3.91 \(\int (\frac{x}{\csc ^{\frac{3}{2}}(e+f x)}-\frac{1}{3} x \sqrt{\csc (e+f x)}) \, dx\)

Optimal. Leaf size=42 \[ \frac{4}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{3 f \sqrt{\csc (e+f x)}} \]

[Out]

4/(9*f^2*Csc[e + f*x]^(3/2)) - (2*x*Cos[e + f*x])/(3*f*Sqrt[Csc[e + f*x]])

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Rubi [A]  time = 0.123271, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {4187, 4189} \[ \frac{4}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{3 f \sqrt{\csc (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(9*f^2*Csc[e + f*x]^(3/2)) - (2*x*Cos[e + f*x])/(3*f*Sqrt[Csc[e + f*x]])

Rule 4187

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

Rule 4189

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[(b*Sin[e + f*x])^n*(b*C
sc[e + f*x])^n, Int[(c + d*x)^m/(b*Sin[e + f*x])^n, x], x] /; FreeQ[{b, c, d, e, f, m, n}, x] &&  !IntegerQ[n]

Rubi steps

\begin{align*} \int \left (\frac{x}{\csc ^{\frac{3}{2}}(e+f x)}-\frac{1}{3} x \sqrt{\csc (e+f x)}\right ) \, dx &=-\left (\frac{1}{3} \int x \sqrt{\csc (e+f x)} \, dx\right )+\int \frac{x}{\csc ^{\frac{3}{2}}(e+f x)} \, dx\\ &=\frac{4}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{3 f \sqrt{\csc (e+f x)}}+\frac{1}{3} \int x \sqrt{\csc (e+f x)} \, dx-\frac{1}{3} \left (\sqrt{\csc (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{x}{\sqrt{\sin (e+f x)}} \, dx\\ &=\frac{4}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{3 f \sqrt{\csc (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.487129, size = 29, normalized size = 0.69 \[ -\frac{2 (3 f x \cot (e+f x)-2)}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-2 + 3*f*x*Cot[e + f*x]))/(9*f^2*Csc[e + f*x]^(3/2))

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \csc \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{x}{3}\sqrt{\csc \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{3} \, x \sqrt{\csc \left (f x + e\right )} + \frac{x}{\csc \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/csc(f*x+e)^(3/2)-1/3*x*csc(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate(-1/3*x*sqrt(csc(f*x + e)) + x/csc(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/csc(f*x+e)^(3/2)-1/3*x*csc(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*} - \frac{\int - \frac{3 x}{\csc ^{\frac{3}{2}}{\left (e + f x \right )}}\, dx + \int x \sqrt{\csc{\left (e + f x \right )}}\, dx}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{3} \, x \sqrt{\csc \left (f x + e\right )} + \frac{x}{\csc \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/csc(f*x+e)^(3/2)-1/3*x*csc(f*x+e)^(1/2),x, algorithm="giac")

[Out]

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