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

Optimal. Leaf size=69 \[ -\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac{\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}+\frac{x}{a^2 c} \]

[Out]

x/(a^2*c) + (Cot[e + f*x]*(3 - 2*Sec[e + f*x]))/(3*a^2*c*f) - (Cot[e + f*x]^3*(1 - Sec[e + f*x]))/(3*a^2*c*f)

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Rubi [A]  time = 0.113739, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3904, 3882, 8} \[ -\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac{\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}+\frac{x}{a^2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(a^2*c) + (Cot[e + f*x]*(3 - 2*Sec[e + f*x]))/(3*a^2*c*f) - (Cot[e + f*x]^3*(1 - Sec[e + f*x]))/(3*a^2*c*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 3882

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))} \, dx &=\frac{\int \cot ^4(e+f x) (c-c \sec (e+f x)) \, dx}{a^2 c^2}\\ &=-\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac{\int \cot ^2(e+f x) (-3 c+2 c \sec (e+f x)) \, dx}{3 a^2 c^2}\\ &=\frac{\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}-\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac{\int 3 c \, dx}{3 a^2 c^2}\\ &=\frac{x}{a^2 c}+\frac{\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}-\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}\\ \end{align*}

Mathematica [A]  time = 0.543341, size = 135, normalized size = 1.96 \[ \frac{\csc \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}\right ) \csc \left (\frac{1}{2} (e+f x)\right ) \sec ^3\left (\frac{1}{2} (e+f x)\right ) (10 \sin (e+f x)+5 \sin (2 (e+f x))-6 \sin (2 e+f x)-8 \sin (e+2 f x)-6 f x \cos (2 e+f x)+3 f x \cos (e+2 f x)-3 f x \cos (3 e+2 f x)-10 \sin (f x)+6 f x \cos (f x))}{96 a^2 c f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[e/2]*Csc[(e + f*x)/2]*Sec[e/2]*Sec[(e + f*x)/2]^3*(6*f*x*Cos[f*x] - 6*f*x*Cos[2*e + f*x] + 3*f*x*Cos[e +
2*f*x] - 3*f*x*Cos[3*e + 2*f*x] - 10*Sin[f*x] + 10*Sin[e + f*x] + 5*Sin[2*(e + f*x)] - 6*Sin[2*e + f*x] - 8*Si
n[e + 2*f*x]))/(96*a^2*c*f)

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Maple [A]  time = 0.056, size = 87, normalized size = 1.3 \begin{align*}{\frac{1}{12\,f{a}^{2}c} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{1}{f{a}^{2}c}\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}}+{\frac{1}{4\,f{a}^{2}c} \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)),x)

[Out]

1/12/f/a^2/c*tan(1/2*f*x+1/2*e)^3-1/f/a^2/c*tan(1/2*f*x+1/2*e)+2/f/a^2/c*arctan(tan(1/2*f*x+1/2*e))+1/4/f/a^2/
c/tan(1/2*f*x+1/2*e)

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Maxima [A]  time = 1.51585, size = 138, normalized size = 2. \begin{align*} -\frac{\frac{\frac{12 \, \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}}}{a^{2} c} - \frac{24 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2} c} - \frac{3 \,{\left (\cos \left (f x + e\right ) + 1\right )}}{a^{2} c \sin \left (f x + e\right )}}{12 \, 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)),x, algorithm="maxima")

[Out]

-1/12*((12*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/(a^2*c) - 24*arctan(sin(f*x
+ e)/(cos(f*x + e) + 1))/(a^2*c) - 3*(cos(f*x + e) + 1)/(a^2*c*sin(f*x + e)))/f

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Fricas [A]  time = 1.04229, size = 180, normalized size = 2.61 \begin{align*} \frac{4 \, \cos \left (f x + e\right )^{2} + 3 \,{\left (f x \cos \left (f x + e\right ) + f x\right )} \sin \left (f x + e\right ) + \cos \left (f x + e\right ) - 2}{3 \,{\left (a^{2} c f \cos \left (f x + e\right ) + a^{2} c 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)),x, algorithm="fricas")

[Out]

1/3*(4*cos(f*x + e)^2 + 3*(f*x*cos(f*x + e) + f*x)*sin(f*x + e) + cos(f*x + e) - 2)/((a^2*c*f*cos(f*x + e) + a
^2*c*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 ^{3}{\left (e + f x \right )} + \sec ^{2}{\left (e + f x \right )} - \sec{\left (e + f x \right )} - 1}\, dx}{a^{2} c} \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)),x)

[Out]

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

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Giac [A]  time = 1.34305, size = 115, normalized size = 1.67 \begin{align*} \frac{\frac{12 \,{\left (f x + e\right )}}{a^{2} c} + \frac{3}{a^{2} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )} + \frac{a^{4} c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 12 \, a^{4} c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a^{6} c^{3}}}{12 \, 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)),x, algorithm="giac")

[Out]

1/12*(12*(f*x + e)/(a^2*c) + 3/(a^2*c*tan(1/2*f*x + 1/2*e)) + (a^4*c^2*tan(1/2*f*x + 1/2*e)^3 - 12*a^4*c^2*tan
(1/2*f*x + 1/2*e))/(a^6*c^3))/f