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

Optimal. Leaf size=38 \[ \frac{4 \sqrt{\sin (e+f x)}}{f^2}-\frac{2 x \cos (e+f x)}{f \sqrt{\sin (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.0613543, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {3315} \[ \frac{4 \sqrt{\sin (e+f x)}}{f^2}-\frac{2 x \cos (e+f x)}{f \sqrt{\sin (e+f x)}} \]

Antiderivative was successfully verified.

[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 3315

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*} \int \left (\frac{x}{\sin ^{\frac{3}{2}}(e+f x)}+x \sqrt{\sin (e+f x)}\right ) \, dx &=\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]  time = 0.408233, size = 33, normalized size = 0.87 \[ \frac{4 \sin (e+f x)-2 f x \cos (e+f x)}{f^2 \sqrt{\sin (e+f x)}} \]

Antiderivative was successfully verified.

[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]])

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Maple [F]  time = 0.119, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}+x\sqrt{\sin \left ( fx+e \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)