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

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Rubi [A]  time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.09, size = 61, normalized size = 1.00 \[ -\frac {2 i \sqrt {a+b \cosh (c+d x)} E\left (\frac {1}{2} i (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cosh (c+d x)}{a+b}}} \]

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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fricas [F]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \cosh \left (d x + c\right ) + a}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cosh \left (d x + c\right ) + a}\,{d x} \]

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)

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maple [B]  time = 0.43, size = 276, normalized size = 4.52 \[ \frac {2 \left (a \EllipticF \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )+b \EllipticF \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )-2 b \EllipticE \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )\right ) \sqrt {-\left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {2 b \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, \sqrt {\left (2 b \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b \right ) \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{\sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 b \left (\sinh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a +b \right ) \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 b \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}\, d} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cosh \left (d x + c\right ) + a}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cosh(c + d*x))^(1/2),x)

[Out]

int((a + b*cosh(c + d*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \cosh {\left (c + d x \right )}}\, dx \]

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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