3.1.8 \(\int \cosh ^{\frac {5}{2}}(a+b x) \, dx\) [8]

Optimal. Leaf size=46 \[ -\frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b}+\frac {2 \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{5 b} \]

[Out]

-6/5*I*(cosh(1/2*a+1/2*b*x)^2)^(1/2)/cosh(1/2*a+1/2*b*x)*EllipticE(I*sinh(1/2*a+1/2*b*x),2^(1/2))/b+2/5*cosh(b
*x+a)^(3/2)*sinh(b*x+a)/b

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Rubi [A]
time = 0.02, antiderivative size = 46, 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 {2 \sinh (a+b x) \cosh ^{\frac {3}{2}}(a+b x)}{5 b}-\frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^(5/2),x]

[Out]

(((-6*I)/5)*EllipticE[(I/2)*(a + b*x), 2])/b + (2*Cosh[a + b*x]^(3/2)*Sinh[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 \cosh ^{\frac {5}{2}}(a+b x) \, dx &=\frac {2 \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{5 b}+\frac {3}{5} \int \sqrt {\cosh (a+b x)} \, dx\\ &=-\frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b}+\frac {2 \cosh ^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{5 b}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 44, normalized size = 0.96 \begin {gather*} \frac {-6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )+\sqrt {\cosh (a+b x)} \sinh (2 (a+b x))}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^(5/2),x]

[Out]

((-6*I)*EllipticE[(I/2)*(a + b*x), 2] + Sqrt[Cosh[a + b*x]]*Sinh[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. \(187\) vs. \(2(66)=132\).
time = 1.02, size = 188, normalized size = 4.09

method	result	size
default	\(\frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (8 \left (\cosh ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-16 \left (\cosh ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+10 \left (\cosh ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-3 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticE \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b}\)	\(188\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(8*cosh(1/2*b*x+1/2*a)^7-16*cosh(1/2*b*x+1/2*a)^
5+10*cosh(1/2*b*x+1/2*a)^3-3*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1/2*a)^2+1)^(1/2)*EllipticE(cosh(
1/2*b*x+1/2*a),2^(1/2))-2*cosh(1/2*b*x+1/2*a))/(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/sinh(1/2*
b*x+1/2*a)/(2*cosh(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(cosh(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(cosh(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 = 0.13, size = 203, normalized size = 4.41 \begin {gather*} -\frac {12 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right ) - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 6 \, {\left (\cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )^{2} - 12 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - 6 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 1\right )} \sqrt {\cosh \left (b x + a\right )}}{10 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(12*(sqrt(2)*cosh(b*x + a)^2 + 2*sqrt(2)*cosh(b*x + a)*sinh(b*x + a) + sqrt(2)*sinh(b*x + a)^2)*weierstr
assZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(b*x + a) + sinh(b*x + a))) - (cosh(b*x + a)^4 + 4*cosh(b*x + a)
*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 6*(cosh(b*x + a)^2 - 2)*sinh(b*x + a)^2 - 12*cosh(b*x + a)^2 + 4*(cosh(b*
x + a)^3 - 6*cosh(b*x + a))*sinh(b*x + a) - 1)*sqrt(cosh(b*x + a)))/(b*cosh(b*x + a)^2 + 2*b*cosh(b*x + a)*sin
h(b*x + a) + b*sinh(b*x + a)^2)

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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(cosh(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(cosh(b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(cosh(b*x + a)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {cosh}\left (a+b\,x\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^(5/2),x)

[Out]

int(cosh(a + b*x)^(5/2), x)

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