3.8 \(\int \frac{\csc (x)}{a+a \sin (x)} \, dx\)

Optimal. Leaf size=20 \[ \frac{\cos (x)}{a \sin (x)+a}-\frac{\tanh ^{-1}(\cos (x))}{a} \]

[Out]

-(ArcTanh[Cos[x]]/a) + Cos[x]/(a + a*Sin[x])

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Rubi [A]  time = 0.0379892, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2747, 3770, 2648} \[ \frac{\cos (x)}{a \sin (x)+a}-\frac{\tanh ^{-1}(\cos (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(a + a*Sin[x]),x]

[Out]

-(ArcTanh[Cos[x]]/a) + Cos[x]/(a + a*Sin[x])

Rule 2747

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[b/(
b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; Fre
eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 3770

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

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{\csc (x)}{a+a \sin (x)} \, dx &=\frac{\int \csc (x) \, dx}{a}-\int \frac{1}{a+a \sin (x)} \, dx\\ &=-\frac{\tanh ^{-1}(\cos (x))}{a}+\frac{\cos (x)}{a+a \sin (x)}\\ \end{align*}

Mathematica [B]  time = 0.0501187, size = 74, normalized size = 3.7 \[ -\frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (\cos \left (\frac{x}{2}\right ) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )+\sin \left (\frac{x}{2}\right ) \left (-\log \left (\sin \left (\frac{x}{2}\right )\right )+\log \left (\cos \left (\frac{x}{2}\right )\right )+2\right )\right )}{a (\sin (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]/(a + a*Sin[x]),x]

[Out]

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

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Maple [A]  time = 0.029, size = 24, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }}+{\frac{1}{a}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)/(a+a*sin(x)),x)

[Out]

2/a/(tan(1/2*x)+1)+1/a*ln(tan(1/2*x))

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Maxima [A]  time = 1.65596, size = 42, normalized size = 2.1 \begin{align*} \frac{\log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} + \frac{2}{a + \frac{a \sin \left (x\right )}{\cos \left (x\right ) + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sin(x)/(cos(x) + 1))/a + 2/(a + a*sin(x)/(cos(x) + 1))

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Fricas [B]  time = 1.91135, size = 204, normalized size = 10.2 \begin{align*} -\frac{{\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2 \, \cos \left (x\right ) + 2 \, \sin \left (x\right ) - 2}{2 \,{\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc{\left (x \right )}}{\sin{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a+a*sin(x)),x)

[Out]

Integral(csc(x)/(sin(x) + 1), x)/a

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Giac [A]  time = 2.23577, size = 32, normalized size = 1.6 \begin{align*} \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{a} + \frac{2}{a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(tan(1/2*x)))/a + 2/(a*(tan(1/2*x) + 1))