\(\int (\frac {x}{\sin ^{\frac {3}{2}}(e+f x)}+x \sqrt {\sin (e+f x)}) \, dx\) [67]

Optimal result

Integrand size = 25, antiderivative size = 38 \[ \int \left (\frac {x}{\sin ^{\frac {3}{2}}(e+f x)}+x \sqrt {\sin (e+f x)}\right ) \, dx=-\frac {2 x \cos (e+f x)}{f \sqrt {\sin (e+f x)}}+\frac {4 \sqrt {\sin (e+f x)}}{f^2} \]

[Out]

-2*x*cos(f*x+e)/f/sin(f*x+e)^(1/2)+4*sin(f*x+e)^(1/2)/f^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {3396} \[ \int \left (\frac {x}{\sin ^{\frac {3}{2}}(e+f x)}+x \sqrt {\sin (e+f x)}\right ) \, dx=\frac {4 \sqrt {\sin (e+f x)}}{f^2}-\frac {2 x \cos (e+f x)}{f \sqrt {\sin (e+f x)}} \]

[In]

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

[Out]

(-2*x*Cos[e + f*x])/(f*Sqrt[Sin[e + f*x]]) + (4*Sqrt[Sin[e + f*x]])/f^2

Rule 3396

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(c + d*x)*Cos[e + f*x]*((b*Si
n[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + (Dist[(n + 2)/(b^2*(n + 1)), Int[(c + d*x)*(b*Sin[e + f*x])^(n + 2),
x], x] - Simp[d*((b*Sin[e + f*x])^(n + 2)/(b^2*f^2*(n + 1)*(n + 2))), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[
n, -1] && NeQ[n, -2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x}{\sin ^{\frac {3}{2}}(e+f x)} \, dx+\int x \sqrt {\sin (e+f x)} \, dx \\ & = -\frac {2 x \cos (e+f x)}{f \sqrt {\sin (e+f x)}}+\frac {4 \sqrt {\sin (e+f x)}}{f^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \left (\frac {x}{\sin ^{\frac {3}{2}}(e+f x)}+x \sqrt {\sin (e+f x)}\right ) \, dx=\frac {-2 f x \cos (e+f x)+4 \sin (e+f x)}{f^2 \sqrt {\sin (e+f x)}} \]

[In]

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

[Out]

(-2*f*x*Cos[e + f*x] + 4*Sin[e + f*x])/(f^2*Sqrt[Sin[e + f*x]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x}{\sin ^{\frac {3}{2}}(e+f x)}+x \sqrt {\sin (e+f x)}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

\[ \int \left (\frac {x}{\sin ^{\frac {3}{2}}(e+f x)}+x \sqrt {\sin (e+f x)}\right ) \, dx=\int \frac {x \left (\sin ^{2}{\left (e + f x \right )} + 1\right )}{\sin ^{\frac {3}{2}}{\left (e + f x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (\frac {x}{\sin ^{\frac {3}{2}}(e+f x)}+x \sqrt {\sin (e+f x)}\right ) \, dx=\int { x \sqrt {\sin \left (f x + e\right )} + \frac {x}{\sin \left (f x + e\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \left (\frac {x}{\sin ^{\frac {3}{2}}(e+f x)}+x \sqrt {\sin (e+f x)}\right ) \, dx=\int { x \sqrt {\sin \left (f x + e\right )} + \frac {x}{\sin \left (f x + e\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.83 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \left (\frac {x}{\sin ^{\frac {3}{2}}(e+f x)}+x \sqrt {\sin (e+f x)}\right ) \, dx=\frac {4\,{\sin \left (e+f\,x\right )}^2-f\,x\,\sin \left (2\,e+2\,f\,x\right )}{f^2\,{\sin \left (e+f\,x\right )}^{3/2}} \]

[In]

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

[Out]

(4*sin(e + f*x)^2 - f*x*sin(2*e + 2*f*x))/(f^2*sin(e + f*x)^(3/2))