\(\int \sqrt {a+b \cosh ^n(x)} \tanh (x) \, dx\) [42]

Optimal result

Integrand size = 15, antiderivative size = 47 \[ \int \sqrt {a+b \cosh ^n(x)} \tanh (x) \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^n(x)}}{\sqrt {a}}\right )}{n}+\frac {2 \sqrt {a+b \cosh ^n(x)}}{n} \] Output:

-2*a^(1/2)*arctanh((a+b*cosh(x)^n)^(1/2)/a^(1/2))/n+2*(a+b*cosh(x)^n)^(1/2 
)/n
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96 \[ \int \sqrt {a+b \cosh ^n(x)} \tanh (x) \, dx=\frac {-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^n(x)}}{\sqrt {a}}\right )+2 \sqrt {a+b \cosh ^n(x)}}{n} \] Input:

Integrate[Sqrt[a + b*Cosh[x]^n]*Tanh[x],x]
 

Output:

(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^n]/Sqrt[a]] + 2*Sqrt[a + b*Cosh[x]^ 
n])/n
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3042, 26, 3709, 798, 60, 73, 221}

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.

\(\displaystyle \int \tanh (x) \sqrt {a+b \cosh ^n(x)} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3709

\(\displaystyle \int \text {sech}(x) \sqrt {a+b \cosh ^n(x)}d\cosh (x)\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {\int \sqrt {b \cosh ^n(x)+a} \text {sech}(x)d\cosh ^n(x)}{n}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {a \int \frac {\text {sech}(x)}{\sqrt {b \cosh ^n(x)+a}}d\cosh ^n(x)+2 \sqrt {a+b \cosh ^n(x)}}{n}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {2 a \int \frac {1}{\frac {\cosh ^{2 n}(x)}{b}-\frac {a}{b}}d\sqrt {b \cosh ^n(x)+a}}{b}+2 \sqrt {a+b \cosh ^n(x)}}{n}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \sqrt {a+b \cosh ^n(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^n(x)}}{\sqrt {a}}\right )}{n}\)

Input:

Int[Sqrt[a + b*Cosh[x]^n]*Tanh[x],x]
 

Output:

(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^n]/Sqrt[a]] + 2*Sqrt[a + b*Cosh[x]^ 
n])/n
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 3042

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

rule 3709

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( 
f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si 
mp[ff^(m + 1)/f   Subst[Int[x^m*((a + b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 
1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && 
ILtQ[(m - 1)/2, 0]
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {2 \sqrt {a +b \cosh \left (x \right )^{n}}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \cosh \left (x \right )^{n}}}{\sqrt {a}}\right )}{n}\)	\(38\)
default	\(\frac {2 \sqrt {a +b \cosh \left (x \right )^{n}}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \cosh \left (x \right )^{n}}}{\sqrt {a}}\right )}{n}\)	\(38\)

Input:

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

Output:

1/n*(2*(a+b*cosh(x)^n)^(1/2)-2*a^(1/2)*arctanh((a+b*cosh(x)^n)^(1/2)/a^(1/ 
2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 3.26 \[ \int \sqrt {a+b \cosh ^n(x)} \tanh (x) \, dx=\left [\frac {\sqrt {a} \log \left (\frac {b \cosh \left (n \log \left (\cosh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\cosh \left (x\right )\right )\right ) - 2 \, \sqrt {b \cosh \left (n \log \left (\cosh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\cosh \left (x\right )\right )\right ) + a} \sqrt {a} + 2 \, a}{\cosh \left (n \log \left (\cosh \left (x\right )\right )\right ) + \sinh \left (n \log \left (\cosh \left (x\right )\right )\right )}\right ) + 2 \, \sqrt {b \cosh \left (n \log \left (\cosh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\cosh \left (x\right )\right )\right ) + a}}{n}, \frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b \cosh \left (n \log \left (\cosh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\cosh \left (x\right )\right )\right ) + a}}\right ) + \sqrt {b \cosh \left (n \log \left (\cosh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\cosh \left (x\right )\right )\right ) + a}\right )}}{n}\right ] \] Input:

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

Output:

[(sqrt(a)*log((b*cosh(n*log(cosh(x))) + b*sinh(n*log(cosh(x))) - 2*sqrt(b* 
cosh(n*log(cosh(x))) + b*sinh(n*log(cosh(x))) + a)*sqrt(a) + 2*a)/(cosh(n* 
log(cosh(x))) + sinh(n*log(cosh(x))))) + 2*sqrt(b*cosh(n*log(cosh(x))) + b 
*sinh(n*log(cosh(x))) + a))/n, 2*(sqrt(-a)*arctan(sqrt(-a)/sqrt(b*cosh(n*l 
og(cosh(x))) + b*sinh(n*log(cosh(x))) + a)) + sqrt(b*cosh(n*log(cosh(x))) 
+ b*sinh(n*log(cosh(x))) + a))/n]
 

Sympy [F]

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

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

Output:

Integral(sqrt(a + b*cosh(x)**n)*tanh(x), x)
 

Maxima [F]

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

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

Output:

integrate(sqrt(b*cosh(x)^n + a)*tanh(x), x)
 

Giac [F]

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

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

Output:

integrate(sqrt(b*cosh(x)^n + a)*tanh(x), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(cosh(x)**n*b + a)*tanh(x),x)