3.69 \(\int (\frac{x}{\sin ^{\frac{5}{2}}(e+f x)}-\frac{x}{3 \sqrt{\sin (e+f x)}}) \, dx\)

Optimal. Leaf size=42 \[ -\frac{4}{3 f^2 \sqrt{\sin (e+f x)}}-\frac{2 x \cos (e+f x)}{3 f \sin ^{\frac{3}{2}}(e+f x)} \]

[Out]

(-2*x*Cos[e + f*x])/(3*f*Sin[e + f*x]^(3/2)) - 4/(3*f^2*Sqrt[Sin[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.0603062, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {3315} \[ -\frac{4}{3 f^2 \sqrt{\sin (e+f x)}}-\frac{2 x \cos (e+f x)}{3 f \sin ^{\frac{3}{2}}(e+f x)} \]

Antiderivative was successfully verified.

[In]

Int[x/Sin[e + f*x]^(5/2) - x/(3*Sqrt[Sin[e + f*x]]),x]

[Out]

(-2*x*Cos[e + f*x])/(3*f*Sin[e + f*x]^(3/2)) - 4/(3*f^2*Sqrt[Sin[e + f*x]])

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{5}{2}}(e+f x)}-\frac{x}{3 \sqrt{\sin (e+f x)}}\right ) \, dx &=-\left (\frac{1}{3} \int \frac{x}{\sqrt{\sin (e+f x)}} \, dx\right )+\int \frac{x}{\sin ^{\frac{5}{2}}(e+f x)} \, dx\\ &=-\frac{2 x \cos (e+f x)}{3 f \sin ^{\frac{3}{2}}(e+f x)}-\frac{4}{3 f^2 \sqrt{\sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.404757, size = 35, normalized size = 0.83 \[ -\frac{2 (2 \sin (e+f x)+f x \cos (e+f x))}{3 f^2 \sin ^{\frac{3}{2}}(e+f x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sin[e + f*x]^(5/2) - x/(3*Sqrt[Sin[e + f*x]]),x]

[Out]

(-2*(f*x*Cos[e + f*x] + 2*Sin[e + f*x]))/(3*f^2*Sin[e + f*x]^(3/2))

________________________________________________________________________________________

Maple [F]  time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \sin \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{x}{3}{\frac{1}{\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)^(5/2)-1/3*x/sin(f*x+e)^(1/2),x)

[Out]

int(x/sin(f*x+e)^(5/2)-1/3*x/sin(f*x+e)^(1/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x}{3 \, \sqrt{\sin \left (f x + e\right )}} + \frac{x}{\sin \left (f x + e\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sin(f*x+e)^(5/2)-1/3*x/sin(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate(-1/3*x/sqrt(sin(f*x + e)) + x/sin(f*x + e)^(5/2), x)

________________________________________________________________________________________

Fricas [A]  time = 1.72272, size = 117, normalized size = 2.79 \begin{align*} \frac{2 \,{\left (f x \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )} \sqrt{\sin \left (f x + e\right )}}{3 \,{\left (f^{2} \cos \left (f x + e\right )^{2} - f^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sin(f*x+e)^(5/2)-1/3*x/sin(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int - \frac{3 x}{\sin ^{\frac{5}{2}}{\left (e + f x \right )}}\, dx + \int \frac{x}{\sqrt{\sin{\left (e + f x \right )}}}\, dx}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sin(f*x+e)**(5/2)-1/3*x/sin(f*x+e)**(1/2),x)

[Out]

-(Integral(-3*x/sin(e + f*x)**(5/2), x) + Integral(x/sqrt(sin(e + f*x)), x))/3

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x}{3 \, \sqrt{\sin \left (f x + e\right )}} + \frac{x}{\sin \left (f x + e\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sin(f*x+e)^(5/2)-1/3*x/sin(f*x+e)^(1/2),x, algorithm="giac")

[Out]

integrate(-1/3*x/sqrt(sin(f*x + e)) + x/sin(f*x + e)^(5/2), x)