3.9 \(\int \cosh ^{\frac{3}{2}}(a+b x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0194128, 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 = {2635, 2641} \[ \frac{2 \sinh (a+b x) \sqrt{\cosh (a+b x)}}{3 b}-\frac{2 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^(3/2),x]

[Out]

(((-2*I)/3)*EllipticF[(I/2)*(a + b*x), 2])/b + (2*Sqrt[Cosh[a + b*x]]*Sinh[a + b*x])/(3*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 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 \cosh ^{\frac{3}{2}}(a+b x) \, dx &=\frac{2 \sqrt{\cosh (a+b x)} \sinh (a+b x)}{3 b}+\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 \sqrt{\cosh (a+b x)} \sinh (a+b x)}{3 b}\\ \end{align*}

Mathematica [C]  time = 0.0939686, size = 81, normalized size = 1.76 \[ \frac{2 \sqrt{\sinh (2 (a+b x))+\cosh (2 (a+b x))+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\cosh (2 (a+b x))-\sinh (2 (a+b x))\right )+\sinh (2 (a+b x))}{3 b \sqrt{\cosh (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^(3/2),x]

[Out]

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

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Maple [B]  time = 0.033, size = 174, normalized size = 3.8 \begin{align*}{\frac{2}{3\,b}\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}} \left ( 4\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{5}-6\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}+\sqrt{- \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \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 ) \right ){\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 ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^(3/2),x)

[Out]

2/3*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(4*cosh(1/2*b*x+1/2*a)^5-6*cosh(1/2*b*x+1/2*a)^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)*EllipticF(cosh(1/2*b*x+1/2*a),2^(1/2))+2*co
sh(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., size = 0, normalized size = 0. \begin{align*} \int \cosh \left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(cosh(b*x + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cosh \left (b x + a\right )^{\frac{3}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

integral(cosh(b*x + a)^(3/2), 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(cosh(b*x+a)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh \left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate(cosh(b*x + a)^(3/2), x)