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

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

[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))

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Rubi [A]  time = 0.0191879, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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*}

Mathematica [C]  time = 0.0636724, 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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Maple [B]  time = 0.06, size = 217, normalized size = 4.7 \begin{align*}{\frac{2}{3\,b} \left ( 2\,\sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+\sqrt{- \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}\sqrt{-2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cosh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) ,\sqrt{2} \right ) +2\,\cosh \left ( 1/2\,bx+a/2 \right ) \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2} \right ) \sqrt{ \left ( 2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}{\frac{1}{\sqrt{2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}}} \left ( 2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{-{\frac{3}{2}}} \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cosh \left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\cosh \left (b x + a\right )^{\frac{5}{2}}}, x\right ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cosh \left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

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)