3.286 \(\int \frac{\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx\)

Optimal. Leaf size=89 \[ -\frac{\cot ^{11}\left (\frac{1}{2} (e+f x)\right )}{88 c^8 f}+\frac{\cot ^9\left (\frac{1}{2} (e+f x)\right )}{24 c^8 f}-\frac{3 \cot ^7\left (\frac{1}{2} (e+f x)\right )}{56 c^8 f}+\frac{\cot ^5\left (\frac{1}{2} (e+f x)\right )}{40 c^8 f} \]

[Out]

Cot[(e + f*x)/2]^5/(40*c^8*f) - (3*Cot[(e + f*x)/2]^7)/(56*c^8*f) + Cot[(e + f*x)/2]^9/(24*c^8*f) - Cot[(e + f
*x)/2]^11/(88*c^8*f)

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Rubi [A]  time = 0.335691, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {12, 270} \[ -\frac{\cot ^{11}\left (\frac{1}{2} (e+f x)\right )}{88 c^8 f}+\frac{\cot ^9\left (\frac{1}{2} (e+f x)\right )}{24 c^8 f}-\frac{3 \cot ^7\left (\frac{1}{2} (e+f x)\right )}{56 c^8 f}+\frac{\cot ^5\left (\frac{1}{2} (e+f x)\right )}{40 c^8 f} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[e + f*x]*Tan[e + f*x]^4)/(c - c*Sec[e + f*x])^8,x]

[Out]

Cot[(e + f*x)/2]^5/(40*c^8*f) - (3*Cot[(e + f*x)/2]^7)/(56*c^8*f) + Cot[(e + f*x)/2]^9/(24*c^8*f) - Cot[(e + f
*x)/2]^11/(88*c^8*f)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{16 c^8 x^{12}} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{x^{12}} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{8 c^8 f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^{12}}-\frac{3}{x^{10}}+\frac{3}{x^8}-\frac{1}{x^6}\right ) \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{8 c^8 f}\\ &=\frac{\cot ^5\left (\frac{1}{2} (e+f x)\right )}{40 c^8 f}-\frac{3 \cot ^7\left (\frac{1}{2} (e+f x)\right )}{56 c^8 f}+\frac{\cot ^9\left (\frac{1}{2} (e+f x)\right )}{24 c^8 f}-\frac{\cot ^{11}\left (\frac{1}{2} (e+f x)\right )}{88 c^8 f}\\ \end{align*}

Mathematica [A]  time = 1.07155, size = 175, normalized size = 1.97 \[ -\frac{\csc \left (\frac{e}{2}\right ) \left (486024 \sin \left (e+\frac{f x}{2}\right )-351450 \sin \left (e+\frac{3 f x}{2}\right )-299970 \sin \left (2 e+\frac{3 f x}{2}\right )+145695 \sin \left (2 e+\frac{5 f x}{2}\right )+180015 \sin \left (3 e+\frac{5 f x}{2}\right )-63580 \sin \left (3 e+\frac{7 f x}{2}\right )-44990 \sin \left (4 e+\frac{7 f x}{2}\right )+6710 \sin \left (4 e+\frac{9 f x}{2}\right )+15004 \sin \left (5 e+\frac{9 f x}{2}\right )-1975 \sin \left (5 e+\frac{11 f x}{2}\right )+\sin \left (6 e+\frac{11 f x}{2}\right )+425964 \sin \left (\frac{f x}{2}\right )\right ) \csc ^{11}\left (\frac{1}{2} (e+f x)\right )}{15375360 c^8 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[e + f*x]*Tan[e + f*x]^4)/(c - c*Sec[e + f*x])^8,x]

[Out]

-(Csc[e/2]*Csc[(e + f*x)/2]^11*(425964*Sin[(f*x)/2] + 486024*Sin[e + (f*x)/2] - 351450*Sin[e + (3*f*x)/2] - 29
9970*Sin[2*e + (3*f*x)/2] + 145695*Sin[2*e + (5*f*x)/2] + 180015*Sin[3*e + (5*f*x)/2] - 63580*Sin[3*e + (7*f*x
)/2] - 44990*Sin[4*e + (7*f*x)/2] + 6710*Sin[4*e + (9*f*x)/2] + 15004*Sin[5*e + (9*f*x)/2] - 1975*Sin[5*e + (1
1*f*x)/2] + Sin[6*e + (11*f*x)/2]))/(15375360*c^8*f)

