3.301 \(\int \frac{1}{a+b \sin (c+\frac{d}{x})} \, dx\)

Optimal. Leaf size=16 \[ \text{Unintegrable}\left (\frac{1}{a+b \sin \left (c+\frac{d}{x}\right )},x\right ) \]

[Out]

Unintegrable[(a + b*Sin[c + d/x])^(-1), x]

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Rubi [A]  time = 0.0051691, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{a+b \sin \left (c+\frac{d}{x}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sin[c + d/x])^(-1),x]

[Out]

Defer[Int][(a + b*Sin[c + d/x])^(-1), x]

Rubi steps

\begin{align*} \int \frac{1}{a+b \sin \left (c+\frac{d}{x}\right )} \, dx &=\int \frac{1}{a+b \sin \left (c+\frac{d}{x}\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0367629, size = 0, normalized size = 0. \[ \int \frac{1}{a+b \sin \left (c+\frac{d}{x}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sin[c + d/x])^(-1),x]

[Out]

Integrate[(a + b*Sin[c + d/x])^(-1), x]

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Maple [A]  time = 0.062, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sin \left ( c+{\frac{d}{x}} \right ) \right ) ^{-1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(c+d/x)),x)

[Out]

int(1/(a+b*sin(c+d/x)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \sin \left (c + \frac{d}{x}\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(c+d/x)),x, algorithm="maxima")

[Out]

integrate(1/(b*sin(c + d/x) + a), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b \sin \left (\frac{c x + d}{x}\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(c+d/x)),x, algorithm="fricas")

[Out]

integral(1/(b*sin((c*x + d)/x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(c+d/x)),x)

[Out]

Integral(1/(a + b*sin(c + d/x)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \sin \left (c + \frac{d}{x}\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(c+d/x)),x, algorithm="giac")

[Out]

integrate(1/(b*sin(c + d/x) + a), x)