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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{e+f x}{\left (a+b \sin \left (c+\frac{d}{x}\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0165308, 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 = {} \[ \int \frac{e+f x}{\left (a+b \sin \left (c+\frac{d}{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 19.4511, size = 0, normalized size = 0. \[ \int \frac{e+f x}{\left (a+b \sin \left (c+\frac{d}{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 2.381, size = 0, normalized size = 0. \begin{align*} \int{(fx+e) \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)/(a+b*sin(c+d/x))^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(a*b*f*x^3 + a*b*e*x^2)*cos(2*(c*x + d)/x)*cos((c*x + d)/x) + 2*(a*b*f*x^3 + a*b*e*x^2)*cos((c*x + d)/x) +
((a^2*b^2 - b^4)*d*cos(2*(c*x + d)/x)^2 + 4*(a^4 - a^2*b^2)*d*cos((c*x + d)/x)^2 + 4*(a^3*b - a*b^3)*d*cos((c
*x + d)/x)*sin(2*(c*x + d)/x) + (a^2*b^2 - b^4)*d*sin(2*(c*x + d)/x)^2 + 4*(a^4 - a^2*b^2)*d*sin((c*x + d)/x)^
2 + 4*(a^3*b - a*b^3)*d*sin((c*x + d)/x) + (a^2*b^2 - b^4)*d - 2*(2*(a^3*b - a*b^3)*d*sin((c*x + d)/x) + (a^2*
b^2 - b^4)*d)*cos(2*(c*x + d)/x))*integrate(-2*(2*(a^2*d*f*x + a^2*d*e)*cos((c*x + d)/x)^2 + 2*(a^2*d*f*x + a^
2*d*e)*sin((c*x + d)/x)^2 + ((3*a*b*f*x^2 + 2*a*b*e*x)*cos((c*x + d)/x) - (a*b*d*f*x + a*b*d*e)*sin((c*x + d)/
x))*cos(2*(c*x + d)/x) + (3*a*b*f*x^2 + 2*a*b*e*x)*cos((c*x + d)/x) + (3*b^2*f*x^2 + 2*b^2*e*x + (a*b*d*f*x +
a*b*d*e)*cos((c*x + d)/x) + (3*a*b*f*x^2 + 2*a*b*e*x)*sin((c*x + d)/x))*sin(2*(c*x + d)/x) + (a*b*d*f*x + a*b*
d*e)*sin((c*x + d)/x))/((a^2*b^2 - b^4)*d*cos(2*(c*x + d)/x)^2 + 4*(a^4 - a^2*b^2)*d*cos((c*x + d)/x)^2 + 4*(a
^3*b - a*b^3)*d*cos((c*x + d)/x)*sin(2*(c*x + d)/x) + (a^2*b^2 - b^4)*d*sin(2*(c*x + d)/x)^2 + 4*(a^4 - a^2*b^
2)*d*sin((c*x + d)/x)^2 + 4*(a^3*b - a*b^3)*d*sin((c*x + d)/x) + (a^2*b^2 - b^4)*d - 2*(2*(a^3*b - a*b^3)*d*si
n((c*x + d)/x) + (a^2*b^2 - b^4)*d)*cos(2*(c*x + d)/x)), x) + 2*(b^2*f*x^3 + b^2*e*x^2 + (a*b*f*x^3 + a*b*e*x^
2)*sin((c*x + d)/x))*sin(2*(c*x + d)/x))/((a^2*b^2 - b^4)*d*cos(2*(c*x + d)/x)^2 + 4*(a^4 - a^2*b^2)*d*cos((c*
x + d)/x)^2 + 4*(a^3*b - a*b^3)*d*cos((c*x + d)/x)*sin(2*(c*x + d)/x) + (a^2*b^2 - b^4)*d*sin(2*(c*x + d)/x)^2
+ 4*(a^4 - a^2*b^2)*d*sin((c*x + d)/x)^2 + 4*(a^3*b - a*b^3)*d*sin((c*x + d)/x) + (a^2*b^2 - b^4)*d - 2*(2*(a
^3*b - a*b^3)*d*sin((c*x + d)/x) + (a^2*b^2 - b^4)*d)*cos(2*(c*x + d)/x))

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

[Out]

integral(-(f*x + e)/(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)/(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{f x + e}{{\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)/(a+b*sin(c+d/x))^2,x, algorithm="giac")

[Out]

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