3.33 \(\int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^7} \, dx\)

Optimal. Leaf size=162 \[ -\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{3003 c^3 f (c-c \sec (e+f x))^4}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{429 c^2 f (c-c \sec (e+f x))^5}-\frac{3 \tan (e+f x) (a \sec (e+f x)+a)^3}{143 c f (c-c \sec (e+f x))^6}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7} \]

[Out]

-((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(13*f*(c - c*Sec[e + f*x])^7) - (3*(a + a*Sec[e + f*x])^3*Tan[e + f*x])
/(143*c*f*(c - c*Sec[e + f*x])^6) - (2*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(429*c^2*f*(c - c*Sec[e + f*x])^5)
- (2*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(3003*c^3*f*(c - c*Sec[e + f*x])^4)

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Rubi [A]  time = 0.316715, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {3951, 3950} \[ -\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{3003 c^3 f (c-c \sec (e+f x))^4}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{429 c^2 f (c-c \sec (e+f x))^5}-\frac{3 \tan (e+f x) (a \sec (e+f x)+a)^3}{143 c f (c-c \sec (e+f x))^6}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^7,x]

[Out]

-((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(13*f*(c - c*Sec[e + f*x])^7) - (3*(a + a*Sec[e + f*x])^3*Tan[e + f*x])
/(143*c*f*(c - c*Sec[e + f*x])^6) - (2*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(429*c^2*f*(c - c*Sec[e + f*x])^5)
- (2*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(3003*c^3*f*(c - c*Sec[e + f*x])^4)

Rule 3951

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))
^(n_.), x_Symbol] :> Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/(a*f*(2*m + 1)), x] +
Dist[(m + n + 1)/(a*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n, x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[m + n + 1, 0] && NeQ[2
*m + 1, 0] &&  !LtQ[n, 0] &&  !(IGtQ[n + 1/2, 0] && LtQ[n + 1/2, -(m + n)])

Rule 3950

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))
^(n_.), x_Symbol] :> Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/(a*f*(2*m + 1)), x] /
; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[2*m
+ 1, 0]

Rubi steps

\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^7} \, dx &=-\frac{(a+a \sec (e+f x))^3 \tan (e+f x)}{13 f (c-c \sec (e+f x))^7}+\frac{3 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^6} \, dx}{13 c}\\ &=-\frac{(a+a \sec (e+f x))^3 \tan (e+f x)}{13 f (c-c \sec (e+f x))^7}-\frac{3 (a+a \sec (e+f x))^3 \tan (e+f x)}{143 c f (c-c \sec (e+f x))^6}+\frac{6 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^5} \, dx}{143 c^2}\\ &=-\frac{(a+a \sec (e+f x))^3 \tan (e+f x)}{13 f (c-c \sec (e+f x))^7}-\frac{3 (a+a \sec (e+f x))^3 \tan (e+f x)}{143 c f (c-c \sec (e+f x))^6}-\frac{2 (a+a \sec (e+f x))^3 \tan (e+f x)}{429 c^2 f (c-c \sec (e+f x))^5}+\frac{2 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx}{429 c^3}\\ &=-\frac{(a+a \sec (e+f x))^3 \tan (e+f x)}{13 f (c-c \sec (e+f x))^7}-\frac{3 (a+a \sec (e+f x))^3 \tan (e+f x)}{143 c f (c-c \sec (e+f x))^6}-\frac{2 (a+a \sec (e+f x))^3 \tan (e+f x)}{429 c^2 f (c-c \sec (e+f x))^5}-\frac{2 (a+a \sec (e+f x))^3 \tan (e+f x)}{3003 c^3 f (c-c \sec (e+f x))^4}\\ \end{align*}

Mathematica [A]  time = 0.575218, size = 193, normalized size = 1.19 \[ -\frac{a^3 \csc \left (\frac{e}{2}\right ) \left (246246 \sin \left (e+\frac{f x}{2}\right )-182754 \sin \left (e+\frac{3 f x}{2}\right )-216216 \sin \left (2 e+\frac{3 f x}{2}\right )+122551 \sin \left (2 e+\frac{5 f x}{2}\right )+99099 \sin \left (3 e+\frac{5 f x}{2}\right )-37609 \sin \left (3 e+\frac{7 f x}{2}\right )-51051 \sin \left (4 e+\frac{7 f x}{2}\right )+15171 \sin \left (4 e+\frac{9 f x}{2}\right )+9009 \sin \left (5 e+\frac{9 f x}{2}\right )-1027 \sin \left (5 e+\frac{11 f x}{2}\right )-3003 \sin \left (6 e+\frac{11 f x}{2}\right )+310 \sin \left (6 e+\frac{13 f x}{2}\right )+285714 \sin \left (\frac{f x}{2}\right )\right ) \csc ^{13}\left (\frac{1}{2} (e+f x)\right )}{12300288 c^7 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^7,x]

[Out]

