3.81 \(\int \sqrt{a+b \cosh (c+d x)} \, dx\)

Optimal. Leaf size=61 \[ -\frac{2 i \sqrt{a+b \cosh (c+d x)} E\left (\frac{1}{2} i (c+d x)|\frac{2 b}{a+b}\right )}{d \sqrt{\frac{a+b \cosh (c+d x)}{a+b}}} \]

[Out]

((-2*I)*Sqrt[a + b*Cosh[c + d*x]]*EllipticE[(I/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cosh[c + d*x])/(a
+ b)])

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Rubi [A]  time = 0.0394434, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2655, 2653} \[ -\frac{2 i \sqrt{a+b \cosh (c+d x)} E\left (\frac{1}{2} i (c+d x)|\frac{2 b}{a+b}\right )}{d \sqrt{\frac{a+b \cosh (c+d x)}{a+b}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Cosh[c + d*x]],x]

[Out]

((-2*I)*Sqrt[a + b*Cosh[c + d*x]]*EllipticE[(I/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cosh[c + d*x])/(a
+ b)])

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \sqrt{a+b \cosh (c+d x)} \, dx &=\frac{\sqrt{a+b \cosh (c+d x)} \int \sqrt{\frac{a}{a+b}+\frac{b \cosh (c+d x)}{a+b}} \, dx}{\sqrt{\frac{a+b \cosh (c+d x)}{a+b}}}\\ &=-\frac{2 i \sqrt{a+b \cosh (c+d x)} E\left (\frac{1}{2} i (c+d x)|\frac{2 b}{a+b}\right )}{d \sqrt{\frac{a+b \cosh (c+d x)}{a+b}}}\\ \end{align*}

Mathematica [A]  time = 0.0850281, size = 61, normalized size = 1. \[ -\frac{2 i \sqrt{a+b \cosh (c+d x)} E\left (\frac{1}{2} i (c+d x)|\frac{2 b}{a+b}\right )}{d \sqrt{\frac{a+b \cosh (c+d x)}{a+b}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Cosh[c + d*x]],x]

[Out]

((-2*I)*Sqrt[a + b*Cosh[c + d*x]]*EllipticE[(I/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cosh[c + d*x])/(a
+ b)])

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Maple [B]  time = 0.062, size = 276, normalized size = 4.5 \begin{align*} 2\,{\frac{\sqrt{- \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{ \left ( 2\, \left ( \cosh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a-b \right ) \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}{\sqrt{2\,b \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sinh \left ( 1/2\,dx+c/2 \right ) \sqrt{2\, \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b}d} \left ( a{\it EllipticF} \left ( \cosh \left ( 1/2\,dx+c/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) +b{\it EllipticF} \left ( \cosh \left ( 1/2\,dx+c/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) -2\,b{\it EllipticE} \left ( \cosh \left ( 1/2\,dx+c/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) \right ) \sqrt{{\frac{2\, \left ( \cosh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a-b}{a-b}}}{\frac{1}{\sqrt{-2\,{\frac{b}{a-b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a*EllipticF(cosh(1/2*d*x+1/2*c)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+b*EllipticF(cosh(1/2*d*x+1/2*c)*
(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))-2*b*EllipticE(cosh(1/2*d*x+1/2*c)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/
b)^(1/2)))*(-sinh(1/2*d*x+1/2*c)^2)^(1/2)*((2*cosh(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)*((2*cosh(1/2*d*x+1/2*c
)^2*b+a-b)*sinh(1/2*d*x+1/2*c)^2)^(1/2)/(-2*b/(a-b))^(1/2)/(2*b*sinh(1/2*d*x+1/2*c)^4+(a+b)*sinh(1/2*d*x+1/2*c
)^2)^(1/2)/sinh(1/2*d*x+1/2*c)/(2*sinh(1/2*d*x+1/2*c)^2*b+a+b)^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cosh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*cosh(d*x + c) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*cosh(d*x + c) + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \cosh{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*cosh(c + d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*cosh(d*x + c) + a), x)