3.143 \(\int \frac{1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{x^2 (a \sin (e+f x)+a)^{3/2}},x\right ) \]

[Out]

Unintegrable[1/(x^2*(a + a*Sin[e + f*x])^(3/2)), x]

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Rubi [A]  time = 0.08029, 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 = {} \[ \int \frac{1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*(a + a*Sin[e + f*x])^(3/2)),x]

[Out]

Defer[Int][1/(x^2*(a + a*Sin[e + f*x])^(3/2)), x]

Rubi steps

\begin{align*} \int \frac{1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx &=\int \frac{1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx\\ \end{align*}

Mathematica [A]  time = 17.4709, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^2*(a + a*Sin[e + f*x])^(3/2)),x]

[Out]

Integrate[1/(x^2*(a + a*Sin[e + f*x])^(3/2)), x]

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Maple [A]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+a*sin(f*x+e))^(3/2),x)

[Out]

int(1/x^2/(a+a*sin(f*x+e))^(3/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+a*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((a*sin(f*x + e) + a)^(3/2)*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 (f x + e\right ) + a}}{a^{2} x^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} x^{2} \sin \left (f x + e\right ) - 2 \, a^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+a*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (a \left (\sin{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+a*sin(f*x+e))**(3/2),x)

[Out]

Integral(1/(x**2*(a*(sin(e + f*x) + 1))**(3/2)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+a*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate(1/((a*sin(f*x + e) + a)^(3/2)*x^2), x)