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

Optimal. Leaf size=92 \[ \frac{2 \cos ^3(a+b x)}{7 b \sqrt{\csc (a+b x)}}+\frac{4 \cos (a+b x)}{7 b \sqrt{\csc (a+b x)}}+\frac{8 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{7 b} \]

[Out]

(4*Cos[a + b*x])/(7*b*Sqrt[Csc[a + b*x]]) + (2*Cos[a + b*x]^3)/(7*b*Sqrt[Csc[a + b*x]]) + (8*Sqrt[Csc[a + b*x]
]*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(7*b)

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Rubi [A]  time = 0.0801452, 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, 2641} \[ \frac{2 \cos ^3(a+b x)}{7 b \sqrt{\csc (a+b x)}}+\frac{4 \cos (a+b x)}{7 b \sqrt{\csc (a+b x)}}+\frac{8 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{7 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Cos[a + b*x])/(7*b*Sqrt[Csc[a + b*x]]) + (2*Cos[a + b*x]^3)/(7*b*Sqrt[Csc[a + b*x]]) + (8*Sqrt[Csc[a + b*x]
]*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(7*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 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \cos ^4(a+b x) \sqrt{\csc (a+b x)} \, dx &=\frac{2 \cos ^3(a+b x)}{7 b \sqrt{\csc (a+b x)}}+\frac{6}{7} \int \cos ^2(a+b x) \sqrt{\csc (a+b x)} \, dx\\ &=\frac{4 \cos (a+b x)}{7 b \sqrt{\csc (a+b x)}}+\frac{2 \cos ^3(a+b x)}{7 b \sqrt{\csc (a+b x)}}+\frac{4}{7} \int \sqrt{\csc (a+b x)} \, dx\\ &=\frac{4 \cos (a+b x)}{7 b \sqrt{\csc (a+b x)}}+\frac{2 \cos ^3(a+b x)}{7 b \sqrt{\csc (a+b x)}}+\frac{1}{7} \left (4 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx\\ &=\frac{4 \cos (a+b x)}{7 b \sqrt{\csc (a+b x)}}+\frac{2 \cos ^3(a+b x)}{7 b \sqrt{\csc (a+b x)}}+\frac{8 \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.147228, size = 63, normalized size = 0.68 \[ \frac{\sqrt{\csc (a+b x)} \left (10 \sin (2 (a+b x))+\sin (4 (a+b x))-32 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{28 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Csc[a + b*x]]*(-32*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + 10*Sin[2*(a + b*x)] + Sin[4*
(a + b*x)]))/(28*b)

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Maple [A]  time = 1.016, size = 100, normalized size = 1.1 \begin{align*}{\frac{1}{\cos \left ( bx+a \right ) b} \left ({\frac{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{5}}{7}}-{\frac{8\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{7}}+{\frac{6\,\sin \left ( bx+a \right ) }{7}}+{\frac{4}{7}\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 ) } \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]

(2/7*sin(b*x+a)^5-8/7*sin(b*x+a)^3+6/7*sin(b*x+a)+4/7*(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)))/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 \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 (\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(-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)**4*csc(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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)