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

Optimal. Leaf size=36 \[ -\frac {\tan (e+f x) (a \sec (e+f x)+a)}{3 f (c-c \sec (e+f x))^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {3950} \[ -\frac {\tan (e+f x) (a \sec (e+f x)+a)}{3 f (c-c \sec (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.25, size = 50, normalized size = 1.39 \[ \frac {a \csc \left (\frac {e}{2}\right ) \left (\sin \left (e+\frac {3 f x}{2}\right )-3 \sin \left (e+\frac {f x}{2}\right )\right ) \csc ^3\left (\frac {1}{2} (e+f x)\right )}{12 c^2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Csc[e/2]*Csc[(e + f*x)/2]^3*(-3*Sin[e + (f*x)/2] + Sin[e + (3*f*x)/2]))/(12*c^2*f)

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 51, normalized size = 1.42 \[ \frac {a \cos \left (f x + e\right )^{2} + 2 \, a \cos \left (f x + e\right ) + a}{3 \, {\left (c^{2} f \cos \left (f x + e\right ) - c^{2} f\right )} \sin \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))/(c-c*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.58, size = 21, normalized size = 0.58 \[ -\frac {a}{3 \, c^{2} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))/(c-c*sec(f*x+e))^2,x, algorithm="giac")

[Out]

-1/3*a/(c^2*f*tan(1/2*f*x + 1/2*e)^3)

________________________________________________________________________________________

maple [A]  time = 0.79, size = 21, normalized size = 0.58 \[ -\frac {a}{3 f \,c^{2} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(a+a*sec(f*x+e))/(c-c*sec(f*x+e))^2,x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.60, size = 97, normalized size = 2.69 \[ -\frac {\frac {a {\left (\frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{3}}{c^{2} \sin \left (f x + e\right )^{3}} - \frac {a {\left (\frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{3}}{c^{2} \sin \left (f x + e\right )^{3}}}{6 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))/(c-c*sec(f*x+e))^2,x, algorithm="maxima")

[Out]

-1/6*(a*(3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)*(cos(f*x + e) + 1)^3/(c^2*sin(f*x + e)^3) - a*(3*sin(f*x +
e)^2/(cos(f*x + e) + 1)^2 - 1)*(cos(f*x + e) + 1)^3/(c^2*sin(f*x + e)^3))/f

________________________________________________________________________________________

mupad [B]  time = 2.06, size = 20, normalized size = 0.56 \[ -\frac {a\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,c^2\,f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cos(e + f*x))/(cos(e + f*x)*(c - c/cos(e + f*x))^2),x)

[Out]

-(a*cot(e/2 + (f*x)/2)^3)/(3*c^2*f)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} - 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} - 2 \sec {\left (e + f x \right )} + 1}\, dx\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________