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

Optimal. Leaf size=210 \[ \frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac{\cot (e+f x)}{a^3 c^5 f}+\frac{2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac{8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac{12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac{8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac{2 \csc (e+f x)}{a^3 c^5 f}+\frac{x}{a^3 c^5} \]

[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)

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Rubi [A]  time = 0.236406, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3904, 3886, 3473, 8, 2606, 194, 2607, 30} \[ \frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac{\cot (e+f x)}{a^3 c^5 f}+\frac{2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac{8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac{12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac{8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac{2 \csc (e+f x)}{a^3 c^5 f}+\frac{x}{a^3 c^5} \]

Antiderivative was successfully verified.

[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 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])

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 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 8

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

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 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 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 30

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

Rubi steps

\begin{align*} \int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx &=-\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{\operatorname{Subst}\left (\int x^8 \, dx,x,-\cot (e+f x)\right )}{a^3 c^5 f}+\frac{2 \operatorname{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 \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^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 [B]  time = 1.72925, size = 441, normalized size = 2.1 \[ \frac{\csc \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}\right ) \tan (e+f x) \sec ^7(e+f x) (-1152405 \sin (e+f x)+512180 \sin (2 (e+f x))+486571 \sin (3 (e+f x))-409744 \sin (4 (e+f x))-25609 \sin (5 (e+f x))+102436 \sin (6 (e+f x))-25609 \sin (7 (e+f x))-825216 \sin (2 e+f x)+622976 \sin (e+2 f x)+142464 \sin (3 e+2 f x)+297088 \sin (2 e+3 f x)+430080 \sin (4 e+3 f x)-424192 \sin (3 e+4 f x)-188160 \sin (5 e+4 f x)+2048 \sin (4 e+5 f x)-40320 \sin (6 e+5 f x)+112768 \sin (5 e+6 f x)+40320 \sin (7 e+6 f x)-38272 \sin (6 e+7 f x)-453600 f x \cos (2 e+f x)-201600 f x \cos (e+2 f x)+201600 f x \cos (3 e+2 f x)-191520 f x \cos (2 e+3 f x)+191520 f x \cos (4 e+3 f x)+161280 f x \cos (3 e+4 f x)-161280 f x \cos (5 e+4 f x)+10080 f x \cos (4 e+5 f x)-10080 f x \cos (6 e+5 f x)-40320 f x \cos (5 e+6 f x)+40320 f x \cos (7 e+6 f x)+10080 f x \cos (6 e+7 f x)-10080 f x \cos (8 e+7 f x)+259584 \sin (e)-897024 \sin (f x)+453600 f x \cos (f x))}{2580480 a^3 c^5 f (\sec (e+f x)-1)^5 (\sec (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[e/2]*Sec[e/2]*Sec[e + f*x]^7*(453600*f*x*Cos[f*x] - 453600*f*x*Cos[2*e + f*x] - 201600*f*x*Cos[e + 2*f*x]
+ 201600*f*x*Cos[3*e + 2*f*x] - 191520*f*x*Cos[2*e + 3*f*x] + 191520*f*x*Cos[4*e + 3*f*x] + 161280*f*x*Cos[3*
e + 4*f*x] - 161280*f*x*Cos[5*e + 4*f*x] + 10080*f*x*Cos[4*e + 5*f*x] - 10080*f*x*Cos[6*e + 5*f*x] - 40320*f*x
*Cos[5*e + 6*f*x] + 40320*f*x*Cos[7*e + 6*f*x] + 10080*f*x*Cos[6*e + 7*f*x] - 10080*f*x*Cos[8*e + 7*f*x] + 259
584*Sin[e] - 897024*Sin[f*x] - 1152405*Sin[e + f*x] + 512180*Sin[2*(e + f*x)] + 486571*Sin[3*(e + f*x)] - 4097
44*Sin[4*(e + f*x)] - 25609*Sin[5*(e + f*x)] + 102436*Sin[6*(e + f*x)] - 25609*Sin[7*(e + f*x)] - 825216*Sin[2
*e + f*x] + 622976*Sin[e + 2*f*x] + 142464*Sin[3*e + 2*f*x] + 297088*Sin[2*e + 3*f*x] + 430080*Sin[4*e + 3*f*x
] - 424192*Sin[3*e + 4*f*x] - 188160*Sin[5*e + 4*f*x] + 2048*Sin[4*e + 5*f*x] - 40320*Sin[6*e + 5*f*x] + 11276
8*Sin[5*e + 6*f*x] + 40320*Sin[7*e + 6*f*x] - 38272*Sin[6*e + 7*f*x])*Tan[e + f*x])/(2580480*a^3*c^5*f*(-1 + S
ec[e + f*x])^5*(1 + Sec[e + f*x])^3)

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Maple [A]  time = 0.075, size = 197, normalized size = 0.9 \begin{align*} -{\frac{1}{640\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}+{\frac{3}{128\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{37}{128\,f{a}^{3}{c}^{5}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{3}{c}^{5}}}+{\frac{1}{1152\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-9}}-{\frac{9}{896\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}}+{\frac{37}{640\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}-{\frac{31}{128\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}}+{\frac{163}{128\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-1}} \end{align*}

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))^5,x)

[Out]

-1/640/f/a^3/c^5*tan(1/2*f*x+1/2*e)^5+3/128/f/a^3/c^5*tan(1/2*f*x+1/2*e)^3-37/128/f/a^3/c^5*tan(1/2*f*x+1/2*e)
+2/f/a^3/c^5*arctan(tan(1/2*f*x+1/2*e))+1/1152/f/a^3/c^5/tan(1/2*f*x+1/2*e)^9-9/896/f/a^3/c^5/tan(1/2*f*x+1/2*
e)^7+37/640/f/a^3/c^5/tan(1/2*f*x+1/2*e)^5-31/128/f/a^3/c^5/tan(1/2*f*x+1/2*e)^3+163/128/f/a^3/c^5/tan(1/2*f*x
+1/2*e)

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Maxima [A]  time = 1.54884, size = 277, normalized size = 1.32 \begin{align*} -\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} \end{align*}

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))^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

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Fricas [A]  time = 1.16328, size = 687, normalized size = 3.27 \begin{align*} \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 )} \end{align*}

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))^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))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.43953, size = 220, normalized size = 1.05 \begin{align*} \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} \end{align*}

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))^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