3.865 \(\int \sqrt{\csc (x)} (x \cos (x)-4 \sec (x) \tan (x)) \, dx\)

Optimal. Leaf size=20 \[ \frac{2 x}{\sqrt{\csc (x)}}-\frac{4 \sec (x)}{\csc ^{\frac{3}{2}}(x)} \]

[Out]

(2*x)/Sqrt[Csc[x]] - (4*Sec[x])/Csc[x]^(3/2)

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Rubi [A]  time = 0.151274, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6742, 4213, 3771, 2639, 2626} \[ \frac{2 x}{\sqrt{\csc (x)}}-\frac{4 \sec (x)}{\csc ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Csc[x]]*(x*Cos[x] - 4*Sec[x]*Tan[x]),x]

[Out]

(2*x)/Sqrt[Csc[x]] - (4*Sec[x])/Csc[x]^(3/2)

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 4213

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*Csc[(a_.) + (b_.)*(x_)^(n_.)]^(p_)*(x_)^(m_.), x_Symbol] :> -Simp[(x^(m - n
+ 1)*Csc[a + b*x^n]^(p - 1))/(b*n*(p - 1)), x] + Dist[(m - n + 1)/(b*n*(p - 1)), Int[x^(m - n)*Csc[a + b*x^n]^
(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 2626

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*b*(a*Csc[
e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1))/(f*(n - 1)), x] + Dist[(b^2*(m + n - 2))/(n - 1), Int[(a*Csc[e + f
*x])^m*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n]

Rubi steps

\begin{align*} \int \sqrt{\csc (x)} (x \cos (x)-4 \sec (x) \tan (x)) \, dx &=\int \left (x \cos (x) \sqrt{\csc (x)}-\frac{4 \sec ^2(x)}{\sqrt{\csc (x)}}\right ) \, dx\\ &=-\left (4 \int \frac{\sec ^2(x)}{\sqrt{\csc (x)}} \, dx\right )+\int x \cos (x) \sqrt{\csc (x)} \, dx\\ &=\frac{2 x}{\sqrt{\csc (x)}}-\frac{4 \sec (x)}{\csc ^{\frac{3}{2}}(x)}\\ \end{align*}

Mathematica [A]  time = 0.447148, size = 17, normalized size = 0.85 \[ \frac{2 (x \csc (x)-2 \sec (x))}{\csc ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Csc[x]]*(x*Cos[x] - 4*Sec[x]*Tan[x]),x]

[Out]

(2*(x*Csc[x] - 2*Sec[x]))/Csc[x]^(3/2)

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Maple [F]  time = 0.188, size = 0, normalized size = 0. \begin{align*} \int \sqrt{\csc \left ( x \right ) } \left ( x\cos \left ( x \right ) -4\,\sec \left ( x \right ) \tan \left ( x \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^(1/2)*(x*cos(x)-4*sec(x)*tan(x)),x)

[Out]

int(csc(x)^(1/2)*(x*cos(x)-4*sec(x)*tan(x)),x)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^(1/2)*(x*cos(x)-4*sec(x)*tan(x)),x, algorithm="maxima")

[Out]

Timed out

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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(csc(x)^(1/2)*(x*cos(x)-4*sec(x)*tan(x)),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**(1/2)*(x*cos(x)-4*sec(x)*tan(x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x \cos \left (x\right ) - 4 \, \sec \left (x\right ) \tan \left (x\right )\right )} \sqrt{\csc \left (x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^(1/2)*(x*cos(x)-4*sec(x)*tan(x)),x, algorithm="giac")

[Out]

integrate((x*cos(x) - 4*sec(x)*tan(x))*sqrt(csc(x)), x)