\(\int \sqrt {\sinh (a+\frac {2 \log (c x^n)}{n})} \, dx\) [286]

Optimal result

Integrand size = 18, antiderivative size = 103 \[ \int \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\frac {1}{2} x \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}+\frac {e^{-a} x \left (c x^n\right )^{-2/n} \csc ^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}}{2 \sqrt {1-e^{-2 a} \left (c x^n\right )^{-4/n}}} \] Output:

1/2*x*sinh(a+2*ln(c*x^n)/n)^(1/2)+1/2*x*arccsc(exp(a)*(c*x^n)^(2/n))*sinh( 
a+2*ln(c*x^n)/n)^(1/2)/exp(a)/((c*x^n)^(2/n))/(1-1/exp(2*a)/((c*x^n)^(4/n) 
))^(1/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.72 \[ \int \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\frac {1}{2} x \left (1-\frac {\arctan \left (\sqrt {-1+e^{2 a} \left (c x^n\right )^{4/n}}\right )}{\sqrt {-1+e^{2 a} \left (c x^n\right )^{4/n}}}\right ) \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \] Input:

Integrate[Sqrt[Sinh[a + (2*Log[c*x^n])/n]],x]
 

Output:

(x*(1 - ArcTan[Sqrt[-1 + E^(2*a)*(c*x^n)^(4/n)]]/Sqrt[-1 + E^(2*a)*(c*x^n) 
^(4/n)])*Sqrt[Sinh[a + (2*Log[c*x^n])/n]])/2
 

Rubi [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6051, 6059, 868, 773, 247, 223}

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.

\(\displaystyle \int \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx\)

\(\Big \downarrow \) 6051

\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 6059

\(\displaystyle \frac {x \left (c x^n\right )^{-2/n} \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \int \left (c x^n\right )^{\frac {2}{n}-1} \sqrt {1-e^{-2 a} \left (c x^n\right )^{-4/n}}d\left (c x^n\right )}{n \sqrt {1-e^{-2 a} \left (c x^n\right )^{-4/n}}}\)

\(\Big \downarrow \) 868

\(\displaystyle \frac {x \left (c x^n\right )^{-2/n} \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \int \sqrt {1-\frac {e^{-2 a} x^{-2 n}}{c^2}}d\left (c x^n\right )^{2/n}}{2 \sqrt {1-e^{-2 a} \left (c x^n\right )^{-4/n}}}\)

\(\Big \downarrow \) 773

\(\displaystyle -\frac {x \left (c x^n\right )^{-2/n} \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \int \frac {x^{-2 n} \sqrt {1-c^2 e^{-2 a} x^{2 n}}}{c^2}d\frac {x^{-n}}{c}}{2 \sqrt {1-e^{-2 a} \left (c x^n\right )^{-4/n}}}\)

\(\Big \downarrow \) 247

\(\displaystyle -\frac {x \left (c x^n\right )^{-2/n} \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \left (-e^{-2 a} \int \frac {1}{\sqrt {1-c^2 e^{-2 a} x^{2 n}}}d\frac {x^{-n}}{c}-\frac {x^{-n} \sqrt {1-e^{-2 a} c^2 x^{2 n}}}{c}\right )}{2 \sqrt {1-e^{-2 a} \left (c x^n\right )^{-4/n}}}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {x \left (c x^n\right )^{-2/n} \left (-e^{-a} \arcsin \left (\frac {e^{-a} x^{-n}}{c}\right )-\frac {x^{-n} \sqrt {1-e^{-2 a} c^2 x^{2 n}}}{c}\right ) \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}}{2 \sqrt {1-e^{-2 a} \left (c x^n\right )^{-4/n}}}\)

Input:

Int[Sqrt[Sinh[a + (2*Log[c*x^n])/n]],x]
 

Output:

