3.29 \(\int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4} \, dx\)

Optimal. Leaf size=166 \[ -\frac{2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac{\cot ^5(e+f x)}{5 a^2 c^4 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^4 f}+\frac{\cot (e+f x)}{a^2 c^4 f}-\frac{2 \csc ^7(e+f x)}{7 a^2 c^4 f}+\frac{6 \csc ^5(e+f x)}{5 a^2 c^4 f}-\frac{2 \csc ^3(e+f x)}{a^2 c^4 f}+\frac{2 \csc (e+f x)}{a^2 c^4 f}+\frac{x}{a^2 c^4} \]

[Out]

x/(a^2*c^4) + Cot[e + f*x]/(a^2*c^4*f) - Cot[e + f*x]^3/(3*a^2*c^4*f) + Cot[e + f*x]^5/(5*a^2*c^4*f) - (2*Cot[
e + f*x]^7)/(7*a^2*c^4*f) + (2*Csc[e + f*x])/(a^2*c^4*f) - (2*Csc[e + f*x]^3)/(a^2*c^4*f) + (6*Csc[e + f*x]^5)
/(5*a^2*c^4*f) - (2*Csc[e + f*x]^7)/(7*a^2*c^4*f)

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Rubi [A]  time = 0.211247, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 13, 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} \[ -\frac{2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac{\cot ^5(e+f x)}{5 a^2 c^4 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^4 f}+\frac{\cot (e+f x)}{a^2 c^4 f}-\frac{2 \csc ^7(e+f x)}{7 a^2 c^4 f}+\frac{6 \csc ^5(e+f x)}{5 a^2 c^4 f}-\frac{2 \csc ^3(e+f x)}{a^2 c^4 f}+\frac{2 \csc (e+f x)}{a^2 c^4 f}+\frac{x}{a^2 c^4} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4),x]

[Out]

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

Mathematica [A]  time = 1.2566, size = 315, normalized size = 1.9 \[ \frac{\csc \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}\right ) \csc ^7\left (\frac{1}{2} (e+f x)\right ) \sec ^3\left (\frac{1}{2} (e+f x)\right ) (-16002 \sin (e+f x)+9144 \sin (2 (e+f x))+3429 \sin (3 (e+f x))-4572 \sin (4 (e+f x))+1143 \sin (5 (e+f x))-11760 \sin (2 e+f x)+8864 \sin (e+2 f x)+3360 \sin (3 e+2 f x)+2064 \sin (2 e+3 f x)+2520 \sin (4 e+3 f x)-4432 \sin (3 e+4 f x)-1680 \sin (5 e+4 f x)+1528 \sin (4 e+5 f x)-5880 f x \cos (2 e+f x)-3360 f x \cos (e+2 f x)+3360 f x \cos (3 e+2 f x)-1260 f x \cos (2 e+3 f x)+1260 f x \cos (4 e+3 f x)+1680 f x \cos (3 e+4 f x)-1680 f x \cos (5 e+4 f x)-420 f x \cos (4 e+5 f x)+420 f x \cos (6 e+5 f x)+4032 \sin (e)-9632 \sin (f x)+5880 f x \cos (f x))}{860160 a^2 c^4 f} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4),x]

[Out]

(Csc[e/2]*Csc[(e + f*x)/2]^7*Sec[e/2]*Sec[(e + f*x)/2]^3*(5880*f*x*Cos[f*x] - 5880*f*x*Cos[2*e + f*x] - 3360*f
*x*Cos[e + 2*f*x] + 3360*f*x*Cos[3*e + 2*f*x] - 1260*f*x*Cos[2*e + 3*f*x] + 1260*f*x*Cos[4*e + 3*f*x] + 1680*f
*x*Cos[3*e + 4*f*x] - 1680*f*x*Cos[5*e + 4*f*x] - 420*f*x*Cos[4*e + 5*f*x] + 420*f*x*Cos[6*e + 5*f*x] + 4032*S
in[e] - 9632*Sin[f*x] - 16002*Sin[e + f*x] + 9144*Sin[2*(e + f*x)] + 3429*Sin[3*(e + f*x)] - 4572*Sin[4*(e + f
*x)] + 1143*Sin[5*(e + f*x)] - 11760*Sin[2*e + f*x] + 8864*Sin[e + 2*f*x] + 3360*Sin[3*e + 2*f*x] + 2064*Sin[2
*e + 3*f*x] + 2520*Sin[4*e + 3*f*x] - 4432*Sin[3*e + 4*f*x] - 1680*Sin[5*e + 4*f*x] + 1528*Sin[4*e + 5*f*x]))/
(860160*a^2*c^4*f)

