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

Optimal. Leaf size=162 \[ -\frac{4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}-\frac{\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{17 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{4 \csc ^7(e+f x)}{a^3 c^6 f}+\frac{22 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{8 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{\csc (e+f x)}{a^3 c^6 f} \]

[Out]

-Cot[e + f*x]^9/(9*a^3*c^6*f) - (4*Cot[e + f*x]^11)/(11*a^3*c^6*f) + Csc[e + f*x]/(a^3*c^6*f) - (8*Csc[e + f*x
]^3)/(3*a^3*c^6*f) + (22*Csc[e + f*x]^5)/(5*a^3*c^6*f) - (4*Csc[e + f*x]^7)/(a^3*c^6*f) + (17*Csc[e + f*x]^9)/
(9*a^3*c^6*f) - (4*Csc[e + f*x]^11)/(11*a^3*c^6*f)

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Rubi [A]  time = 0.259866, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {3958, 2606, 194, 2607, 30, 270, 14} \[ -\frac{4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}-\frac{\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{17 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{4 \csc ^7(e+f x)}{a^3 c^6 f}+\frac{22 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{8 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{\csc (e+f x)}{a^3 c^6 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cot[e + f*x]^9/(9*a^3*c^6*f) - (4*Cot[e + f*x]^11)/(11*a^3*c^6*f) + Csc[e + f*x]/(a^3*c^6*f) - (8*Csc[e + f*x
]^3)/(3*a^3*c^6*f) + (22*Csc[e + f*x]^5)/(5*a^3*c^6*f) - (4*Csc[e + f*x]^7)/(a^3*c^6*f) + (17*Csc[e + f*x]^9)/
(9*a^3*c^6*f) - (4*Csc[e + f*x]^11)/(11*a^3*c^6*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]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx &=\frac{\int \left (a^3 \cot ^{11}(e+f x) \csc (e+f x)+3 a^3 \cot ^{10}(e+f x) \csc ^2(e+f x)+3 a^3 \cot ^9(e+f x) \csc ^3(e+f x)+a^3 \cot ^8(e+f x) \csc ^4(e+f x)\right ) \, dx}{a^6 c^6}\\ &=\frac{\int \cot ^{11}(e+f x) \csc (e+f x) \, dx}{a^3 c^6}+\frac{\int \cot ^8(e+f x) \csc ^4(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{3 \int \cot ^9(e+f x) \csc ^3(e+f x) \, dx}{a^3 c^6}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right )^5 \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f}+\frac{\operatorname{Subst}\left (\int x^8 \left (1+x^2\right ) \, dx,x,-\cot (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 x^2 \left (-1+x^2\right )^4 \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f}\\ &=-\frac{3 \cot ^{11}(e+f x)}{11 a^3 c^6 f}-\frac{\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{\operatorname{Subst}\left (\int \left (x^8+x^{10}\right ) \, dx,x,-\cot (e+f x)\right )}{a^3 c^6 f}-\frac{3 \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{\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{\csc (e+f x)}{a^3 c^6 f}-\frac{8 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{22 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{4 \csc ^7(e+f x)}{a^3 c^6 f}+\frac{17 \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 [A]  time = 2.15696, size = 289, normalized size = 1.78 \[ \frac{\csc (e) (-3440690 \sin (e+f x)+2064414 \sin (2 (e+f x))+1063486 \sin (3 (e+f x))-1563950 \sin (4 (e+f x))+312790 \sin (5 (e+f x))+312790 \sin (6 (e+f x))-187674 \sin (7 (e+f x))+31279 \sin (8 (e+f x))-1499520 \sin (2 e+f x)+1051776 \sin (e+2 f x)+4224 \sin (3 e+2 f x)-85376 \sin (2 e+3 f x)+629376 \sin (4 e+3 f x)-483200 \sin (3 e+4 f x)-316800 \sin (5 e+4 f x)+392320 \sin (4 e+5 f x)-232320 \sin (6 e+5 f x)-30080 \sin (5 e+6 f x)+190080 \sin (7 e+6 f x)-32640 \sin (6 e+7 f x)-63360 \sin (8 e+7 f x)+16000 \sin (7 e+8 f x)+1119360 \sin (e)-260480 \sin (f x)) \tan (e+f x) \sec ^8(e+f x)}{8110080 a^3 c^6 f (\sec (e+f x)-1)^6 (\sec (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[e]*Sec[e + f*x]^8*(1119360*Sin[e] - 260480*Sin[f*x] - 3440690*Sin[e + f*x] + 2064414*Sin[2*(e + f*x)] + 1
063486*Sin[3*(e + f*x)] - 1563950*Sin[4*(e + f*x)] + 312790*Sin[5*(e + f*x)] + 312790*Sin[6*(e + f*x)] - 18767
4*Sin[7*(e + f*x)] + 31279*Sin[8*(e + f*x)] - 1499520*Sin[2*e + f*x] + 1051776*Sin[e + 2*f*x] + 4224*Sin[3*e +
2*f*x] - 85376*Sin[2*e + 3*f*x] + 629376*Sin[4*e + 3*f*x] - 483200*Sin[3*e + 4*f*x] - 316800*Sin[5*e + 4*f*x]
+ 392320*Sin[4*e + 5*f*x] - 232320*Sin[6*e + 5*f*x] - 30080*Sin[5*e + 6*f*x] + 190080*Sin[7*e + 6*f*x] - 3264
0*Sin[6*e + 7*f*x] - 63360*Sin[8*e + 7*f*x] + 16000*Sin[7*e + 8*f*x])*Tan[e + f*x])/(8110080*a^3*c^6*f*(-1 + S
ec[e + f*x])^6*(1 + Sec[e + f*x])^3)

