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]

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

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Rubi [A]  time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.46, 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))

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fricas [A]  time = 0.64, size = 48, normalized size = 1.14 \[ \frac {2 \, {\left (f x \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )} \sqrt {\sin \left (f x + e\right )}}{3 \, {\left (f^{2} \cos \left (f x + e\right )^{2} - f^{2}\right )}} \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x}{3 \, \sqrt {\sin \left (f x + e\right )}} + \frac {x}{\sin \left (f x + e\right )^{\frac {5}{2}}}\,{d x} \]

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)

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maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sin \left (f x +e \right )^{\frac {5}{2}}}-\frac {x}{3 \sqrt {\sin \left (f x +e \right )}}\, dx \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x}{3 \, \sqrt {\sin \left (f x + e\right )}} + \frac {x}{\sin \left (f x + e\right )^{\frac {5}{2}}}\,{d x} \]

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)

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mupad [B]  time = 3.11, size = 140, normalized size = 3.33 \[ -\frac {4\,\sqrt {\sin \left (e+f\,x\right )}\,\left (20\,\sin \left (e+f\,x\right )-10\,\sin \left (3\,e+3\,f\,x\right )+2\,\sin \left (5\,e+5\,f\,x\right )-2\,f\,x\,\left (2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )+3\,f\,x\,\left (2\,{\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}^2-1\right )-f\,x\,\left (2\,{\sin \left (\frac {5\,e}{2}+\frac {5\,f\,x}{2}\right )}^2-1\right )\right )}{3\,f^2\,\left (30\,{\sin \left (e+f\,x\right )}^2-12\,{\sin \left (2\,e+2\,f\,x\right )}^2+2\,{\sin \left (3\,e+3\,f\,x\right )}^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \left (- \frac {3 x}{\sin ^{\frac {5}{2}}{\left (e + f x \right )}}\right )\, dx + \int \frac {x}{\sqrt {\sin {\left (e + f x \right )}}}\, dx}{3} \]

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

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