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

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 30

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

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])

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*}

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Mathematica [A]  time = 0.02, size = 15, normalized size = 1.00 \[ \frac {2}{b \sqrt {\csc (a+b x)}} \]

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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fricas [A]  time = 0.64, size = 13, normalized size = 0.87 \[ \frac {2 \, \sqrt {\sin \left (b x + a\right )}}{b} \]

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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giac [A]  time = 0.46, size = 13, normalized size = 0.87 \[ \frac {2 \, \sqrt {\sin \left (b x + a\right )}}{b} \]

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*sqrt(sin(b*x + a))/b

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maple [A]  time = 0.04, size = 14, normalized size = 0.93 \[ \frac {2}{b \sqrt {\csc \left (b x +a \right )}} \]

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 = 0.64, size = 13, normalized size = 0.87 \[ \frac {2 \, \sqrt {\sin \left (b x + a\right )}}{b} \]

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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mupad [B]  time = 0.30, size = 15, normalized size = 1.00 \[ \frac {2}{b\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(b*(1/sin(a + b*x))^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\left (a + b x \right )} \sqrt {\csc {\left (a + b x \right )}}\, dx \]

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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