3.20 \(\int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^6} \, dx\)

Optimal. Leaf size=163 \[ -\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{1155 f \left (c^2-c^2 \sec (e+f x)\right )^3}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{231 c^2 f (c-c \sec (e+f x))^4}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{33 c f (c-c \sec (e+f x))^5}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6} \]

[Out]

-((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(11*f*(c - c*Sec[e + f*x])^6) - ((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(
33*c*f*(c - c*Sec[e + f*x])^5) - (2*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(231*c^2*f*(c - c*Sec[e + f*x])^4) -
(2*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(1155*f*(c^2 - c^2*Sec[e + f*x])^3)

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Rubi [A]  time = 0.314677, antiderivative size = 163, 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} \[ -\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{1155 f \left (c^2-c^2 \sec (e+f x)\right )^3}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{231 c^2 f (c-c \sec (e+f x))^4}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{33 c f (c-c \sec (e+f x))^5}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{11 f (c-c \sec (e+f x))^6} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^6,x]

[Out]

-((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(11*f*(c - c*Sec[e + f*x])^6) - ((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(
33*c*f*(c - c*Sec[e + f*x])^5) - (2*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(231*c^2*f*(c - c*Sec[e + f*x])^4) -
(2*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(1155*f*(c^2 - c^2*Sec[e + f*x])^3)

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))^2}{(c-c \sec (e+f x))^6} \, dx &=-\frac{(a+a \sec (e+f x))^2 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}+\frac{3 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^5} \, dx}{11 c}\\ &=-\frac{(a+a \sec (e+f x))^2 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}-\frac{(a+a \sec (e+f x))^2 \tan (e+f x)}{33 c f (c-c \sec (e+f x))^5}+\frac{2 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx}{33 c^2}\\ &=-\frac{(a+a \sec (e+f x))^2 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}-\frac{(a+a \sec (e+f x))^2 \tan (e+f x)}{33 c f (c-c \sec (e+f x))^5}-\frac{2 (a+a \sec (e+f x))^2 \tan (e+f x)}{231 c^2 f (c-c \sec (e+f x))^4}+\frac{2 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^3} \, dx}{231 c^3}\\ &=-\frac{(a+a \sec (e+f x))^2 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}-\frac{(a+a \sec (e+f x))^2 \tan (e+f x)}{33 c f (c-c \sec (e+f x))^5}-\frac{2 (a+a \sec (e+f x))^2 \tan (e+f x)}{231 c^2 f (c-c \sec (e+f x))^4}-\frac{2 (a+a \sec (e+f x))^2 \tan (e+f x)}{1155 f \left (c^2-c^2 \sec (e+f x)\right )^3}\\ \end{align*}

Mathematica [A]  time = 0.703718, size = 167, normalized size = 1.02 \[ -\frac{a^2 \csc \left (\frac{e}{2}\right ) \left (37422 \sin \left (e+\frac{f x}{2}\right )-27060 \sin \left (e+\frac{3 f x}{2}\right )-23100 \sin \left (2 e+\frac{3 f x}{2}\right )+11220 \sin \left (2 e+\frac{5 f x}{2}\right )+13860 \sin \left (3 e+\frac{5 f x}{2}\right )-4895 \sin \left (3 e+\frac{7 f x}{2}\right )-3465 \sin \left (4 e+\frac{7 f x}{2}\right )+517 \sin \left (4 e+\frac{9 f x}{2}\right )+1155 \sin \left (5 e+\frac{9 f x}{2}\right )-152 \sin \left (5 e+\frac{11 f x}{2}\right )+32802 \sin \left (\frac{f x}{2}\right )\right ) \csc ^{11}\left (\frac{1}{2} (e+f x)\right )}{1182720 c^6 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^6,x]

[Out]

-(a^2*Csc[e/2]*Csc[(e + f*x)/2]^11*(32802*Sin[(f*x)/2] + 37422*Sin[e + (f*x)/2] - 27060*Sin[e + (3*f*x)/2] - 2
3100*Sin[2*e + (3*f*x)/2] + 11220*Sin[2*e + (5*f*x)/2] + 13860*Sin[3*e + (5*f*x)/2] - 4895*Sin[3*e + (7*f*x)/2
] - 3465*Sin[4*e + (7*f*x)/2] + 517*Sin[4*e + (9*f*x)/2] + 1155*Sin[5*e + (9*f*x)/2] - 152*Sin[5*e + (11*f*x)/
2]))/(1182720*c^6*f)

