\(\int \sqrt {\cosh (a+b x)} \, dx\) [10]

Optimal result

Integrand size = 10, antiderivative size = 20 \[ \int \sqrt {\cosh (a+b x)} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2719} \[ \int \sqrt {\cosh (a+b x)} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b} \]

[In]

Int[Sqrt[Cosh[a + b*x]],x]

[Out]

((-2*I)*EllipticE[(I/2)*(a + b*x), 2])/b

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \sqrt {\cosh (a+b x)} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b} \]

[In]

Integrate[Sqrt[Cosh[a + b*x]],x]

[Out]

((-2*I)*EllipticE[(I/2)*(a + b*x), 2])/b

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(134\) vs. \(2(46)=92\).

Time = 0.38 (sec) , antiderivative size = 135, normalized size of antiderivative = 6.75

method	result	size
default	\(-\frac {2 \sqrt {\left (-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, b}\)	\(135\)
risch	\(\frac {\sqrt {2}\, \sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}}{b}+\frac {\left (-\frac {2 \left (1+{\mathrm e}^{2 b x +2 a}\right )}{\sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}}+\frac {i \sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{b x +a}-i\right )}\, \sqrt {i {\mathrm e}^{b x +a}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}+{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}\, \sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )}\)	\(230\)

[In]

int(cosh(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \sqrt {\cosh (a+b x)} \, dx=-\frac {2 \, {\left (\sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right ) + \sqrt {\cosh \left (b x + a\right )}\right )}}{b} \]

[In]

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

[Out]

-2*(sqrt(2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(b*x + a) + sinh(b*x + a))) + sqrt(cosh(b*x
+ a)))/b

Sympy [F]

\[ \int \sqrt {\cosh (a+b x)} \, dx=\int \sqrt {\cosh {\left (a + b x \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(cosh(a + b*x)), x)

Maxima [F]

\[ \int \sqrt {\cosh (a+b x)} \, dx=\int { \sqrt {\cosh \left (b x + a\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(cosh(b*x + a)), x)

Giac [F]

\[ \int \sqrt {\cosh (a+b x)} \, dx=\int { \sqrt {\cosh \left (b x + a\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(cosh(b*x + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\cosh (a+b x)} \, dx=\int \sqrt {\mathrm {cosh}\left (a+b\,x\right )} \,d x \]

[In]

int(cosh(a + b*x)^(1/2),x)

[Out]

int(cosh(a + b*x)^(1/2), x)