\(\int x \sqrt {a+a \cosh (x)} \, dx\) [129]

Optimal result

Integrand size = 12, antiderivative size = 32 \[ \int x \sqrt {a+a \cosh (x)} \, dx=-4 \sqrt {a+a \cosh (x)}+2 x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right ) \] Output:

-4*(a+a*cosh(x))^(1/2)+2*x*(a+a*cosh(x))^(1/2)*tanh(1/2*x)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int x \sqrt {a+a \cosh (x)} \, dx=2 \sqrt {a (1+\cosh (x))} \left (-2+x \tanh \left (\frac {x}{2}\right )\right ) \] Input:

Integrate[x*Sqrt[a + a*Cosh[x]],x]
 

Output:

2*Sqrt[a*(1 + Cosh[x])]*(-2 + x*Tanh[x/2])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3800, 3042, 3777, 26, 3042, 26, 3118}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \sqrt {a \cosh (x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x \sqrt {a+a \sin \left (\frac {\pi }{2}+i x\right )}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \int x \cosh \left (\frac {x}{2}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \int x \sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx\)

\(\Big \downarrow \) 3777

\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x \sinh \left (\frac {x}{2}\right )-2 i \int -i \sinh \left (\frac {x}{2}\right )dx\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x \sinh \left (\frac {x}{2}\right )-2 \int \sinh \left (\frac {x}{2}\right )dx\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x \sinh \left (\frac {x}{2}\right )-2 \int -i \sin \left (\frac {i x}{2}\right )dx\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x \sinh \left (\frac {x}{2}\right )+2 i \int \sin \left (\frac {i x}{2}\right )dx\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (2 x \sinh \left (\frac {x}{2}\right )-4 \cosh \left (\frac {x}{2}\right )\right )\)

Input:

Int[x*Sqrt[a + a*Cosh[x]],x]
 

Output:

Sqrt[a + a*Cosh[x]]*Sech[x/2]*(-4*Cosh[x/2] + 2*x*Sinh[x/2])
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3118

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

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e 
/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n]))   Int[(c + d*x)^m*Sin[e/2 + a 
*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19

method	result	size
risch	\(\frac {\sqrt {2}\, \sqrt {a \left ({\mathrm e}^{x}+1\right )^{2} {\mathrm e}^{-x}}\, \left (x \,{\mathrm e}^{x}-x -2 \,{\mathrm e}^{x}-2\right )}{{\mathrm e}^{x}+1}\)	\(38\)

Input:

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

Output:

2^(1/2)*(a*(exp(x)+1)^2*exp(-x))^(1/2)/(exp(x)+1)*(x*exp(x)-x-2*exp(x)-2)
 

Fricas [F(-2)]

Exception generated. \[ \int x \sqrt {a+a \cosh (x)} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

\[ \int x \sqrt {a+a \cosh (x)} \, dx=\int x \sqrt {a \left (\cosh {\left (x \right )} + 1\right )}\, dx \] Input:

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

Output:

Integral(x*sqrt(a*(cosh(x) + 1)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int x \sqrt {a+a \cosh (x)} \, dx=-{\left (\sqrt {2} \sqrt {a} x - {\left (\sqrt {2} \sqrt {a} x - 2 \, \sqrt {2} \sqrt {a}\right )} e^{x} + 2 \, \sqrt {2} \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, x\right )} \] Input:

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

Output:

-(sqrt(2)*sqrt(a)*x - (sqrt(2)*sqrt(a)*x - 2*sqrt(2)*sqrt(a))*e^x + 2*sqrt 
(2)*sqrt(a))*e^(-1/2*x)
 

Giac [F]

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

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

Output:

integrate(sqrt(a*cosh(x) + a)*x, x)
 

Mupad [B] (verification not implemented)

Time = 1.81 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int x \sqrt {a+a \cosh (x)} \, dx=-\frac {\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}\,\left (2\,x+4\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x+4\right )}{{\mathrm {e}}^x+1} \] Input:

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

Output:

-((a + a*(exp(-x)/2 + exp(x)/2))^(1/2)*(2*x + 4*exp(x) - 2*x*exp(x) + 4))/ 
(exp(x) + 1)
 

Reduce [F]

\[ \int x \sqrt {a+a \cosh (x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\cosh \left (x \right )+1}\, x d x \right ) \] Input:

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

Output:

sqrt(a)*int(sqrt(cosh(x) + 1)*x,x)