Integrand size = 15, antiderivative size = 29 \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh ^n(x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^n(x)}}{\sqrt {a}}\right )}{\sqrt {a} n} \] Output:
-2*arctanh((a+b*cosh(x)^n)^(1/2)/a^(1/2))/a^(1/2)/n
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh ^n(x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^n(x)}}{\sqrt {a}}\right )}{\sqrt {a} n} \] Input:
Integrate[Tanh[x]/Sqrt[a + b*Cosh[x]^n],x]
Output:
(-2*ArcTanh[Sqrt[a + b*Cosh[x]^n]/Sqrt[a]])/(Sqrt[a]*n)
Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 26, 3709, 798, 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.
\(\displaystyle \int \frac {\tanh (x)}{\sqrt {a+b \cosh ^n(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\tan \left (\frac {\pi }{2}+i x\right ) \sqrt {a+b \sin \left (\frac {\pi }{2}+i x\right )^n}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\sqrt {b \sin \left (i x+\frac {\pi }{2}\right )^n+a} \tan \left (i x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3709 |
\(\displaystyle \int \frac {\text {sech}(x)}{\sqrt {a+b \cosh ^n(x)}}d\cosh (x)\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \frac {\text {sech}(x)}{\sqrt {b \cosh ^n(x)+a}}d\cosh ^n(x)}{n}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 \int \frac {1}{\frac {\cosh ^{2 n}(x)}{b}-\frac {a}{b}}d\sqrt {b \cosh ^n(x)+a}}{b n}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^n(x)}}{\sqrt {a}}\right )}{\sqrt {a} n}\) |
Input:
Int[Tanh[x]/Sqrt[a + b*Cosh[x]^n],x]
Output:
(-2*ArcTanh[Sqrt[a + b*Cosh[x]^n]/Sqrt[a]])/(Sqrt[a]*n)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
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]
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]]
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]
Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \cosh \left (x \right )^{n}}}{\sqrt {a}}\right )}{\sqrt {a}\, n}\) | \(24\) |
default | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \cosh \left (x \right )^{n}}}{\sqrt {a}}\right )}{\sqrt {a}\, n}\) | \(24\) |
Input:
int(tanh(x)/(a+b*cosh(x)^n)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2*arctanh((a+b*cosh(x)^n)^(1/2)/a^(1/2))/a^(1/2)/n
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.79 \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh ^n(x)}} \, dx=\left [\frac {\log \left (\frac {b \cosh \left (n \log \left (\cosh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\cosh \left (x\right )\right )\right ) - 2 \, \sqrt {b \cosh \left (n \log \left (\cosh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\cosh \left (x\right )\right )\right ) + a} \sqrt {a} + 2 \, a}{\cosh \left (n \log \left (\cosh \left (x\right )\right )\right ) + \sinh \left (n \log \left (\cosh \left (x\right )\right )\right )}\right )}{\sqrt {a} n}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b \cosh \left (n \log \left (\cosh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\cosh \left (x\right )\right )\right ) + a}}\right )}{a n}\right ] \] Input:
integrate(tanh(x)/(a+b*cosh(x)^n)^(1/2),x, algorithm="fricas")
Output:
[log((b*cosh(n*log(cosh(x))) + b*sinh(n*log(cosh(x))) - 2*sqrt(b*cosh(n*lo g(cosh(x))) + b*sinh(n*log(cosh(x))) + a)*sqrt(a) + 2*a)/(cosh(n*log(cosh( x))) + sinh(n*log(cosh(x)))))/(sqrt(a)*n), 2*sqrt(-a)*arctan(sqrt(-a)/sqrt (b*cosh(n*log(cosh(x))) + b*sinh(n*log(cosh(x))) + a))/(a*n)]
\[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh ^n(x)}} \, dx=\int \frac {\tanh {\left (x \right )}}{\sqrt {a + b \cosh ^{n}{\left (x \right )}}}\, dx \] Input:
integrate(tanh(x)/(a+b*cosh(x)**n)**(1/2),x)
Output:
Integral(tanh(x)/sqrt(a + b*cosh(x)**n), x)
\[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh ^n(x)}} \, dx=\int { \frac {\tanh \left (x\right )}{\sqrt {b \cosh \left (x\right )^{n} + a}} \,d x } \] Input:
integrate(tanh(x)/(a+b*cosh(x)^n)^(1/2),x, algorithm="maxima")
Output:
integrate(tanh(x)/sqrt(b*cosh(x)^n + a), x)
\[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh ^n(x)}} \, dx=\int { \frac {\tanh \left (x\right )}{\sqrt {b \cosh \left (x\right )^{n} + a}} \,d x } \] Input:
integrate(tanh(x)/(a+b*cosh(x)^n)^(1/2),x, algorithm="giac")
Output:
integrate(tanh(x)/sqrt(b*cosh(x)^n + a), x)
Timed out. \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh ^n(x)}} \, dx=\int \frac {\mathrm {tanh}\left (x\right )}{\sqrt {a+b\,{\mathrm {cosh}\left (x\right )}^n}} \,d x \] Input:
int(tanh(x)/(a + b*cosh(x)^n)^(1/2),x)
Output:
int(tanh(x)/(a + b*cosh(x)^n)^(1/2), x)
\[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh ^n(x)}} \, dx=\int \frac {\sqrt {\cosh \left (x \right )^{n} b +a}\, \tanh \left (x \right )}{\cosh \left (x \right )^{n} b +a}d x \] Input:
int(tanh(x)/(a+b*cosh(x)^n)^(1/2),x)
Output:
int((sqrt(cosh(x)**n*b + a)*tanh(x))/(cosh(x)**n*b + a),x)