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Maple [A]  time = 0.135, size = 62, normalized size = 0.7 \begin{align*}{\frac{1}{8\,f{c}^{8}} \left ( -{\frac{1}{11} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-11}}+{\frac{1}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}-{\frac{3}{7} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}}+{\frac{1}{3} \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)*tan(f*x+e)^4/(c-c*sec(f*x+e))^8,x)

[Out]

1/8/f/c^8*(-1/11/tan(1/2*f*x+1/2*e)^11+1/5/tan(1/2*f*x+1/2*e)^5-3/7/tan(1/2*f*x+1/2*e)^7+1/3/tan(1/2*f*x+1/2*e
)^9)

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Maxima [A]  time = 0.985534, size = 119, normalized size = 1.34 \begin{align*} \frac{{\left (\frac{385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{495 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{231 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 105\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{9240 \, c^{8} f \sin \left (f x + e\right )^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*tan(f*x+e)^4/(c-c*sec(f*x+e))^8,x, algorithm="maxima")

[Out]

1/9240*(385*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 495*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 231*sin(f*x + e)^6
/(cos(f*x + e) + 1)^6 - 105)*(cos(f*x + e) + 1)^11/(c^8*f*sin(f*x + e)^11)

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Fricas [A]  time = 0.523058, size = 377, normalized size = 4.24 \begin{align*} \frac{152 \, \cos \left (f x + e\right )^{6} + 395 \, \cos \left (f x + e\right )^{5} + 289 \, \cos \left (f x + e\right )^{4} + 15 \, \cos \left (f x + e\right )^{3} - 19 \, \cos \left (f x + e\right )^{2} + 10 \, \cos \left (f x + e\right ) - 2}{1155 \,{\left (c^{8} f \cos \left (f x + e\right )^{5} - 5 \, c^{8} f \cos \left (f x + e\right )^{4} + 10 \, c^{8} f \cos \left (f x + e\right )^{3} - 10 \, c^{8} f \cos \left (f x + e\right )^{2} + 5 \, c^{8} f \cos \left (f x + e\right ) - c^{8} 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)*tan(f*x+e)^4/(c-c*sec(f*x+e))^8,x, algorithm="fricas")

[Out]

1/1155*(152*cos(f*x + e)^6 + 395*cos(f*x + e)^5 + 289*cos(f*x + e)^4 + 15*cos(f*x + e)^3 - 19*cos(f*x + e)^2 +
10*cos(f*x + e) - 2)/((c^8*f*cos(f*x + e)^5 - 5*c^8*f*cos(f*x + e)^4 + 10*c^8*f*cos(f*x + e)^3 - 10*c^8*f*cos
(f*x + e)^2 + 5*c^8*f*cos(f*x + e) - c^8*f)*sin(f*x + e))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tan ^{4}{\left (e + f x \right )} \sec{\left (e + f x \right )}}{\sec ^{8}{\left (e + f x \right )} - 8 \sec ^{7}{\left (e + f x \right )} + 28 \sec ^{6}{\left (e + f x \right )} - 56 \sec ^{5}{\left (e + f x \right )} + 70 \sec ^{4}{\left (e + f x \right )} - 56 \sec ^{3}{\left (e + f x \right )} + 28 \sec ^{2}{\left (e + f x \right )} - 8 \sec{\left (e + f x \right )} + 1}\, dx}{c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*tan(f*x+e)**4/(c-c*sec(f*x+e))**8,x)

[Out]

Integral(tan(e + f*x)**4*sec(e + f*x)/(sec(e + f*x)**8 - 8*sec(e + f*x)**7 + 28*sec(e + f*x)**6 - 56*sec(e + f
*x)**5 + 70*sec(e + f*x)**4 - 56*sec(e + f*x)**3 + 28*sec(e + f*x)**2 - 8*sec(e + f*x) + 1), x)/c**8

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Giac [A]  time = 2.26592, size = 86, normalized size = 0.97 \begin{align*} \frac{231 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 495 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 385 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 105}{9240 \, c^{8} f \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*tan(f*x+e)^4/(c-c*sec(f*x+e))^8,x, algorithm="giac")

[Out]

1/9240*(231*tan(1/2*f*x + 1/2*e)^6 - 495*tan(1/2*f*x + 1/2*e)^4 + 385*tan(1/2*f*x + 1/2*e)^2 - 105)/(c^8*f*tan
(1/2*f*x + 1/2*e)^11)