]

x/(a^2*c^5) + Cot[e + f*x]/(a^2*c^5*f) - Cot[e + f*x]^3/(3*a^2*c^5*f) + Cot[e + f*x]^5/(5*a^2*c^5*f) - Cot[e +
f*x]^7/(7*a^2*c^5*f) + (4*Cot[e + f*x]^9)/(9*a^2*c^5*f) + (3*Csc[e + f*x])/(a^2*c^5*f) - (13*Csc[e + f*x]^3)/
(3*a^2*c^5*f) + (21*Csc[e + f*x]^5)/(5*a^2*c^5*f) - (15*Csc[e + f*x]^7)/(7*a^2*c^5*f) + (4*Csc[e + f*x]^9)/(9*
a^2*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 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))^2 (c-c \sec (e+f x))^5} \, dx &=-\frac {\int \cot ^{10}(e+f x) (a+a \sec (e+f x))^3 \, dx}{a^5 c^5}\\ &=-\frac {\int \left (a^3 \cot ^{10}(e+f x)+3 a^3 \cot ^9(e+f x) \csc (e+f x)+3 a^3 \cot ^8(e+f x) \csc ^2(e+f x)+a^3 \cot ^7(e+f x) \csc ^3(e+f x)\right ) \, dx}{a^5 c^5}\\ &=-\frac {\int \cot ^{10}(e+f x) \, dx}{a^2 c^5}-\frac {\int \cot ^7(e+f x) \csc ^3(e+f x) \, dx}{a^2 c^5}-\frac {3 \int \cot ^9(e+f x) \csc (e+f x) \, dx}{a^2 c^5}-\frac {3 \int \cot ^8(e+f x) \csc ^2(e+f x) \, dx}{a^2 c^5}\\ &=\frac {\cot ^9(e+f x)}{9 a^2 c^5 f}+\frac {\int \cot ^8(e+f x) \, dx}{a^2 c^5}+\frac {\operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\csc (e+f x)\right )}{a^2 c^5 f}-\frac {3 \operatorname {Subst}\left (\int x^8 \, dx,x,-\cot (e+f x)\right )}{a^2 c^5 f}+\frac {3 \operatorname {Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (e+f x)\right )}{a^2 c^5 f}\\ &=-\frac {\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \cot ^9(e+f x)}{9 a^2 c^5 f}-\frac {\int \cot ^6(e+f x) \, dx}{a^2 c^5}+\frac {\operatorname {Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\csc (e+f x)\right )}{a^2 c^5 f}+\frac {3 \operatorname {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^2 c^5 f}\\ &=\frac {\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \cot ^9(e+f x)}{9 a^2 c^5 f}+\frac {3 \csc (e+f x)}{a^2 c^5 f}-\frac {13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac {21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac {15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \csc ^9(e+f x)}{9 a^2 c^5 f}+\frac {\int \cot ^4(e+f x) \, dx}{a^2 c^5}\\ &=-\frac {\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \cot ^9(e+f x)}{9 a^2 c^5 f}+\frac {3 \csc (e+f x)}{a^2 c^5 f}-\frac {13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac {21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac {15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \csc ^9(e+f x)}{9 a^2 c^5 f}-\frac {\int \cot ^2(e+f x) \, dx}{a^2 c^5}\\ &=\frac {\cot (e+f x)}{a^2 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \cot ^9(e+f x)}{9 a^2 c^5 f}+\frac {3 \csc (e+f x)}{a^2 c^5 f}-\frac {13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac {21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac {15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \csc ^9(e+f x)}{9 a^2 c^5 f}+\frac {\int 1 \, dx}{a^2 c^5}\\ &=\frac {x}{a^2 c^5}+\frac {\cot (e+f x)}{a^2 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \cot ^9(e+f x)}{9 a^2 c^5 f}+\frac {3 \csc (e+f x)}{a^2 c^5 f}-\frac {13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac {21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac {15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \csc ^9(e+f x)}{9 a^2 c^5 f}\\ \end {align*}

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Mathematica [A]  time = 1.20, size = 383, normalized size = 1.82 \[ \frac {\csc \left (\frac {e}{2}\right ) \sec \left (\frac {e}{2}\right ) \tan (e+f x) \sec ^6(e+f x) (-675036 \sin (e+f x)+506277 \sin (2 (e+f x))+37502 \sin (3 (e+f x))-225012 \sin (4 (e+f x))+112506 \sin (5 (e+f x))-18751 \sin (6 (e+f x))-431424 \sin (2 e+f x)+375552 \sin (e+2 f x)+201600 \sin (3 e+2 f x)-41248 \sin (2 e+3 f x)+84000 \sin (4 e+3 f x)-155712 \sin (3 e+4 f x)-100800 \sin (5 e+4 f x)+98016 \sin (4 e+5 f x)+30240 \sin (6 e+5 f x)-21376 \sin (5 e+6 f x)-181440 f x \cos (2 e+f x)-136080 f x \cos (e+2 f x)+136080 f x \cos (3 e+2 f x)-10080 f x \cos (2 e+3 f x)+10080 f x \cos (4 e+3 f x)+60480 f x \cos (3 e+4 f x)-60480 f x \cos (5 e+4 f x)-30240 f x \cos (4 e+5 f x)+30240 f x \cos (6 e+5 f x)+5040 f x \cos (5 e+6 f x)-5040 f x \cos (7 e+6 f x)+169344 \sin (e)-338112 \sin (f x)+181440 f x \cos (f x))}{645120 a^2 c^5 f (\sec (e+f x)-1)^5 (\sec (e+f x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5),x]