-(a^3*Csc[e/2]*Csc[(e + f*x)/2]^13*(285714*Sin[(f*x)/2] + 246246*Sin[e + (f*x)/2] - 182754*Sin[e + (3*f*x)/2]
- 216216*Sin[2*e + (3*f*x)/2] + 122551*Sin[2*e + (5*f*x)/2] + 99099*Sin[3*e + (5*f*x)/2] - 37609*Sin[3*e + (7*
f*x)/2] - 51051*Sin[4*e + (7*f*x)/2] + 15171*Sin[4*e + (9*f*x)/2] + 9009*Sin[5*e + (9*f*x)/2] - 1027*Sin[5*e +
(11*f*x)/2] - 3003*Sin[6*e + (11*f*x)/2] + 310*Sin[6*e + (13*f*x)/2]))/(12300288*c^7*f)

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Maple [A]  time = 0.135, size = 65, normalized size = 0.4 \begin{align*}{\frac{{a}^{3}}{8\,f{c}^{7}} \left ({\frac{1}{13} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-13}}-{\frac{3}{11} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-11}}-{\frac{1}{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)*(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^7,x)

[Out]

1/8/f*a^3/c^7*(1/13/tan(1/2*f*x+1/2*e)^13-3/11/tan(1/2*f*x+1/2*e)^11-1/7/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 [B]  time = 1.1346, size = 698, normalized size = 4.31 \begin{align*} \text{result too large to display} \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))^7,x, algorithm="maxima")

[Out]

-1/960960*(a^3*(8190*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5005*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 25740*si
n(f*x + e)^6/(cos(f*x + e) + 1)^6 + 9009*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 30030*sin(f*x + e)^10/(cos(f*x
+ e) + 1)^10 - 45045*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 3465)*(cos(f*x + e) + 1)^13/(c^7*sin(f*x + e)^13)
+ 5*a^3*(1638*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 5005*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 8580*sin(f*x +
e)^6/(cos(f*x + e) + 1)^6 - 9009*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 6006*sin(f*x + e)^10/(cos(f*x + e) + 1
)^10 - 3003*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 231)*(cos(f*x + e) + 1)^13/(c^7*sin(f*x + e)^13) + 35*a^3*
(468*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 715*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1287*sin(f*x + e)^8/(cos(
f*x + e) + 1)^8 - 1716*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 1287*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 99
)*(cos(f*x + e) + 1)^13/(c^7*sin(f*x + e)^13) + 77*a^3*(65*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 117*sin(f*x +
e)^8/(cos(f*x + e) + 1)^8 + 195*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 15)*(cos(f*x + e) + 1)^13/(c^7*sin(f*
x + e)^13))/f

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Fricas [A]  time = 0.474635, size = 486, normalized size = 3. \begin{align*} \frac{310 \, a^{3} \cos \left (f x + e\right )^{7} + 1143 \, a^{3} \cos \left (f x + e\right )^{6} + 1492 \, a^{3} \cos \left (f x + e\right )^{5} + 736 \, a^{3} \cos \left (f x + e\right )^{4} + 34 \, a^{3} \cos \left (f x + e\right )^{3} - 29 \, a^{3} \cos \left (f x + e\right )^{2} + 12 \, a^{3} \cos \left (f x + e\right ) - 2 \, a^{3}}{3003 \,{\left (c^{7} f \cos \left (f x + e\right )^{6} - 6 \, c^{7} f \cos \left (f x + e\right )^{5} + 15 \, c^{7} f \cos \left (f x + e\right )^{4} - 20 \, c^{7} f \cos \left (f x + e\right )^{3} + 15 \, c^{7} f \cos \left (f x + e\right )^{2} - 6 \, 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)*(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^7,x, algorithm="fricas")

[Out]

1/3003*(310*a^3*cos(f*x + e)^7 + 1143*a^3*cos(f*x + e)^6 + 1492*a^3*cos(f*x + e)^5 + 736*a^3*cos(f*x + e)^4 +
34*a^3*cos(f*x + e)^3 - 29*a^3*cos(f*x + e)^2 + 12*a^3*cos(f*x + e) - 2*a^3)/((c^7*f*cos(f*x + e)^6 - 6*c^7*f*
cos(f*x + e)^5 + 15*c^7*f*cos(f*x + e)^4 - 20*c^7*f*cos(f*x + e)^3 + 15*c^7*f*cos(f*x + e)^2 - 6*c^7*f*cos(f*x
+ e) + c^7*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))**7,x)

[Out]

Timed out

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Giac [A]  time = 1.35317, size = 104, normalized size = 0.64 \begin{align*} -\frac{429 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 1001 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 819 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 231 \, a^{3}}{24024 \, c^{7} f \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{13}} \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))^7,x, algorithm="giac")

[Out]

-1/24024*(429*a^3*tan(1/2*f*x + 1/2*e)^6 - 1001*a^3*tan(1/2*f*x + 1/2*e)^4 + 819*a^3*tan(1/2*f*x + 1/2*e)^2 -
231*a^3)/(c^7*f*tan(1/2*f*x + 1/2*e)^13)