3.304 \(\int \frac{(e+f x)^2}{(a+b \sin (c+\frac{d}{x}))^2} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{(e+f x)^2}{\left (a+b \sin \left (c+\frac{d}{x}\right )\right )^2},x\right ) \]

[Out]

Unintegrable[(e + f*x)^2/(a + b*Sin[c + d/x])^2, x]

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Rubi [A]  time = 0.0295132, 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 = {} \[ \int \frac{(e+f x)^2}{\left (a+b \sin \left (c+\frac{d}{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e + f*x)^2/(a + b*Sin[c + d/x])^2,x]

[Out]

Defer[Int][(e + f*x)^2/(a + b*Sin[c + d/x])^2, x]

Rubi steps

\begin{align*} \int \frac{(e+f x)^2}{\left (a+b \sin \left (c+\frac{d}{x}\right )\right )^2} \, dx &=\int \frac{(e+f x)^2}{\left (a+b \sin \left (c+\frac{d}{x}\right )\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 112.861, size = 0, normalized size = 0. \[ \int \frac{(e+f x)^2}{\left (a+b \sin \left (c+\frac{d}{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e + f*x)^2/(a + b*Sin[c + d/x])^2,x]

[Out]

Integrate[(e + f*x)^2/(a + b*Sin[c + d/x])^2, x]

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Maple [A]  time = 3.425, size = 0, normalized size = 0. \begin{align*} \int{ \left ( fx+e \right ) ^{2} \left ( a+b\sin \left ( c+{\frac{d}{x}} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^2/(a+b*sin(c+d/x))^2,x)

[Out]

int((f*x+e)^2/(a+b*sin(c+d/x))^2,x)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2/(a+b*sin(c+d/x))^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{f^{2} x^{2} + 2 \, e f x + e^{2}}{b^{2} \cos \left (\frac{c x + d}{x}\right )^{2} - 2 \, a b \sin \left (\frac{c x + d}{x}\right ) - a^{2} - b^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2/(a+b*sin(c+d/x))^2,x, algorithm="fricas")

[Out]

integral(-(f^2*x^2 + 2*e*f*x + e^2)/(b^2*cos((c*x + d)/x)^2 - 2*a*b*sin((c*x + d)/x) - a^2 - b^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**2/(a+b*sin(c+d/x))**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{2}}{{\left (b \sin \left (c + \frac{d}{x}\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2/(a+b*sin(c+d/x))^2,x, algorithm="giac")

[Out]

integrate((f*x + e)^2/(b*sin(c + d/x) + a)^2, x)