[Out]

(Csc[e/2]*Sec[e/2]*Sec[e + f*x]^6*(181440*f*x*Cos[f*x] - 181440*f*x*Cos[2*e + f*x] - 136080*f*x*Cos[e + 2*f*x]
+ 136080*f*x*Cos[3*e + 2*f*x] - 10080*f*x*Cos[2*e + 3*f*x] + 10080*f*x*Cos[4*e + 3*f*x] + 60480*f*x*Cos[3*e +
4*f*x] - 60480*f*x*Cos[5*e + 4*f*x] - 30240*f*x*Cos[4*e + 5*f*x] + 30240*f*x*Cos[6*e + 5*f*x] + 5040*f*x*Cos[
5*e + 6*f*x] - 5040*f*x*Cos[7*e + 6*f*x] + 169344*Sin[e] - 338112*Sin[f*x] - 675036*Sin[e + f*x] + 506277*Sin[
2*(e + f*x)] + 37502*Sin[3*(e + f*x)] - 225012*Sin[4*(e + f*x)] + 112506*Sin[5*(e + f*x)] - 18751*Sin[6*(e + f
*x)] - 431424*Sin[2*e + f*x] + 375552*Sin[e + 2*f*x] + 201600*Sin[3*e + 2*f*x] - 41248*Sin[2*e + 3*f*x] + 8400
0*Sin[4*e + 3*f*x] - 155712*Sin[3*e + 4*f*x] - 100800*Sin[5*e + 4*f*x] + 98016*Sin[4*e + 5*f*x] + 30240*Sin[6*
e + 5*f*x] - 21376*Sin[5*e + 6*f*x])*Tan[e + f*x])/(645120*a^2*c^5*f*(-1 + Sec[e + f*x])^5*(1 + Sec[e + f*x])^
2)

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fricas [A]  time = 0.42, size = 232, normalized size = 1.10 \[ \frac {668 \, \cos \left (f x + e\right )^{6} - 1059 \, \cos \left (f x + e\right )^{5} - 573 \, \cos \left (f x + e\right )^{4} + 1813 \, \cos \left (f x + e\right )^{3} - 393 \, \cos \left (f x + e\right )^{2} + 315 \, {\left (f x \cos \left (f x + e\right )^{5} - 3 \, f x \cos \left (f x + e\right )^{4} + 2 \, f x \cos \left (f x + e\right )^{3} + 2 \, f x \cos \left (f x + e\right )^{2} - 3 \, f x \cos \left (f x + e\right ) + f x\right )} \sin \left (f x + e\right ) - 789 \, \cos \left (f x + e\right ) + 368}{315 \, {\left (a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 3 \, a^{2} c^{5} f \cos \left (f x + e\right )^{4} + 2 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3} + 2 \, a^{2} c^{5} f \cos \left (f x + e\right )^{2} - 3 \, a^{2} c^{5} f \cos \left (f x + e\right ) + a^{2} c^{5} f\right )} \sin \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(f*x+e))^2/(c-c*sec(f*x+e))^5,x, algorithm="fricas")

[Out]

1/315*(668*cos(f*x + e)^6 - 1059*cos(f*x + e)^5 - 573*cos(f*x + e)^4 + 1813*cos(f*x + e)^3 - 393*cos(f*x + e)^
2 + 315*(f*x*cos(f*x + e)^5 - 3*f*x*cos(f*x + e)^4 + 2*f*x*cos(f*x + e)^3 + 2*f*x*cos(f*x + e)^2 - 3*f*x*cos(f
*x + e) + f*x)*sin(f*x + e) - 789*cos(f*x + e) + 368)/((a^2*c^5*f*cos(f*x + e)^5 - 3*a^2*c^5*f*cos(f*x + e)^4
+ 2*a^2*c^5*f*cos(f*x + e)^3 + 2*a^2*c^5*f*cos(f*x + e)^2 - 3*a^2*c^5*f*cos(f*x + e) + a^2*c^5*f)*sin(f*x + e)
)

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giac [A]  time = 0.48, size = 143, normalized size = 0.68 \[ \frac {\frac {20160 \, {\left (f x + e\right )}}{a^{2} c^{5}} + \frac {31185 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 6720 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 1827 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 360 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 35}{a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}} + \frac {105 \, {\left (a^{4} c^{10} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 24 \, a^{4} c^{10} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} c^{15}}}{20160 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(f*x+e))^2/(c-c*sec(f*x+e))^5,x, algorithm="giac")

