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

Optimal. Leaf size=17 \[ \text {Int}\left (\frac {1}{a+b \sin \left (c+\frac {d}{x}\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{a+b \sin \left (c+\frac {d}{x}\right )} \, dx \]

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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fricas [A]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b \sin \left (\frac {c x + d}{x}\right ) + a}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \sin \left (c + \frac {d}{x}\right ) + a}\,{d x} \]

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)

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maple [A]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {1}{a +b \sin \left (c +\frac {d}{x}\right )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \sin \left (c + \frac {d}{x}\right ) + a}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{a+b\,\sin \left (c+\frac {d}{x}\right )} \,d x \]

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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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a + b \sin {\left (c + \frac {d}{x} \right )}}\, dx \]

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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