3.13 \(\int \frac{\sec (x)}{a+b \csc (x)} \, dx\)

Optimal. Leaf size=54 \[ -\frac{b \log (a \sin (x)+b)}{a^2-b^2}-\frac{\log (1-\sin (x))}{2 (a+b)}+\frac{\log (\sin (x)+1)}{2 (a-b)} \]

[Out]

-Log[1 - Sin[x]]/(2*(a + b)) + Log[1 + Sin[x]]/(2*(a - b)) - (b*Log[b + a*Sin[x]])/(a^2 - b^2)

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Rubi [A]  time = 0.0842033, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3872, 2721, 801} \[ -\frac{b \log (a \sin (x)+b)}{a^2-b^2}-\frac{\log (1-\sin (x))}{2 (a+b)}+\frac{\log (\sin (x)+1)}{2 (a-b)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]/(a + b*Csc[x]),x]

[Out]

-Log[1 - Sin[x]]/(2*(a + b)) + Log[1 + Sin[x]]/(2*(a - b)) - (b*Log[b + a*Sin[x]])/(a^2 - b^2)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2721

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rubi steps

\begin{align*} \int \frac{\sec (x)}{a+b \csc (x)} \, dx &=\int \frac{\tan (x)}{b+a \sin (x)} \, dx\\ &=\operatorname{Subst}\left (\int \frac{x}{(b+x) \left (a^2-x^2\right )} \, dx,x,a \sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{2 (a+b) (a-x)}+\frac{1}{2 (a-b) (a+x)}+\frac{b}{(-a+b) (a+b) (b+x)}\right ) \, dx,x,a \sin (x)\right )\\ &=-\frac{\log (1-\sin (x))}{2 (a+b)}+\frac{\log (1+\sin (x))}{2 (a-b)}-\frac{b \log (b+a \sin (x))}{a^2-b^2}\\ \end{align*}

Mathematica [A]  time = 0.0706628, size = 64, normalized size = 1.19 \[ \frac{-b \log (a \sin (x)+b)+(b-a) \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+(a+b) \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )}{(a-b) (a+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]/(a + b*Csc[x]),x]

[Out]

((-a + b)*Log[Cos[x/2] - Sin[x/2]] + (a + b)*Log[Cos[x/2] + Sin[x/2]] - b*Log[b + a*Sin[x]])/((a - b)*(a + b))

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Maple [A]  time = 0.041, size = 55, normalized size = 1. \begin{align*} -{\frac{b\ln \left ( b+a\sin \left ( x \right ) \right ) }{ \left ( a+b \right ) \left ( a-b \right ) }}+{\frac{\ln \left ( \sin \left ( x \right ) +1 \right ) }{2\,a-2\,b}}-{\frac{\ln \left ( \sin \left ( x \right ) -1 \right ) }{2\,a+2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)/(a+b*csc(x)),x)

[Out]

-b/(a-b)/(a+b)*ln(b+a*sin(x))+1/(2*a-2*b)*ln(sin(x)+1)-1/(2*a+2*b)*ln(sin(x)-1)

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Maxima [A]  time = 0.973521, size = 65, normalized size = 1.2 \begin{align*} -\frac{b \log \left (a \sin \left (x\right ) + b\right )}{a^{2} - b^{2}} + \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \,{\left (a - b\right )}} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{2 \,{\left (a + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)/(a+b*csc(x)),x, algorithm="maxima")

[Out]

-b*log(a*sin(x) + b)/(a^2 - b^2) + 1/2*log(sin(x) + 1)/(a - b) - 1/2*log(sin(x) - 1)/(a + b)

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Fricas [A]  time = 0.520392, size = 128, normalized size = 2.37 \begin{align*} -\frac{2 \, b \log \left (a \sin \left (x\right ) + b\right ) -{\left (a + b\right )} \log \left (\sin \left (x\right ) + 1\right ) +{\left (a - b\right )} \log \left (-\sin \left (x\right ) + 1\right )}{2 \,{\left (a^{2} - b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)/(a+b*csc(x)),x, algorithm="fricas")

[Out]

-1/2*(2*b*log(a*sin(x) + b) - (a + b)*log(sin(x) + 1) + (a - b)*log(-sin(x) + 1))/(a^2 - b^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec{\left (x \right )}}{a + b \csc{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)/(a+b*csc(x)),x)

[Out]

Integral(sec(x)/(a + b*csc(x)), x)

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Giac [A]  time = 1.42877, size = 72, normalized size = 1.33 \begin{align*} -\frac{a b \log \left ({\left | a \sin \left (x\right ) + b \right |}\right )}{a^{3} - a b^{2}} + \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \,{\left (a - b\right )}} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{2 \,{\left (a + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)/(a+b*csc(x)),x, algorithm="giac")

[Out]

-a*b*log(abs(a*sin(x) + b))/(a^3 - a*b^2) + 1/2*log(sin(x) + 1)/(a - b) - 1/2*log(-sin(x) + 1)/(a + b)