3.266 \(\int \cos ^3(a+b x) \sqrt{\csc (a+b x)} \, dx\)

Optimal. Leaf size=33 \[ \frac{2}{b \sqrt{\csc (a+b x)}}-\frac{2}{5 b \csc ^{\frac{5}{2}}(a+b x)} \]

[Out]

-2/(5*b*Csc[a + b*x]^(5/2)) + 2/(b*Sqrt[Csc[a + b*x]])

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Rubi [A]  time = 0.0326656, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2621, 14} \[ \frac{2}{b \sqrt{\csc (a+b x)}}-\frac{2}{5 b \csc ^{\frac{5}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^3*Sqrt[Csc[a + b*x]],x]

[Out]

-2/(5*b*Csc[a + b*x]^(5/2)) + 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \cos ^3(a+b x) \sqrt{\csc (a+b x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{-1+x^2}{x^{7/2}} \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{x^{7/2}}+\frac{1}{x^{3/2}}\right ) \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac{2}{5 b \csc ^{\frac{5}{2}}(a+b x)}+\frac{2}{b \sqrt{\csc (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0576229, size = 27, normalized size = 0.82 \[ \frac{\cos (2 (a+b x))+9}{5 b \sqrt{\csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^3*Sqrt[Csc[a + b*x]],x]

[Out]

(9 + Cos[2*(a + b*x)])/(5*b*Sqrt[Csc[a + b*x]])

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Maple [A]  time = 0.565, size = 26, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ( -{\frac{2}{5} \left ( \sin \left ( bx+a \right ) \right ) ^{{\frac{5}{2}}}}+2\,\sqrt{\sin \left ( bx+a \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^3*csc(b*x+a)^(1/2),x)

[Out]

(-2/5*sin(b*x+a)^(5/2)+2*sin(b*x+a)^(1/2))/b

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Maxima [A]  time = 0.95676, size = 34, normalized size = 1.03 \begin{align*} \frac{2 \,{\left (\frac{5}{\sin \left (b x + a\right )^{2}} - 1\right )} \sin \left (b x + a\right )^{\frac{5}{2}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*csc(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2/5*(5/sin(b*x + a)^2 - 1)*sin(b*x + a)^(5/2)/b

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Fricas [A]  time = 1.07112, size = 63, normalized size = 1.91 \begin{align*} \frac{2 \,{\left (\cos \left (b x + a\right )^{2} + 4\right )} \sqrt{\sin \left (b x + a\right )}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*csc(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/5*(cos(b*x + a)^2 + 4)*sqrt(sin(b*x + a))/b

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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(cos(b*x+a)**3*csc(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.18348, size = 42, normalized size = 1.27 \begin{align*} -\frac{2 \,{\left (\sin \left (b x + a\right )^{\frac{5}{2}} - 5 \, \sqrt{\sin \left (b x + a\right )}\right )} \mathrm{sgn}\left (\sin \left (b x + a\right )\right )}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*csc(b*x+a)^(1/2),x, algorithm="giac")

[Out]

-2/5*(sin(b*x + a)^(5/2) - 5*sqrt(sin(b*x + a)))*sgn(sin(b*x + a))/b