[Out]

1/20160*(20160*(f*x + e)/(a^2*c^5) + (31185*tan(1/2*f*x + 1/2*e)^8 - 6720*tan(1/2*f*x + 1/2*e)^6 + 1827*tan(1/
2*f*x + 1/2*e)^4 - 360*tan(1/2*f*x + 1/2*e)^2 + 35)/(a^2*c^5*tan(1/2*f*x + 1/2*e)^9) + 105*(a^4*c^10*tan(1/2*f
*x + 1/2*e)^3 - 24*a^4*c^10*tan(1/2*f*x + 1/2*e))/(a^6*c^15))/f

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maple [A]  time = 0.94, size = 175, normalized size = 0.83 \[ \frac {\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )}{192 f \,a^{2} c^{5}}-\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{8 f \,a^{2} c^{5}}+\frac {1}{576 f \,a^{2} c^{5} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{9}}-\frac {1}{56 f \,a^{2} c^{5} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{7}}+\frac {29}{320 f \,a^{2} c^{5} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{5}}-\frac {1}{3 f \,a^{2} c^{5} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{3}}+\frac {99}{64 f \,a^{2} c^{5} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}+\frac {2 \arctan \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f \,a^{2} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sec(f*x+e))^2/(c-c*sec(f*x+e))^5,x)

[Out]

1/192/f/a^2/c^5*tan(1/2*e+1/2*f*x)^3-1/8/f/a^2/c^5*tan(1/2*e+1/2*f*x)+1/576/f/a^2/c^5/tan(1/2*e+1/2*f*x)^9-1/5
6/f/a^2/c^5/tan(1/2*e+1/2*f*x)^7+29/320/f/a^2/c^5/tan(1/2*e+1/2*f*x)^5-1/3/f/a^2/c^5/tan(1/2*e+1/2*f*x)^3+99/6
4/f/a^2/c^5/tan(1/2*e+1/2*f*x)+2/f/a^2/c^5*arctan(tan(1/2*e+1/2*f*x))

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maxima [A]  time = 0.43, size = 186, normalized size = 0.89 \[ -\frac {\frac {105 \, {\left (\frac {24 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} c^{5}} - \frac {40320 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2} c^{5}} + \frac {{\left (\frac {360 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {1827 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {6720 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {31185 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{a^{2} c^{5} \sin \left (f x + e\right )^{9}}}{20160 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(f*x+e))^2/(c-c*sec(f*x+e))^5,x, algorithm="maxima")

[Out]

-1/20160*(105*(24*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/(a^2*c^5) - 40320*arc
tan(sin(f*x + e)/(cos(f*x + e) + 1))/(a^2*c^5) + (360*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 1827*sin(f*x + e)^
4/(cos(f*x + e) + 1)^4 + 6720*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 31185*sin(f*x + e)^8/(cos(f*x + e) + 1)^8
- 35)*(cos(f*x + e) + 1)^9/(a^2*c^5*sin(f*x + e)^9))/f

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mupad [B]  time = 1.78, size = 209, normalized size = 1.00 \[ \frac {35\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+105\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-2520\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+31185\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-6720\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+1827\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-360\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+20160\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (e+f\,x\right )}{20160\,a^2\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + a/cos(e + f*x))^2*(c - c/cos(e + f*x))^5),x)

[Out]

(35*cos(e/2 + (f*x)/2)^12 + 105*sin(e/2 + (f*x)/2)^12 - 2520*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2)^10 + 3118
5*cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2)^8 - 6720*cos(e/2 + (f*x)/2)^6*sin(e/2 + (f*x)/2)^6 + 1827*cos(e/2 +
(f*x)/2)^8*sin(e/2 + (f*x)/2)^4 - 360*cos(e/2 + (f*x)/2)^10*sin(e/2 + (f*x)/2)^2 + 20160*cos(e/2 + (f*x)/2)^3*
sin(e/2 + (f*x)/2)^9*(e + f*x))/(20160*a^2*c^5*f*cos(e/2 + (f*x)/2)^3*sin(e/2 + (f*x)/2)^9)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {1}{\sec ^{7}{\left (e + f x \right )} - 3 \sec ^{6}{\left (e + f x \right )} + \sec ^{5}{\left (e + f x \right )} + 5 \sec ^{4}{\left (e + f x \right )} - 5 \sec ^{3}{\left (e + f x \right )} - \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} - 1}\, dx}{a^{2} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(f*x+e))**2/(c-c*sec(f*x+e))**5,x)

[Out]

-Integral(1/(sec(e + f*x)**7 - 3*sec(e + f*x)**6 + sec(e + f*x)**5 + 5*sec(e + f*x)**4 - 5*sec(e + f*x)**3 - s
ec(e + f*x)**2 + 3*sec(e + f*x) - 1), x)/(a**2*c**5)

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