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

Optimal. Leaf size=120 \[ \frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac{2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac{\csc ^7(e+f x)}{a^3 c^5 f}+\frac{9 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac{5 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac{\csc (e+f x)}{a^3 c^5 f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.204163, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3958, 2606, 194, 2607, 30, 270} \[ \frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac{2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac{\csc ^7(e+f x)}{a^3 c^5 f}+\frac{9 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac{5 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac{\csc (e+f x)}{a^3 c^5 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3958

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
)^(n_.), x_Symbol] :> Dist[(-(a*c))^m, Int[ExpandTrig[csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n
- m), x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m,
n] && GeQ[n - m, 0] && GtQ[m*n, 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 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]

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]

Rubi steps

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

Mathematica [B]  time = 1.56247, size = 257, normalized size = 2.14 \[ -\frac{\csc (e) (76455 \sin (e+f x)-33980 \sin (2 (e+f x))-32281 \sin (3 (e+f x))+27184 \sin (4 (e+f x))+1699 \sin (5 (e+f x))-6796 \sin (6 (e+f x))+1699 \sin (7 (e+f x))+22656 \sin (2 e+f x)-17216 \sin (e+2 f x)+4416 \sin (3 e+2 f x)+3200 \sin (2 e+3 f x)-15360 \sin (4 e+3 f x)+12160 \sin (3 e+4 f x)-1920 \sin (5 e+4 f x)-5120 \sin (4 e+5 f x)+5760 \sin (6 e+5 f x)+320 \sin (5 e+6 f x)-2880 \sin (7 e+6 f x)+640 \sin (6 e+7 f x)-33024 \sin (e)+6144 \sin (f x)) \tan (e+f x) \sec ^7(e+f x)}{184320 a^3 c^5 f (\sec (e+f x)-1)^5 (\sec (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csc[e]*Sec[e + f*x]^7*(-33024*Sin[e] + 6144*Sin[f*x] + 76455*Sin[e + f*x] - 33980*Sin[2*(e + f*x)] - 32281*S
in[3*(e + f*x)] + 27184*Sin[4*(e + f*x)] + 1699*Sin[5*(e + f*x)] - 6796*Sin[6*(e + f*x)] + 1699*Sin[7*(e + f*x
)] + 22656*Sin[2*e + f*x] - 17216*Sin[e + 2*f*x] + 4416*Sin[3*e + 2*f*x] + 3200*Sin[2*e + 3*f*x] - 15360*Sin[4
*e + 3*f*x] + 12160*Sin[3*e + 4*f*x] - 1920*Sin[5*e + 4*f*x] - 5120*Sin[4*e + 5*f*x] + 5760*Sin[6*e + 5*f*x] +
320*Sin[5*e + 6*f*x] - 2880*Sin[7*e + 6*f*x] + 640*Sin[6*e + 7*f*x])*Tan[e + f*x])/(184320*a^3*c^5*f*(-1 + Se
c[e + f*x])^5*(1 + Sec[e + f*x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.065, size = 115, normalized size = 1. \begin{align*}{\frac{1}{128\,f{a}^{3}{c}^{5}} \left ({\frac{1}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}-{\frac{7}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+21\,\tan \left ( 1/2\,fx+e/2 \right ) -{\frac{35}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}}+35\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{-1}+{\frac{21}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}- \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}+{\frac{1}{9} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-9}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128/f/a^3/c^5*(1/5*tan(1/2*f*x+1/2*e)^5-7/3*tan(1/2*f*x+1/2*e)^3+21*tan(1/2*f*x+1/2*e)-35/3/tan(1/2*f*x+1/2*
e)^3+35/tan(1/2*f*x+1/2*e)+21/5/tan(1/2*f*x+1/2*e)^5-1/tan(1/2*f*x+1/2*e)^7+1/9/tan(1/2*f*x+1/2*e)^9)

________________________________________________________________________________________

Maxima [A]  time = 1.03045, size = 244, normalized size = 2.03 \begin{align*} \frac{\frac{3 \,{\left (\frac{315 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{35 \, \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^{5}} - \frac{{\left (\frac{45 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{189 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{525 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac{1575 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 5\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{a^{3} c^{5} \sin \left (f x + e\right )^{9}}}{5760 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5760*(3*(315*sin(f*x + e)/(cos(f*x + e) + 1) - 35*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(co
s(f*x + e) + 1)^5)/(a^3*c^5) - (45*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 189*sin(f*x + e)^4/(cos(f*x + e) + 1)
^4 + 525*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1575*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5)*(cos(f*x + e) + 1
)^9/(a^3*c^5*sin(f*x + e)^9))/f

________________________________________________________________________________________

Fricas [A]  time = 0.476116, size = 462, normalized size = 3.85 \begin{align*} \frac{10 \, \cos \left (f x + e\right )^{7} + 25 \, \cos \left (f x + e\right )^{6} - 60 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{4} + 80 \, \cos \left (f x + e\right )^{3} - 24 \, \cos \left (f x + e\right )^{2} - 32 \, \cos \left (f x + e\right ) + 16}{45 \,{\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(sec(f*x+e)/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^5,x, algorithm="fricas")

[Out]

1/45*(10*cos(f*x + e)^7 + 25*cos(f*x + e)^6 - 60*cos(f*x + e)^5 - 10*cos(f*x + e)^4 + 80*cos(f*x + e)^3 - 24*c
os(f*x + e)^2 - 32*cos(f*x + e) + 16)/((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*cos(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(-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(sec(f*x+e)/(a+a*sec(f*x+e))**3/(c-c*sec(f*x+e))**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.40693, size = 203, normalized size = 1.69 \begin{align*} \frac{\frac{1575 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 525 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 189 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 45 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 5}{a^{3} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9}} + \frac{3 \,{\left (3 \, a^{12} c^{20} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 35 \, a^{12} c^{20} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 315 \, a^{12} c^{20} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{15} c^{25}}}{5760 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5760*((1575*tan(1/2*f*x + 1/2*e)^8 - 525*tan(1/2*f*x + 1/2*e)^6 + 189*tan(1/2*f*x + 1/2*e)^4 - 45*tan(1/2*f*
x + 1/2*e)^2 + 5)/(a^3*c^5*tan(1/2*f*x + 1/2*e)^9) + 3*(3*a^12*c^20*tan(1/2*f*x + 1/2*e)^5 - 35*a^12*c^20*tan(
1/2*f*x + 1/2*e)^3 + 315*a^12*c^20*tan(1/2*f*x + 1/2*e))/(a^15*c^25))/f