3.13 \(\int \frac {1}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx\)

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

[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 = {2636, 2641} \[ \frac {2 \sinh (a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b} \]

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 2636

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 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, 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]  time = 0.07, size = 84, normalized size = 1.83 \[ \frac {2 \left (\cosh (a+b x) \sqrt {\sinh (2 (a+b x))+\cosh (2 (a+b x))+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\cosh (2 (a+b x))-\sinh (2 (a+b x))\right )+\sinh (a+b x)\right )}{3 b \cosh ^{\frac {3}{2}}(a+b x)} \]

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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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\cosh \left (b x + a\right )^{\frac {5}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/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 \[ \int \frac {1}{\cosh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]

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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maple [B]  time = 0.33, size = 217, normalized size = 4.72 \[ \frac {2 \left (2 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right ) \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{3 \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )^{\frac {3}{2}} \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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+(-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)*sinh(1/2*b*x+1/2*a)^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 \[ \int \frac {1}{\cosh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]

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

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

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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