3.12 \(\int \frac{\cos (x)}{a+b \csc (x)} \, dx\)

Optimal. Leaf size=20 \[ \frac{\sin (x)}{a}-\frac{b \log (a \sin (x)+b)}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0631133, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3872, 2833, 12, 43} \[ \frac{\sin (x)}{a}-\frac{b \log (a \sin (x)+b)}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/(a + b*Csc[x]),x]

[Out]

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

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 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0120148, size = 19, normalized size = 0.95 \[ \frac{a \sin (x)-b \log (a \sin (x)+b)}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/(a + b*Csc[x]),x]

[Out]

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

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Maple [A]  time = 0.039, size = 31, normalized size = 1.6 \begin{align*} -{\frac{b\ln \left ( a+b\csc \left ( x \right ) \right ) }{{a}^{2}}}+{\frac{1}{a\csc \left ( x \right ) }}+{\frac{b\ln \left ( \csc \left ( x \right ) \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.955774, size = 27, normalized size = 1.35 \begin{align*} -\frac{b \log \left (a \sin \left (x\right ) + b\right )}{a^{2}} + \frac{\sin \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*log(a*sin(x) + b)/a^2 + sin(x)/a

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Fricas [A]  time = 0.489971, size = 53, normalized size = 2.65 \begin{align*} -\frac{b \log \left (a \sin \left (x\right ) + b\right ) - a \sin \left (x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*log(a*sin(x) + b) - a*sin(x))/a^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(x)/(a + b*csc(x)), x)

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Giac [A]  time = 1.34851, size = 28, normalized size = 1.4 \begin{align*} -\frac{b \log \left ({\left | a \sin \left (x\right ) + b \right |}\right )}{a^{2}} + \frac{\sin \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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