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

Optimal. Leaf size=19 \[ \text{Unintegrable}\left (\frac{1}{x \left (a+\frac{1}{2} b \sin (2 x)\right )},x\right ) \]

[Out]

Unintegrable[1/(x*(a + (b*Sin[2*x])/2)), x]

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Rubi [A]  time = 0.0853994, 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 = {} \[ \int \frac{1}{x (a+b \cos (x) \sin (x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(a + b*Cos[x]*Sin[x])),x]

[Out]

Defer[Int][1/(x*(a + (b*Sin[2*x])/2)), x]

Rubi steps

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

Mathematica [A]  time = 1.62831, size = 0, normalized size = 0. \[ \int \frac{1}{x (a+b \cos (x) \sin (x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*(a + b*Cos[x]*Sin[x])),x]

[Out]

Integrate[1/(x*(a + b*Cos[x]*Sin[x])), x]

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Maple [A]  time = 0.107, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b\cos \left ( x \right ) \sin \left ( x \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*(a + b*sin(x)*cos(x))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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