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

Optimal. Leaf size=42 \[ -\frac{a \tanh ^{-1}(\sin (e+f x))}{c f}-\frac{2 a \tan (e+f x)}{f (c-c \sec (e+f x))} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0536762, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3957, 3770} \[ -\frac{a \tanh ^{-1}(\sin (e+f x))}{c f}-\frac{2 a \tan (e+f x)}{f (c-c \sec (e+f x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3957

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0672002, size = 77, normalized size = 1.83 \[ -\frac{a \left (-\frac{2 \cot \left (\frac{1}{2} (e+f x)\right )}{f}-\frac{\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}{f}+\frac{\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}{f}\right )}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*((-2*Cot[(e + f*x)/2])/f - Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]/f + Log[Cos[(e + f*x)/2] + Sin[(e + f
*x)/2]]/f))/c)

________________________________________________________________________________________

Maple [A]  time = 0.063, size = 63, normalized size = 1.5 \begin{align*} -{\frac{a}{fc}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) }+{\frac{a}{fc}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) }+2\,{\frac{a}{fc\tan \left ( 1/2\,fx+e/2 \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))/(c-c*sec(f*x+e)),x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 0.995552, size = 136, normalized size = 3.24 \begin{align*} -\frac{a{\left (\frac{\log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c} - \frac{\log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c} - \frac{\cos \left (f x + e\right ) + 1}{c \sin \left (f x + e\right )}\right )} - \frac{a{\left (\cos \left (f x + e\right ) + 1\right )}}{c \sin \left (f x + e\right )}}{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))/(c-c*sec(f*x+e)),x, algorithm="maxima")

[Out]

-(a*(log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/c - log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/c - (cos(f*x + e) +
1)/(c*sin(f*x + e))) - a*(cos(f*x + e) + 1)/(c*sin(f*x + e)))/f

________________________________________________________________________________________

Fricas [A]  time = 0.471912, size = 174, normalized size = 4.14 \begin{align*} -\frac{a \log \left (\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - a \log \left (-\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - 4 \, a \cos \left (f x + e\right ) - 4 \, a}{2 \, c f \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))/(c-c*sec(f*x+e)),x, algorithm="fricas")

[Out]

-1/2*(a*log(sin(f*x + e) + 1)*sin(f*x + e) - a*log(-sin(f*x + e) + 1)*sin(f*x + e) - 4*a*cos(f*x + e) - 4*a)/(
c*f*sin(f*x + e))

________________________________________________________________________________________

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

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)),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.38624, size = 85, normalized size = 2.02 \begin{align*} -\frac{\frac{a \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{c} - \frac{a \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{c} - \frac{2 \, a}{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}}{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))/(c-c*sec(f*x+e)),x, algorithm="giac")

[Out]

-(a*log(abs(tan(1/2*f*x + 1/2*e) + 1))/c - a*log(abs(tan(1/2*f*x + 1/2*e) - 1))/c - 2*a/(c*tan(1/2*f*x + 1/2*e
)))/f