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Maple [A]  time = 0.068, size = 128, normalized size = 0.8 \begin{align*}{\frac{1}{256\,f{a}^{3}{c}^{6}} \left ({\frac{1}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}-{\frac{8}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+28\,\tan \left ( 1/2\,fx+e/2 \right ) -{\frac{70}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}}-{\frac{1}{11} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-11}}+56\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{-1}+{\frac{56}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}-4\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{-7}+{\frac{8}{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))^6,x)

[Out]

1/256/f/a^3/c^6*(1/5*tan(1/2*f*x+1/2*e)^5-8/3*tan(1/2*f*x+1/2*e)^3+28*tan(1/2*f*x+1/2*e)-70/3/tan(1/2*f*x+1/2*
e)^3-1/11/tan(1/2*f*x+1/2*e)^11+56/tan(1/2*f*x+1/2*e)+56/5/tan(1/2*f*x+1/2*e)^5-4/tan(1/2*f*x+1/2*e)^7+8/9/tan
(1/2*f*x+1/2*e)^9)

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Maxima [A]  time = 1.01738, size = 270, normalized size = 1.67 \begin{align*} \frac{\frac{33 \,{\left (\frac{420 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{40 \, \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{{\left (\frac{440 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{1980 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{5544 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac{11550 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac{27720 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 45\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{a^{3} c^{6} \sin \left (f x + e\right )^{11}}}{126720 \, 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))^6,x, algorithm="maxima")

[Out]

1/126720*(33*(420*sin(f*x + e)/(cos(f*x + e) + 1) - 40*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) + (440*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 1980*sin(f*x + e)^4/(cos(f*x + e)
+ 1)^4 + 5544*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 11550*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 27720*sin(f*x
+ e)^10/(cos(f*x + e) + 1)^10 - 45)*(cos(f*x + e) + 1)^11/(a^3*c^6*sin(f*x + e)^11))/f

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Fricas [A]  time = 0.491307, size = 540, normalized size = 3.33 \begin{align*} \frac{125 \, \cos \left (f x + e\right )^{8} + 120 \, \cos \left (f x + e\right )^{7} - 680 \, \cos \left (f x + e\right )^{6} + 400 \, \cos \left (f x + e\right )^{5} + 720 \, \cos \left (f x + e\right )^{4} - 832 \, \cos \left (f x + e\right )^{3} - 64 \, \cos \left (f x + e\right )^{2} + 384 \, \cos \left (f x + e\right ) - 128}{495 \,{\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 )} \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))^6,x, algorithm="fricas")

[Out]

1/495*(125*cos(f*x + e)^8 + 120*cos(f*x + e)^7 - 680*cos(f*x + e)^6 + 400*cos(f*x + e)^5 + 720*cos(f*x + e)^4
- 832*cos(f*x + e)^3 - 64*cos(f*x + e)^2 + 384*cos(f*x + e) - 128)/((a^3*c^6*f*cos(f*x + e)^7 - 3*a^3*c^6*f*co
s(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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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))**6,x)

[Out]

Timed out

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Giac [A]  time = 1.26152, size = 221, normalized size = 1.36 \begin{align*} \frac{\frac{27720 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 11550 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} + 5544 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 1980 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 440 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 45}{a^{3} c^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11}} + \frac{33 \,{\left (3 \, a^{12} c^{24} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 40 \, a^{12} c^{24} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 420 \, a^{12} c^{24} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{15} c^{30}}}{126720 \, 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))^6,x, algorithm="giac")

[Out]

1/126720*((27720*tan(1/2*f*x + 1/2*e)^10 - 11550*tan(1/2*f*x + 1/2*e)^8 + 5544*tan(1/2*f*x + 1/2*e)^6 - 1980*t
an(1/2*f*x + 1/2*e)^4 + 440*tan(1/2*f*x + 1/2*e)^2 - 45)/(a^3*c^6*tan(1/2*f*x + 1/2*e)^11) + 33*(3*a^12*c^24*t
an(1/2*f*x + 1/2*e)^5 - 40*a^12*c^24*tan(1/2*f*x + 1/2*e)^3 + 420*a^12*c^24*tan(1/2*f*x + 1/2*e))/(a^15*c^30))
/f