3.281 \(\int \frac{1}{x^2 (a+b \sin (c+d (f+g x)^n))} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right )},x\right ) \]

[Out]

Unintegrable[1/(x^2*(a + b*Sin[c + d*(f + g*x)^n])), x]

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Rubi [A]  time = 0.0274604, 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 = {} \[ \int \frac{1}{x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*(a + b*Sin[c + d*(f + g*x)^n])),x]

[Out]

Defer[Int][1/(x^2*(a + b*Sin[c + d*(f + g*x)^n])), x]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right )} \, dx &=\int \frac{1}{x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 0.813269, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^2*(a + b*Sin[c + d*(f + g*x)^n])),x]

[Out]

Integrate[1/(x^2*(a + b*Sin[c + d*(f + g*x)^n])), x]

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Maple [A]  time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b\sin \left ( c+d \left ( gx+f \right ) ^{n} \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b*sin(c+d*(g*x+f)^n)),x)

[Out]

int(1/x^2/(a+b*sin(c+d*(g*x+f)^n)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="maxima")

[Out]

integrate(1/((b*sin((g*x + f)^n*d + c) + a)*x^2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b x^{2} \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="fricas")

[Out]

integral(1/(b*x^2*sin((g*x + f)^n*d + c) + a*x^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(1/x**2/(a+b*sin(c+d*(g*x+f)**n)),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="giac")

[Out]

integrate(1/((b*sin((g*x + f)^n*d + c) + a)*x^2), x)