\(\int \frac {1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx\) [143]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx=\text {Int}\left (\frac {1}{x^2 (a+a \sin (e+f x))^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 13.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

\[\int \frac {1}{x^{2} \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.11 \[ \int \frac {1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 3.63 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{x^{2} \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 1.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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