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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{x (a \sin (e+f x)+a)^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 35.57, size = 0, normalized size = 0.00 \[ \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {a \sin \left (f x + e\right ) + a}}{a^{2} x \cos \left (f x + e\right )^{2} - 2 \, a^{2} x \sin \left (f x + e\right ) - 2 \, a^{2} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{x\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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