3.1.10 \(\int \cos ^{\frac {5}{2}}(a+b x) \, dx\) [10]

Optimal. Leaf size=42 \[ \frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b}+\frac {2 \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{5 b} \]

[Out]

6/5*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1/2*a+1/2*b*x)*EllipticE(sin(1/2*a+1/2*b*x),2^(1/2))/b+2/5*cos(b*x+a)^(3/
2)*sin(b*x+a)/b

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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2715, 2719} \begin {gather*} \frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b}+\frac {2 \sin (a+b x) \cos ^{\frac {3}{2}}(a+b x)}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^(5/2),x]

[Out]

(6*EllipticE[(a + b*x)/2, 2])/(5*b) + (2*Cos[a + b*x]^(3/2)*Sin[a + b*x])/(5*b)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2719

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

Rubi steps

\begin {align*} \int \cos ^{\frac {5}{2}}(a+b x) \, dx &=\frac {2 \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{5 b}+\frac {3}{5} \int \sqrt {\cos (a+b x)} \, dx\\ &=\frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b}+\frac {2 \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{5 b}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 40, normalized size = 0.95 \begin {gather*} \frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sqrt {\cos (a+b x)} \sin (2 (a+b x))}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^(5/2),x]

[Out]

(6*EllipticE[(a + b*x)/2, 2] + Sqrt[Cos[a + b*x]]*Sin[2*(a + b*x)])/(5*b)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(62)=124\).
time = 0.03, size = 202, normalized size = 4.81

method	result	size
default	\(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (-8 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+8 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right )}{5 \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/5*((2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*(-8*cos(1/2*b*x+1/2*a)*sin(1/2*b*x+1/2*a)^6+8*sin
(1/2*b*x+1/2*a)^4*cos(1/2*b*x+1/2*a)-2*sin(1/2*b*x+1/2*a)^2*cos(1/2*b*x+1/2*a)-3*(sin(1/2*b*x+1/2*a)^2)^(1/2)*
(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2)))/(-2*sin(1/2*b*x+1/2*a)^4+sin(1/2*b*x+1
/2*a)^2)^(1/2)/sin(1/2*b*x+1/2*a)/(2*cos(1/2*b*x+1/2*a)^2-1)^(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)^(5/2),x, algorithm="maxima")

[Out]

integrate(cos(b*x + a)^(5/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 74, normalized size = 1.76 \begin {gather*} \frac {2 \, \cos \left (b x + a\right )^{\frac {3}{2}} \sin \left (b x + a\right ) + 3 i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - 3 i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

1/5*(2*cos(b*x + a)^(3/2)*sin(b*x + a) + 3*I*sqrt(2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x
+ a) + I*sin(b*x + a))) - 3*I*sqrt(2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(
b*x + a))))/b

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3063 deep

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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)^(5/2),x, algorithm="giac")

[Out]

integrate(cos(b*x + a)^(5/2), x)

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Mupad [B]
time = 0.17, size = 42, normalized size = 1.00 \begin {gather*} -\frac {2\,{\cos \left (a+b\,x\right )}^{7/2}\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (a+b\,x\right )}^2\right )}{7\,b\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^(5/2),x)

[Out]

-(2*cos(a + b*x)^(7/2)*sin(a + b*x)*hypergeom([1/2, 7/4], 11/4, cos(a + b*x)^2))/(7*b*(sin(a + b*x)^2)^(1/2))

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