3.3.68 \(\int \cos ^4(a+b x) \sqrt {\csc (a+b x)} \, dx\) [268]

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

[Out]

4/7*cos(b*x+a)/b/csc(b*x+a)^(1/2)+2/7*cos(b*x+a)^3/b/csc(b*x+a)^(1/2)-8/7*(sin(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/2)/
sin(1/2*a+1/4*Pi+1/2*b*x)*EllipticF(cos(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))*csc(b*x+a)^(1/2)*sin(b*x+a)^(1/2)/b

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Rubi [A]
time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, 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 = {2708, 3856, 2720} \begin {gather*} \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} \end {gather*}

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 2708

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] && Integers
Q[2*m, 2*n]

Rule 2720

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

Rule 3856

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]

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*}

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Mathematica [A]
time = 0.09, size = 63, normalized size = 0.68 \begin {gather*} \frac {\sqrt {\csc (a+b x)} \left (-32 F\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sqrt {\sin (a+b x)}+10 \sin (2 (a+b x))+\sin (4 (a+b x))\right )}{28 b} \end {gather*}

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 = 0.10, size = 100, normalized size = 1.09

method	result	size
default	\(\frac {\frac {2 \left (\sin ^{5}\left (b x +a \right )\right )}{7}-\frac {8 \left (\sin ^{3}\left (b x +a \right )\right )}{7}+\frac {6 \sin \left (b x +a \right )}{7}+\frac {4 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{7}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 78, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left ({\left (\cos \left (b x + a\right )^{3} + 2 \, \cos \left (b x + a\right )\right )} \sqrt {\sin \left (b x + a\right )} - 2 i \, \sqrt {2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 2 i \, \sqrt {-2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}}{7 \, b} \end {gather*}

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]

2/7*((cos(b*x + a)^3 + 2*cos(b*x + a))*sqrt(sin(b*x + a)) - 2*I*sqrt(2*I)*weierstrassPInverse(4, 0, cos(b*x +
a) + I*sin(b*x + a)) + 2*I*sqrt(-2*I)*weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b*x + a)))/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cos ^{4}{\left (a + b x \right )} \sqrt {\csc {\left (a + b x \right )}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\cos \left (a+b\,x\right )}^4\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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