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

Optimal. Leaf size=35 \[ \frac{2}{3 b \csc ^{\frac{3}{2}}(a+b x)}-\frac{2}{7 b \csc ^{\frac{7}{2}}(a+b x)} \]

[Out]

-2/(7*b*Csc[a + b*x]^(7/2)) + 2/(3*b*Csc[a + b*x]^(3/2))

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Rubi [A]  time = 0.0327947, antiderivative size = 35, 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} \[ \frac{2}{3 b \csc ^{\frac{3}{2}}(a+b x)}-\frac{2}{7 b \csc ^{\frac{7}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(7*b*Csc[a + b*x]^(7/2)) + 2/(3*b*Csc[a + b*x]^(3/2))

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 \frac{\cos ^3(a+b x)}{\sqrt{\csc (a+b x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{-1+x^2}{x^{9/2}} \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{x^{9/2}}+\frac{1}{x^{5/2}}\right ) \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac{2}{7 b \csc ^{\frac{7}{2}}(a+b x)}+\frac{2}{3 b \csc ^{\frac{3}{2}}(a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0637448, size = 29, normalized size = 0.83 \[ \frac{2 \left (7 \csc ^2(a+b x)-3\right )}{21 b \csc ^{\frac{7}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-3 + 7*Csc[a + b*x]^2))/(21*b*Csc[a + b*x]^(7/2))

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Maple [A]  time = 0.53, size = 26, normalized size = 0.7 \begin{align*}{\frac{1}{b} \left ( -{\frac{2}{7} \left ( \sin \left ( bx+a \right ) \right ) ^{{\frac{7}{2}}}}+{\frac{2}{3} \left ( \sin \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}}} \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]

(-2/7*sin(b*x+a)^(7/2)+2/3*sin(b*x+a)^(3/2))/b

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Maxima [A]  time = 0.998336, size = 34, normalized size = 0.97 \begin{align*} \frac{2 \,{\left (\frac{7}{\sin \left (b x + a\right )^{2}} - 3\right )} \sin \left (b x + a\right )^{\frac{7}{2}}}{21 \, 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]

2/21*(7/sin(b*x + a)^2 - 3)*sin(b*x + a)^(7/2)/b

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Fricas [A]  time = 1.10725, size = 95, normalized size = 2.71 \begin{align*} -\frac{2 \,{\left (3 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2} - 4\right )}}{21 \, b \sqrt{\sin \left (b x + a\right )}} \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]

-2/21*(3*cos(b*x + a)^4 + cos(b*x + a)^2 - 4)/(b*sqrt(sin(b*x + a)))

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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 = 1.25266, size = 34, normalized size = 0.97 \begin{align*} \frac{2 \,{\left (\frac{7}{\sin \left (b x + a\right )^{2}} - 3\right )} \sin \left (b x + a\right )^{\frac{7}{2}}}{21 \, 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]

2/21*(7/sin(b*x + a)^2 - 3)*sin(b*x + a)^(7/2)/b