3.8 \(\int \cosh ^{\frac {5}{2}}(a+b x) \, dx\)

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

[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 = {2635, 2639} \[ \frac {2 \sinh (a+b x) \cosh ^{\frac {3}{2}}(a+b x)}{5 b}-\frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b} \]

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 2635

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] && Integer
Q[2*n]

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 \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 = 0.06, size = 44, normalized size = 0.96 \[ \frac {\sinh (2 (a+b x)) \sqrt {\cosh (a+b x)}-6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b} \]

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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fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b x + a\right )^{\frac {5}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cosh(b*x + a)^(5/2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh \left (b x + a\right )^{\frac {5}{2}}\,{d x} \]

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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maple [B]  time = 0.30, size = 188, normalized size = 4.09 \[ \frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (8 \left (\cosh ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-16 \left (\cosh ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+10 \left (\cosh ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-3 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticE \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \[ \int \cosh \left (b x + a\right )^{\frac {5}{2}}\,{d x} \]

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

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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