\(\int (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}) \, dx\) [69]

Optimal result

Integrand size = 28, antiderivative size = 42 \[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=-\frac {2 x \cos (e+f x)}{3 f \sin ^{\frac {3}{2}}(e+f x)}-\frac {4}{3 f^2 \sqrt {\sin (e+f x)}} \]

[Out]

-2/3*x*cos(f*x+e)/f/sin(f*x+e)^(3/2)-4/3/f^2/sin(f*x+e)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {3396} \[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=-\frac {4}{3 f^2 \sqrt {\sin (e+f x)}}-\frac {2 x \cos (e+f x)}{3 f \sin ^{\frac {3}{2}}(e+f x)} \]

[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 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}& = -\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] (verified)

Time = 0.66 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83 \[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=-\frac {2 (f x \cos (e+f x)+2 \sin (e+f x))}{3 f^2 \sin ^{\frac {3}{2}}(e+f x)} \]

[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]

\[\int \left (\frac {x}{\sin \left (f x +e \right )^{\frac {5}{2}}}-\frac {x}{3 \sqrt {\sin \left (f x +e \right )}}\right )d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14 \[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=\frac {2 \, {\left (f x \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )} \sqrt {\sin \left (f x + e\right )}}{3 \, {\left (f^{2} \cos \left (f x + e\right )^{2} - f^{2}\right )}} \]

[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]

\[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=- \frac {\int \left (- \frac {3 x}{\sin ^{\frac {5}{2}}{\left (e + f x \right )}}\right )\, dx + \int \frac {x}{\sqrt {\sin {\left (e + f x \right )}}}\, dx}{3} \]

[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

Maxima [F]

\[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=\int { -\frac {x}{3 \, \sqrt {\sin \left (f x + e\right )}} + \frac {x}{\sin \left (f x + e\right )^{\frac {5}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=\int { -\frac {x}{3 \, \sqrt {\sin \left (f x + e\right )}} + \frac {x}{\sin \left (f x + e\right )^{\frac {5}{2}}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 2.89 (sec) , antiderivative size = 140, normalized size of antiderivative = 3.33 \[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=-\frac {4\,\sqrt {\sin \left (e+f\,x\right )}\,\left (20\,\sin \left (e+f\,x\right )-10\,\sin \left (3\,e+3\,f\,x\right )+2\,\sin \left (5\,e+5\,f\,x\right )-2\,f\,x\,\left (2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )+3\,f\,x\,\left (2\,{\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}^2-1\right )-f\,x\,\left (2\,{\sin \left (\frac {5\,e}{2}+\frac {5\,f\,x}{2}\right )}^2-1\right )\right )}{3\,f^2\,\left (30\,{\sin \left (e+f\,x\right )}^2-12\,{\sin \left (2\,e+2\,f\,x\right )}^2+2\,{\sin \left (3\,e+3\,f\,x\right )}^2\right )} \]

[In]

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

[Out]

-(4*sin(e + f*x)^(1/2)*(20*sin(e + f*x) - 10*sin(3*e + 3*f*x) + 2*sin(5*e + 5*f*x) - 2*f*x*(2*sin(e/2 + (f*x)/
2)^2 - 1) + 3*f*x*(2*sin((3*e)/2 + (3*f*x)/2)^2 - 1) - f*x*(2*sin((5*e)/2 + (5*f*x)/2)^2 - 1)))/(3*f^2*(2*sin(
3*e + 3*f*x)^2 - 12*sin(2*e + 2*f*x)^2 + 30*sin(e + f*x)^2))