3.285 \(\int \frac{\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^7} \, dx\)

Optimal. Leaf size=67 \[ \frac{\cot ^9\left (\frac{1}{2} (e+f x)\right )}{36 c^7 f}-\frac{\cot ^7\left (\frac{1}{2} (e+f x)\right )}{14 c^7 f}+\frac{\cot ^5\left (\frac{1}{2} (e+f x)\right )}{20 c^7 f} \]

[Out]

Cot[(e + f*x)/2]^5/(20*c^7*f) - Cot[(e + f*x)/2]^7/(14*c^7*f) + Cot[(e + f*x)/2]^9/(36*c^7*f)

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Rubi [A]  time = 0.298053, antiderivative size = 67, 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} \[ \frac{\cot ^9\left (\frac{1}{2} (e+f x)\right )}{36 c^7 f}-\frac{\cot ^7\left (\frac{1}{2} (e+f x)\right )}{14 c^7 f}+\frac{\cot ^5\left (\frac{1}{2} (e+f x)\right )}{20 c^7 f} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[e + f*x]*Tan[e + f*x]^4)/(c - c*Sec[e + f*x])^7,x]

[Out]

Cot[(e + f*x)/2]^5/(20*c^7*f) - Cot[(e + f*x)/2]^7/(14*c^7*f) + Cot[(e + f*x)/2]^9/(36*c^7*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))^7} \, dx &=\frac{2 \operatorname{Subst}\left (\int -\frac{\left (1-x^2\right )^2}{8 c^7 x^{10}} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{f}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^{10}} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{4 c^7 f}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^{10}}-\frac{2}{x^8}+\frac{1}{x^6}\right ) \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{4 c^7 f}\\ &=\frac{\cot ^5\left (\frac{1}{2} (e+f x)\right )}{20 c^7 f}-\frac{\cot ^7\left (\frac{1}{2} (e+f x)\right )}{14 c^7 f}+\frac{\cot ^9\left (\frac{1}{2} (e+f x)\right )}{36 c^7 f}\\ \end{align*}

Mathematica [B]  time = 0.914968, size = 151, normalized size = 2.25 \[ \frac{\csc \left (\frac{e}{2}\right ) \left (-718830 \sin \left (e+\frac{f x}{2}\right )+467208 \sin \left (e+\frac{3 f x}{2}\right )+659400 \sin \left (2 e+\frac{3 f x}{2}\right )-303192 \sin \left (2 e+\frac{5 f x}{2}\right )-179640 \sin \left (3 e+\frac{5 f x}{2}\right )+30753 \sin \left (3 e+\frac{7 f x}{2}\right )+89955 \sin \left (4 e+\frac{7 f x}{2}\right )-13427 \sin \left (4 e+\frac{9 f x}{2}\right )+15 \sin \left (5 e+\frac{9 f x}{2}\right )-971082 \sin \left (\frac{f x}{2}\right )\right ) \csc ^9\left (\frac{1}{2} (e+f x)\right )}{23063040 c^7 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[e + f*x]*Tan[e + f*x]^4)/(c - c*Sec[e + f*x])^7,x]

[Out]

(Csc[e/2]*Csc[(e + f*x)/2]^9*(-971082*Sin[(f*x)/2] - 718830*Sin[e + (f*x)/2] + 467208*Sin[e + (3*f*x)/2] + 659
400*Sin[2*e + (3*f*x)/2] - 303192*Sin[2*e + (5*f*x)/2] - 179640*Sin[3*e + (5*f*x)/2] + 30753*Sin[3*e + (7*f*x)
/2] + 89955*Sin[4*e + (7*f*x)/2] - 13427*Sin[4*e + (9*f*x)/2] + 15*Sin[5*e + (9*f*x)/2]))/(23063040*c^7*f)

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Maple [A]  time = 0.12, size = 49, normalized size = 0.7 \begin{align*}{\frac{1}{4\,f{c}^{7}} \left ({\frac{1}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}-{\frac{2}{7} \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)*tan(f*x+e)^4/(c-c*sec(f*x+e))^7,x)

[Out]

1/4/f/c^7*(1/5/tan(1/2*f*x+1/2*e)^5-2/7/tan(1/2*f*x+1/2*e)^7+1/9/tan(1/2*f*x+1/2*e)^9)

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Maxima [A]  time = 1.02752, size = 92, normalized size = 1.37 \begin{align*} -\frac{{\left (\frac{90 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{63 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 35\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{1260 \, c^{7} f \sin \left (f x + e\right )^{9}} \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))^7,x, algorithm="maxima")

[Out]

-1/1260*(90*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 63*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 35)*(cos(f*x + e) +
1)^9/(c^7*f*sin(f*x + e)^9)

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Fricas [B]  time = 0.664516, size = 309, normalized size = 4.61 \begin{align*} \frac{47 \, \cos \left (f x + e\right )^{5} + 127 \, \cos \left (f x + e\right )^{4} + 101 \, \cos \left (f x + e\right )^{3} + 11 \, \cos \left (f x + e\right )^{2} - 8 \, \cos \left (f x + e\right ) + 2}{315 \,{\left (c^{7} f \cos \left (f x + e\right )^{4} - 4 \, c^{7} f \cos \left (f x + e\right )^{3} + 6 \, c^{7} f \cos \left (f x + e\right )^{2} - 4 \, c^{7} f \cos \left (f x + e\right ) + c^{7} 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))^7,x, algorithm="fricas")

[Out]

1/315*(47*cos(f*x + e)^5 + 127*cos(f*x + e)^4 + 101*cos(f*x + e)^3 + 11*cos(f*x + e)^2 - 8*cos(f*x + e) + 2)/(
(c^7*f*cos(f*x + e)^4 - 4*c^7*f*cos(f*x + e)^3 + 6*c^7*f*cos(f*x + e)^2 - 4*c^7*f*cos(f*x + e) + c^7*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 ^{7}{\left (e + f x \right )} - 7 \sec ^{6}{\left (e + f x \right )} + 21 \sec ^{5}{\left (e + f x \right )} - 35 \sec ^{4}{\left (e + f x \right )} + 35 \sec ^{3}{\left (e + f x \right )} - 21 \sec ^{2}{\left (e + f x \right )} + 7 \sec{\left (e + f x \right )} - 1}\, dx}{c^{7}} \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))**7,x)

[Out]

-Integral(tan(e + f*x)**4*sec(e + f*x)/(sec(e + f*x)**7 - 7*sec(e + f*x)**6 + 21*sec(e + f*x)**5 - 35*sec(e +
f*x)**4 + 35*sec(e + f*x)**3 - 21*sec(e + f*x)**2 + 7*sec(e + f*x) - 1), x)/c**7

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Giac [A]  time = 2.18048, size = 68, normalized size = 1.01 \begin{align*} \frac{63 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 90 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 35}{1260 \, c^{7} f \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9}} \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))^7,x, algorithm="giac")

[Out]

1/1260*(63*tan(1/2*f*x + 1/2*e)^4 - 90*tan(1/2*f*x + 1/2*e)^2 + 35)/(c^7*f*tan(1/2*f*x + 1/2*e)^9)