3.1.13 \(\int \frac {1}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx\) [13]

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

[Out]

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

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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 = {2716, 2720} \begin {gather*} \frac {2 \sinh (a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^(-5/2),x]

[Out]

(((-2*I)/3)*EllipticF[(I/2)*(a + b*x), 2])/b + (2*Sinh[a + b*x])/(3*b*Cosh[a + b*x]^(3/2))

Rule 2716

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

Rubi steps

\begin {align*} \int \frac {1}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx &=\frac {2 \sinh (a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}+\frac {1}{3} \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx\\ &=-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b}+\frac {2 \sinh (a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.05, size = 84, normalized size = 1.83 \begin {gather*} \frac {2 \left (\sinh (a+b x)+\cosh (a+b x) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\cosh (2 (a+b x))-\sinh (2 (a+b x))\right ) \sqrt {1+\cosh (2 (a+b x))+\sinh (2 (a+b x))}\right )}{3 b \cosh ^{\frac {3}{2}}(a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^(-5/2),x]

[Out]

(2*(Sinh[a + b*x] + Cosh[a + b*x]*Hypergeometric2F1[1/4, 1/2, 5/4, -Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]]*Sqr
t[1 + Cosh[2*(a + b*x)] + Sinh[2*(a + b*x)]]))/(3*b*Cosh[a + b*x]^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(216\) vs. \(2(66)=132\).
time = 1.07, size = 217, normalized size = 4.72

method	result	size
default	\(\frac {2 \left (2 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right ) \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{3 \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )^{\frac {3}{2}} \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) b}\)	\(217\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cosh(b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/3*(2*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*sinh(1/2*b*x+1/2*a)^2-1)^(1/2)*EllipticF(cosh(1/2*b*x+1/2*a),2^(1/2)
)*sinh(1/2*b*x+1/2*a)^2+2*cosh(1/2*b*x+1/2*a)*sinh(1/2*b*x+1/2*a)^2+(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*sinh(1/
2*b*x+1/2*a)^2-1)^(1/2)*EllipticF(cosh(1/2*b*x+1/2*a),2^(1/2)))*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*
a)^2)^(1/2)/(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/(2*cosh(1/2*b*x+1/2*a)^2-1)^(3/2)/sinh(1/2*b
*x+1/2*a)/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(1/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 = 0.12, size = 310, normalized size = 6.74 \begin {gather*} \frac {2 \, {\left ({\left (\sqrt {2} \cosh \left (b x + a\right )^{4} + 4 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sqrt {2} \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \left (b x + a\right )^{2} + \sqrt {2}\right )} \sinh \left (b x + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b x + a\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{3} + \sqrt {2} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 2 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )}\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cosh(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\cosh ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cosh(b*x+a)**(5/2),x)

[Out]

Integral(cosh(a + b*x)**(-5/2), 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(1/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 \frac {1}{{\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(1/cosh(a + b*x)^(5/2),x)

[Out]

int(1/cosh(a + b*x)^(5/2), x)

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