3.269 \(\int \frac{\cos ^4(a+b x)}{\sqrt{\csc (a+b x)}} \, dx\)

Optimal. Leaf size=92 \[ \frac{2 \cos ^3(a+b x)}{9 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{15 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{8 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{15 b} \]

[Out]

(4*Cos[a + b*x])/(15*b*Csc[a + b*x]^(3/2)) + (2*Cos[a + b*x]^3)/(9*b*Csc[a + b*x]^(3/2)) + (8*Sqrt[Csc[a + b*x
]]*EllipticE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(15*b)

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Rubi [A]  time = 0.0788764, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2628, 3771, 2639} \[ \frac{2 \cos ^3(a+b x)}{9 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{15 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{8 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{15 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^4/Sqrt[Csc[a + b*x]],x]

[Out]

(4*Cos[a + b*x])/(15*b*Csc[a + b*x]^(3/2)) + (2*Cos[a + b*x]^3)/(9*b*Csc[a + b*x]^(3/2)) + (8*Sqrt[Csc[a + b*x
]]*EllipticE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(15*b)

Rule 2628

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(a*(a*Csc[e
+ f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1))/(b*f*(m + n)), x] + Dist[(n + 1)/(b^2*(m + n)), Int[(a*Csc[e + f*x]
)^m*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m + n, 0] && IntegersQ[
2*m, 2*n]

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]

Rubi steps

\begin{align*} \int \frac{\cos ^4(a+b x)}{\sqrt{\csc (a+b x)}} \, dx &=\frac{2 \cos ^3(a+b x)}{9 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{2}{3} \int \frac{\cos ^2(a+b x)}{\sqrt{\csc (a+b x)}} \, dx\\ &=\frac{4 \cos (a+b x)}{15 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{2 \cos ^3(a+b x)}{9 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{4}{15} \int \frac{1}{\sqrt{\csc (a+b x)}} \, dx\\ &=\frac{4 \cos (a+b x)}{15 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{2 \cos ^3(a+b x)}{9 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{1}{15} \left (4 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \sqrt{\sin (a+b x)} \, dx\\ &=\frac{4 \cos (a+b x)}{15 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{2 \cos ^3(a+b x)}{9 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{8 \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{15 b}\\ \end{align*}

Mathematica [A]  time = 0.412331, size = 63, normalized size = 0.68 \[ \frac{39 \cos (a+b x)+5 \cos (3 (a+b x))-\frac{48 E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )}{\sin ^{\frac{3}{2}}(a+b x)}}{90 b \csc ^{\frac{3}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^4/Sqrt[Csc[a + b*x]],x]

[Out]

(39*Cos[a + b*x] + 5*Cos[3*(a + b*x)] - (48*EllipticE[(-2*a + Pi - 2*b*x)/4, 2])/Sin[a + b*x]^(3/2))/(90*b*Csc
[a + b*x]^(3/2))

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Maple [A]  time = 1.095, size = 152, normalized size = 1.7 \begin{align*}{\frac{1}{\cos \left ( bx+a \right ) b} \left ( -{\frac{2\, \left ( \cos \left ( bx+a \right ) \right ) ^{6}}{9}}-{\frac{8}{15}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticE} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }+{\frac{4}{15}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }-{\frac{2\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{45}}+{\frac{4\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{15}} \right ){\frac{1}{\sqrt{\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^4/csc(b*x+a)^(1/2),x)

[Out]

(-2/9*cos(b*x+a)^6-8/15*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*EllipticE((sin(b*x+a)
+1)^(1/2),1/2*2^(1/2))+4/15*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*EllipticF((sin(b*
x+a)+1)^(1/2),1/2*2^(1/2))-2/45*cos(b*x+a)^4+4/15*cos(b*x+a)^2)/cos(b*x+a)/sin(b*x+a)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )^{4}}{\sqrt{\csc \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4/csc(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(b*x + a)^4/sqrt(csc(b*x + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (b x + a\right )^{4}}{\sqrt{\csc \left (b x + a\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4/csc(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

integral(cos(b*x + a)^4/sqrt(csc(b*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{4}{\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)**4/csc(b*x+a)**(1/2),x)

[Out]

Integral(cos(a + b*x)**4/sqrt(csc(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )^{4}}{\sqrt{\csc \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4/csc(b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate(cos(b*x + a)^4/sqrt(csc(b*x + a)), x)