3.2.42 \(\int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx\) [142]

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

[Out]

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

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Rubi [A]
time = 0.06, 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 = {} \begin {gather*} \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx \end {gather*}

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 = 20.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx \end {gather*}

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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Maple [A]
time = 0.01, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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

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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