3.122 \(\int \frac{1}{a+b \sin (x)} \, dx\)

Optimal. Leaf size=40 \[ \frac{2 \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}} \]

[Out]

(2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2]

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Rubi [A]  time = 0.0705924, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{2 \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sin[x])^(-1),x]

[Out]

(2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2]

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Rubi in Sympy [A]  time = 3.27996, size = 31, normalized size = 0.78 \[ \frac{2 \operatorname{atan}{\left (\frac{a \tan{\left (\frac{x}{2} \right )} + b}{\sqrt{a^{2} - b^{2}}} \right )}}{\sqrt{a^{2} - b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*sin(x)),x)

[Out]

2*atan((a*tan(x/2) + b)/sqrt(a**2 - b**2))/sqrt(a**2 - b**2)

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Mathematica [A]  time = 0.0465735, size = 40, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sin[x])^(-1),x]

[Out]

(2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2]

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Maple [A]  time = 0.014, size = 39, normalized size = 1. \[ 2\,{\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( x/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*sin(x)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*sin(x) + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234622, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{2 \,{\left (a^{3} - a b^{2}\right )} \cos \left (x\right ) \sin \left (x\right ) + 2 \,{\left (a^{2} b - b^{3}\right )} \cos \left (x\right ) -{\left ({\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}\right )} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right )}{2 \, \sqrt{-a^{2} + b^{2}}}, -\frac{\arctan \left (-\frac{a \sin \left (x\right ) + b}{\sqrt{a^{2} - b^{2}} \cos \left (x\right )}\right )}{\sqrt{a^{2} - b^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*sin(x) + a),x, algorithm="fricas")

[Out]

[1/2*log(-(2*(a^3 - a*b^2)*cos(x)*sin(x) + 2*(a^2*b - b^3)*cos(x) - ((2*a^2 - b^
2)*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*
sin(x) - a^2 - b^2))/sqrt(-a^2 + b^2), -arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*
cos(x)))/sqrt(a^2 - b^2)]

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Sympy [A]  time = 27.0663, size = 114, normalized size = 2.85 \[ \begin{cases} \tilde{\infty } \log{\left (\tan{\left (\frac{x}{2} \right )} \right )} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2}{b \tan{\left (\frac{x}{2} \right )} - b} & \text{for}\: a = - b \\- \frac{2}{b \tan{\left (\frac{x}{2} \right )} + b} & \text{for}\: a = b \\\frac{\log{\left (\tan{\left (\frac{x}{2} \right )} \right )}}{b} & \text{for}\: a = 0 \\- \frac{\sqrt{- a^{2} + b^{2}} \log{\left (\tan{\left (\frac{x}{2} \right )} + \frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right )}}{a^{2} - b^{2}} + \frac{\sqrt{- a^{2} + b^{2}} \log{\left (\tan{\left (\frac{x}{2} \right )} + \frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right )}}{a^{2} - b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*sin(x)),x)

[Out]

Piecewise((zoo*log(tan(x/2)), Eq(a, 0) & Eq(b, 0)), (2/(b*tan(x/2) - b), Eq(a, -
b)), (-2/(b*tan(x/2) + b), Eq(a, b)), (log(tan(x/2))/b, Eq(a, 0)), (-sqrt(-a**2
+ b**2)*log(tan(x/2) + b/a - sqrt(-a**2 + b**2)/a)/(a**2 - b**2) + sqrt(-a**2 +
b**2)*log(tan(x/2) + b/a + sqrt(-a**2 + b**2)/a)/(a**2 - b**2), True))

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GIAC/XCAS [A]  time = 0.201879, size = 65, normalized size = 1.62 \[ \frac{2 \,{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor{\rm sign}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, x\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{\sqrt{a^{2} - b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*sin(x) + a),x, algorithm="giac")

[Out]

2*(pi*floor(1/2*x/pi + 1/2)*sign(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2))
)/sqrt(a^2 - b^2)