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Maple [A]  time = 0.071, size = 153, normalized size = 0.9 \begin{align*}{\frac{1}{96\,f{a}^{2}{c}^{4}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{7}{32\,f{a}^{2}{c}^{4}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{2}{c}^{4}}}-{\frac{1}{224\,f{a}^{2}{c}^{4}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}}+{\frac{7}{160\,f{a}^{2}{c}^{4}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}-{\frac{11}{48\,f{a}^{2}{c}^{4}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}}+{\frac{21}{16\,f{a}^{2}{c}^{4}} \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))^2/(c-c*sec(f*x+e))^4,x)

[Out]

1/96/f/a^2/c^4*tan(1/2*f*x+1/2*e)^3-7/32/f/a^2/c^4*tan(1/2*f*x+1/2*e)+2/f/a^2/c^4*arctan(tan(1/2*f*x+1/2*e))-1
/224/f/a^2/c^4/tan(1/2*f*x+1/2*e)^7+7/160/f/a^2/c^4/tan(1/2*f*x+1/2*e)^5-11/48/f/a^2/c^4/tan(1/2*f*x+1/2*e)^3+
21/16/f/a^2/c^4/tan(1/2*f*x+1/2*e)

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Maxima [A]  time = 1.52891, size = 225, normalized size = 1.36 \begin{align*} -\frac{\frac{35 \,{\left (\frac{21 \, \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^{4}} - \frac{6720 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2} c^{4}} - \frac{{\left (\frac{147 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{770 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{4410 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 15\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{a^{2} c^{4} \sin \left (f x + e\right )^{7}}}{3360 \, f} \end{align*}

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))^4,x, algorithm="maxima")

[Out]

-1/3360*(35*(21*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/(a^2*c^4) - 6720*arctan
(sin(f*x + e)/(cos(f*x + e) + 1))/(a^2*c^4) - (147*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 770*sin(f*x + e)^4/(c
os(f*x + e) + 1)^4 + 4410*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 15)*(cos(f*x + e) + 1)^7/(a^2*c^4*sin(f*x + e)
^7))/f

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Fricas [A]  time = 1.08481, size = 424, normalized size = 2.55 \begin{align*} \frac{191 \, \cos \left (f x + e\right )^{5} - 172 \, \cos \left (f x + e\right )^{4} - 253 \, \cos \left (f x + e\right )^{3} + 258 \, \cos \left (f x + e\right )^{2} + 105 \,{\left (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 ) - f x\right )} \sin \left (f x + e\right ) + 87 \, \cos \left (f x + e\right ) - 96}{105 \,{\left (a^{2} c^{4} f \cos \left (f x + e\right )^{4} - 2 \, a^{2} c^{4} f \cos \left (f x + e\right )^{3} + 2 \, a^{2} c^{4} f \cos \left (f x + e\right ) - a^{2} c^{4} 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))^2/(c-c*sec(f*x+e))^4,x, algorithm="fricas")

[Out]

1/105*(191*cos(f*x + e)^5 - 172*cos(f*x + e)^4 - 253*cos(f*x + e)^3 + 258*cos(f*x + e)^2 + 105*(f*x*cos(f*x +
e)^4 - 2*f*x*cos(f*x + e)^3 + 2*f*x*cos(f*x + e) - f*x)*sin(f*x + e) + 87*cos(f*x + e) - 96)/((a^2*c^4*f*cos(f
*x + e)^4 - 2*a^2*c^4*f*cos(f*x + e)^3 + 2*a^2*c^4*f*cos(f*x + e) - a^2*c^4*f)*sin(f*x + e))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\sec ^{6}{\left (e + f x \right )} - 2 \sec ^{5}{\left (e + f x \right )} - \sec ^{4}{\left (e + f x \right )} + 4 \sec ^{3}{\left (e + f x \right )} - \sec ^{2}{\left (e + f x \right )} - 2 \sec{\left (e + f x \right )} + 1}\, dx}{a^{2} c^{4}} \end{align*}

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))**4,x)

[Out]

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

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Giac [A]  time = 1.43285, size = 174, normalized size = 1.05 \begin{align*} \frac{\frac{3360 \,{\left (f x + e\right )}}{a^{2} c^{4}} + \frac{4410 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 770 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 147 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 15}{a^{2} c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7}} + \frac{35 \,{\left (a^{4} c^{8} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 21 \, a^{4} c^{8} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{6} c^{12}}}{3360 \, f} \end{align*}

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))^4,x, algorithm="giac")

[Out]

1/3360*(3360*(f*x + e)/(a^2*c^4) + (4410*tan(1/2*f*x + 1/2*e)^6 - 770*tan(1/2*f*x + 1/2*e)^4 + 147*tan(1/2*f*x
+ 1/2*e)^2 - 15)/(a^2*c^4*tan(1/2*f*x + 1/2*e)^7) + 35*(a^4*c^8*tan(1/2*f*x + 1/2*e)^3 - 21*a^4*c^8*tan(1/2*f
*x + 1/2*e))/(a^6*c^12))/f