-1/2*(x*(-(Sqrt[1 - (c^2*x^(2*n))/E^(2*a)]/(c*x^n)) - ArcSin[1/(c*E^a*x^n) 
]/E^a)*Sqrt[Sinh[a + (2*Log[c*x^n])/n]])/((c*x^n)^(2/n)*Sqrt[1 - 1/(E^(2*a 
)*(c*x^n)^(4/n))])
 

Defintions of rubi rules used

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, 
m, p, x]
 

rule 773

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^ 
2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] &&  !IntegerQ[p]
 

rule 868

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

rule 6051

Int[Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> S 
imp[x/(n*(c*x^n)^(1/n))   Subst[Int[x^(1/n - 1)*Sinh[d*(a + b*Log[x])]^p, x 
], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1] 
)
 

rule 6059

Int[((e_.)*(x_))^(m_.)*Sinh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_), x_Symbol] 
:> Simp[Sinh[d*(a + b*Log[x])]^p/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p 
)   Int[(e*x)^m*x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p, x], x] /; FreeQ[ 
{a, b, d, e, m, p}, x] &&  !IntegerQ[p]
 

Maple [F]

\[\int \sqrt {\sinh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )}d x\]

Input:

int(sinh(a+2*ln(c*x^n)/n)^(1/2),x)
 

Output:

int(sinh(a+2*ln(c*x^n)/n)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.14 \[ \int \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\frac {1}{4} \, {\left (2 \, \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 1}{x^{2}}} e^{\left (\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )} - \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 1}{x^{2}}}\right ) e^{\left (\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )}\right )} e^{\left (-\frac {a n + 2 \, \log \left (c\right )}{n}\right )} \] Input:

integrate(sinh(a+2*log(c*x^n)/n)^(1/2),x, algorithm="fricas")
 

Output:

1/4*(2*sqrt(1/2)*x*sqrt((x^4*e^(2*(a*n + 2*log(c))/n) - 1)/x^2)*e^(1/2*(a* 
n + 2*log(c))/n) - sqrt(2)*arctan(sqrt(2)*sqrt(1/2)*x*sqrt((x^4*e^(2*(a*n 
+ 2*log(c))/n) - 1)/x^2))*e^(1/2*(a*n + 2*log(c))/n))*e^(-(a*n + 2*log(c)) 
/n)
 

Sympy [F]

\[ \int \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\int \sqrt {\sinh {\left (a + \frac {2 \log {\left (c x^{n} \right )}}{n} \right )}}\, dx \] Input:

integrate(sinh(a+2*ln(c*x**n)/n)**(1/2),x)
 

Output:

Integral(sqrt(sinh(a + 2*log(c*x**n)/n)), x)
 

Maxima [F]

\[ \int \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\int { \sqrt {\sinh \left (a + \frac {2 \, \log \left (c x^{n}\right )}{n}\right )} \,d x } \] Input:

integrate(sinh(a+2*log(c*x^n)/n)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(sinh(a + 2*log(c*x^n)/n)), x)
 

Giac [F]

\[ \int \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\int { \sqrt {\sinh \left (a + \frac {2 \, \log \left (c x^{n}\right )}{n}\right )} \,d x } \] Input:

integrate(sinh(a+2*log(c*x^n)/n)^(1/2),x, algorithm="giac")
 

Output:

sage0*x
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\int \sqrt {\mathrm {sinh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )} \,d x \] Input:

int(sinh(a + (2*log(c*x^n))/n)^(1/2),x)
 

Output:

int(sinh(a + (2*log(c*x^n))/n)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\sqrt {\sinh \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right )+a n}{n}\right )}\, x -\left (\int \frac {\sqrt {\sinh \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right )+a n}{n}\right )}\, \cosh \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right )+a n}{n}\right )}{\sinh \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right )+a n}{n}\right )}d x \right ) \] Input:

int(sinh(a+2*log(c*x^n)/n)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

sqrt(sinh((2*log(x**n*c) + a*n)/n))*x - int((sqrt(sinh((2*log(x**n*c) + a* 
n)/n))*cosh((2*log(x**n*c) + a*n)/n))/sinh((2*log(x**n*c) + a*n)/n),x)