3.19 \(\int \frac{\csc (x)}{(a \cos (x)+b \sin (x))^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{b \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2}}-\frac{\tanh ^{-1}(\cos (x))}{a^2}+\frac{1}{a (a \cos (x)+b \sin (x))} \]

[Out]

-(ArcTanh[Cos[x]]/a^2) + (b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]) + 1/(a*(a*Co
s[x] + b*Sin[x]))

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Rubi [A]  time = 0.0601325, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3093, 3770, 3074, 206} \[ \frac{b \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2}}-\frac{\tanh ^{-1}(\cos (x))}{a^2}+\frac{1}{a (a \cos (x)+b \sin (x))} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(a*Cos[x] + b*Sin[x])^2,x]

[Out]

-(ArcTanh[Cos[x]]/a^2) + (b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]) + 1/(a*(a*Co
s[x] + b*Sin[x]))

Rule 3093

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

Rule 3770

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

Rule 3074

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.295683, size = 72, normalized size = 1.14 \[ \frac{-\frac{2 b \tanh ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )-b}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}}+\frac{a \csc (x)}{a \cot (x)+b}+\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]/(a*Cos[x] + b*Sin[x])^2,x]

[Out]

((-2*b*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] + (a*Csc[x])/(b + a*Cot[x]) - Log[Cos[x/2]]
+ Log[Sin[x/2]])/a^2

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Maple [A]  time = 0.126, size = 106, normalized size = 1.7 \begin{align*} -2\,{\frac{b\tan \left ( x/2 \right ) }{{a}^{2} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}a-2\,b\tan \left ( x/2 \right ) -a \right ) }}-2\,{\frac{1}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}a-2\,b\tan \left ( x/2 \right ) -a \right ) }}-2\,{\frac{b}{\sqrt{{a}^{2}+{b}^{2}}{a}^{2}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tan \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }+{\frac{1}{{a}^{2}}\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*cos(x)+b*sin(x))^2,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a*cos(x)+b*sin(x))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.595954, size = 563, normalized size = 8.94 \begin{align*} \frac{2 \, a^{3} + 2 \, a b^{2} +{\left (a b \cos \left (x\right ) + b^{2} \sin \left (x\right )\right )} \sqrt{a^{2} + b^{2}} \log \left (\frac{2 \, a b \cos \left (x\right ) \sin \left (x\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) -{\left ({\left (a^{3} + a b^{2}\right )} \cos \left (x\right ) +{\left (a^{2} b + b^{3}\right )} \sin \left (x\right )\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) +{\left ({\left (a^{3} + a b^{2}\right )} \cos \left (x\right ) +{\left (a^{2} b + b^{3}\right )} \sin \left (x\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{2 \,{\left ({\left (a^{5} + a^{3} b^{2}\right )} \cos \left (x\right ) +{\left (a^{4} b + a^{2} b^{3}\right )} \sin \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a*cos(x)+b*sin(x))^2,x, algorithm="fricas")

[Out]

1/2*(2*a^3 + 2*a*b^2 + (a*b*cos(x) + b^2*sin(x))*sqrt(a^2 + b^2)*log((2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)
^2 - 2*a^2 - b^2 - 2*sqrt(a^2 + b^2)*(b*cos(x) - a*sin(x)))/(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2)
) - ((a^3 + a*b^2)*cos(x) + (a^2*b + b^3)*sin(x))*log(1/2*cos(x) + 1/2) + ((a^3 + a*b^2)*cos(x) + (a^2*b + b^3
)*sin(x))*log(-1/2*cos(x) + 1/2))/((a^5 + a^3*b^2)*cos(x) + (a^4*b + a^2*b^3)*sin(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc{\left (x \right )}}{\left (a \cos{\left (x \right )} + b \sin{\left (x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a*cos(x)+b*sin(x))**2,x)

[Out]

Integral(csc(x)/(a*cos(x) + b*sin(x))**2, x)

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Giac [A]  time = 1.21133, size = 147, normalized size = 2.33 \begin{align*} \frac{b \log \left (\frac{{\left | 2 \, a \tan \left (\frac{1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac{1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} a^{2}} + \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{a^{2}} - \frac{2 \,{\left (b \tan \left (\frac{1}{2} \, x\right ) + a\right )}}{{\left (a \tan \left (\frac{1}{2} \, x\right )^{2} - 2 \, b \tan \left (\frac{1}{2} \, x\right ) - a\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a*cos(x)+b*sin(x))^2,x, algorithm="giac")

[Out]

b*log(abs(2*a*tan(1/2*x) - 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2*x) - 2*b + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 +
b^2)*a^2) + log(abs(tan(1/2*x)))/a^2 - 2*(b*tan(1/2*x) + a)/((a*tan(1/2*x)^2 - 2*b*tan(1/2*x) - a)*a^2)