3.138 \(\int \frac{1}{x^2 \sqrt{a+a \sin (c+d x)}} \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{a \sin (c+d x)+a}},x\right ) \]

[Out]

Unintegrable[1/(x^2*Sqrt[a + a*Sin[c + d*x]]), x]

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Rubi [A]  time = 0.0735011, 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 = {} \[ \int \frac{1}{x^2 \sqrt{a+a \sin (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*Sqrt[a + a*Sin[c + d*x]]),x]

[Out]

Defer[Int][1/(x^2*Sqrt[a + a*Sin[c + d*x]]), x]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \sqrt{a+a \sin (c+d x)}} \, dx &=\int \frac{1}{x^2 \sqrt{a+a \sin (c+d x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 0.747242, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \sqrt{a+a \sin (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^2*Sqrt[a + a*Sin[c + d*x]]),x]

[Out]

Integrate[1/(x^2*Sqrt[a + a*Sin[c + d*x]]), x]

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Maple [A]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+a*sin(d*x+c))^(1/2),x)

[Out]

int(1/x^2/(a+a*sin(d*x+c))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a*sin(d*x + c) + a)*x^2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \sin \left (d x + c\right ) + a}}{a x^{2} \sin \left (d x + 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+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*sin(d*x + c) + a)/(a*x^2*sin(d*x + c) + a*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Integral(1/(x**2*sqrt(a*(sin(c + d*x) + 1))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a*sin(d*x + c) + a)*x^2), x)