3.261 \(\int \cos (a+b x) \sqrt{\csc (a+b x)} \, dx\)

Optimal. Leaf size=15 \[ \frac{2}{b \sqrt{\csc (a+b x)}} \]

[Out]

2/(b*Sqrt[Csc[a + b*x]])

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Rubi [A]  time = 0.0246537, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2621, 30} \[ \frac{2}{b \sqrt{\csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]*Sqrt[Csc[a + b*x]],x]

[Out]

2/(b*Sqrt[Csc[a + b*x]])

Rule 2621

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \cos (a+b x) \sqrt{\csc (a+b x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac{2}{b \sqrt{\csc (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0205176, size = 15, normalized size = 1. \[ \frac{2}{b \sqrt{\csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]*Sqrt[Csc[a + b*x]],x]

[Out]

2/(b*Sqrt[Csc[a + b*x]])

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Maple [A]  time = 0.042, size = 14, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{b\sqrt{\csc \left ( bx+a \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)*csc(b*x+a)^(1/2),x)

[Out]

2/b/csc(b*x+a)^(1/2)

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Maxima [A]  time = 0.938317, size = 18, normalized size = 1.2 \begin{align*} \frac{2 \, \sqrt{\sin \left (b x + a\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*csc(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(sin(b*x + a))/b

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Fricas [A]  time = 1.10671, size = 32, normalized size = 2.13 \begin{align*} \frac{2 \, \sqrt{\sin \left (b x + a\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*csc(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(sin(b*x + a))/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (a + b x \right )} \sqrt{\csc{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*csc(b*x+a)**(1/2),x)

[Out]

Integral(cos(a + b*x)*sqrt(csc(a + b*x)), x)

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Giac [A]  time = 1.14225, size = 27, normalized size = 1.8 \begin{align*} \frac{2 \, \mathrm{sgn}\left (\sin \left (b x + a\right )\right ) \sqrt{\sin \left (b x + a\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*csc(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2*sgn(sin(b*x + a))*sqrt(sin(b*x + a))/b