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Maple [A]  time = 0.115, size = 65, normalized size = 0.4 \begin{align*}{\frac{{a}^{2}}{8\,f{c}^{6}} \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)*(a+a*sec(f*x+e))^2/(c-c*sec(f*x+e))^6,x)

[Out]

1/8/f*a^2/c^6*(-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 [B]  time = 1.05086, size = 525, normalized size = 3.22 \begin{align*} \frac{\frac{a^{2}{\left (\frac{385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{990 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac{1386 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac{1155 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac{3465 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 315\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{c^{6} \sin \left (f x + e\right )^{11}} + \frac{6 \, a^{2}{\left (\frac{385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{330 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac{462 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac{1155 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac{1155 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 105\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{c^{6} \sin \left (f x + e\right )^{11}} + \frac{5 \, a^{2}{\left (\frac{385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{990 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{1386 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac{1155 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac{693 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 63\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{c^{6} \sin \left (f x + e\right )^{11}}}{110880 \, 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))^2/(c-c*sec(f*x+e))^6,x, algorithm="maxima")

[Out]

1/110880*(a^2*(385*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 990*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1386*sin(f*
x + e)^6/(cos(f*x + e) + 1)^6 - 1155*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 3465*sin(f*x + e)^10/(cos(f*x + e)
+ 1)^10 - 315)*(cos(f*x + e) + 1)^11/(c^6*sin(f*x + e)^11) + 6*a^2*(385*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 -
330*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1155*sin(f*x + e)^8/(cos(f
*x + e) + 1)^8 - 1155*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 105)*(cos(f*x + e) + 1)^11/(c^6*sin(f*x + e)^11)
+ 5*a^2*(385*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 990*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1386*sin(f*x + e
)^6/(cos(f*x + e) + 1)^6 - 1155*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 693*sin(f*x + e)^10/(cos(f*x + e) + 1)^1
0 - 63)*(cos(f*x + e) + 1)^11/(c^6*sin(f*x + e)^11))/f

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Fricas [A]  time = 0.462013, size = 414, normalized size = 2.54 \begin{align*} \frac{152 \, a^{2} \cos \left (f x + e\right )^{6} + 395 \, a^{2} \cos \left (f x + e\right )^{5} + 289 \, a^{2} \cos \left (f x + e\right )^{4} + 15 \, a^{2} \cos \left (f x + e\right )^{3} - 19 \, a^{2} \cos \left (f x + e\right )^{2} + 10 \, a^{2} \cos \left (f x + e\right ) - 2 \, a^{2}}{1155 \,{\left (c^{6} f \cos \left (f x + e\right )^{5} - 5 \, c^{6} f \cos \left (f x + e\right )^{4} + 10 \, c^{6} f \cos \left (f x + e\right )^{3} - 10 \, c^{6} f \cos \left (f x + e\right )^{2} + 5 \, c^{6} f \cos \left (f x + e\right ) - 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))^2/(c-c*sec(f*x+e))^6,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2} \left (\int \frac{\sec{\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{2 \sec ^{2}{\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{\sec ^{3}{\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} + 15 \sec ^{4}{\left (e + f x \right )} - 20 \sec ^{3}{\left (e + f x \right )} + 15 \sec ^{2}{\left (e + f x \right )} - 6 \sec{\left (e + f x \right )} + 1}\, dx\right )}{c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))**2/(c-c*sec(f*x+e))**6,x)

[Out]

a**2*(Integral(sec(e + f*x)/(sec(e + f*x)**6 - 6*sec(e + f*x)**5 + 15*sec(e + f*x)**4 - 20*sec(e + f*x)**3 + 1
5*sec(e + f*x)**2 - 6*sec(e + f*x) + 1), x) + Integral(2*sec(e + f*x)**2/(sec(e + f*x)**6 - 6*sec(e + f*x)**5
+ 15*sec(e + f*x)**4 - 20*sec(e + f*x)**3 + 15*sec(e + f*x)**2 - 6*sec(e + f*x) + 1), x) + Integral(sec(e + f*
x)**3/(sec(e + f*x)**6 - 6*sec(e + f*x)**5 + 15*sec(e + f*x)**4 - 20*sec(e + f*x)**3 + 15*sec(e + f*x)**2 - 6*
sec(e + f*x) + 1), x))/c**6

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Giac [A]  time = 1.32409, size = 104, normalized size = 0.64 \begin{align*} \frac{231 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 495 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 385 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 105 \, a^{2}}{9240 \, c^{6} 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)*(a+a*sec(f*x+e))^2/(c-c*sec(f*x+e))^6,x, algorithm="giac")

[Out]

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