\(\int \cosh ^{\frac {3}{2}}(a+\frac {2 \log (c x^n)}{n}) \, dx\) [259]

Optimal result

Integrand size = 18, antiderivative size = 170 \[ \int \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\frac {1}{4} x \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )-\frac {3 e^{-2 a} x \left (c x^n\right )^{-4/n} \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )}+\frac {3 e^{-2 a} x \left (c x^n\right )^{-4/n} \text {arctanh}\left (\sqrt {1+e^{-2 a} \left (c x^n\right )^{-4/n}}\right ) \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{3/2}} \] Output:

1/4*x*cosh(a+2*ln(c*x^n)/n)^(3/2)-3/4*x*cosh(a+2*ln(c*x^n)/n)^(3/2)/exp(2* 
a)/((c*x^n)^(4/n))/(1+1/exp(2*a)/((c*x^n)^(4/n)))+3/4*x*arctanh((1+1/exp(2 
*a)/((c*x^n)^(4/n)))^(1/2))*cosh(a+2*ln(c*x^n)/n)^(3/2)/exp(2*a)/((c*x^n)^ 
(4/n))/(1+1/exp(2*a)/((c*x^n)^(4/n)))^(3/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.44 \[ \int \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=-\frac {x \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{2},\frac {1}{2},-e^{2 a+\frac {4 \log \left (c x^n\right )}{n}}\right )}{2 \left (1+e^{2 a+\frac {4 \log \left (c x^n\right )}{n}}\right )^{3/2}} \] Input:

Integrate[Cosh[a + (2*Log[c*x^n])/n]^(3/2),x]
 

Output:

-1/2*(x*Cosh[a + (2*Log[c*x^n])/n]^(3/2)*Hypergeometric2F1[-3/2, -1/2, 1/2 
, -E^(2*a + (4*Log[c*x^n])/n)])/(1 + E^(2*a + (4*Log[c*x^n])/n))^(3/2)
 

Rubi [A] (warning: unable to verify)

Time = 0.38 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6052, 6060, 798, 51, 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.

\(\displaystyle \int \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx\)

\(\Big \downarrow \) 6052

\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 6060

\(\displaystyle \frac {x \left (c x^n\right )^{-4/n} \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \int \left (c x^n\right )^{\frac {4}{n}-1} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}d\left (c x^n\right )}{n \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}\)

\(\Big \downarrow \) 798

\(\displaystyle -\frac {x \left (c x^n\right )^{-4/n} \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \int \frac {x^{-2 n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}{c^2}d\left (c x^n\right )^{-4/n}}{4 \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}\)

\(\Big \downarrow \) 51

\(\displaystyle -\frac {x \left (c x^n\right )^{-4/n} \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {3}{2} e^{-2 a} \int \frac {x^{-n} \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}}{c}d\left (c x^n\right )^{-4/n}-\frac {x^{-n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}{c}\right )}{4 \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {x \left (c x^n\right )^{-4/n} \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {3}{2} e^{-2 a} \left (\int \frac {x^{-n}}{c \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}}d\left (c x^n\right )^{-4/n}+2 \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}\right )-\frac {x^{-n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}{c}\right )}{4 \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {x \left (c x^n\right )^{-4/n} \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {3}{2} e^{-2 a} \left (2 e^{2 a} \int \frac {1}{c^2 e^{2 a} x^{2 n}-e^{2 a}}d\sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}+2 \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}\right )-\frac {x^{-n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}{c}\right )}{4 \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {x \left (c x^n\right )^{-4/n} \left (\frac {3}{2} e^{-2 a} \left (2 \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}-2 \text {arctanh}\left (\sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}\right )\right )-\frac {x^{-n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}{c}\right ) \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}}\)

Input:

Int[Cosh[a + (2*Log[c*x^n])/n]^(3/2),x]
 

Output:

-1/4*(x*(-((1 + 1/(E^(2*a)*(c*x^n)^(4/n)))^(3/2)/(c*x^n)) + (3*(2*Sqrt[1 + 
 1/(E^(2*a)*(c*x^n)^(4/n))] - 2*ArcTanh[Sqrt[1 + 1/(E^(2*a)*(c*x^n)^(4/n)) 
]]))/(2*E^(2*a)))*Cosh[a + (2*Log[c*x^n])/n]^(3/2))/((c*x^n)^(4/n)*(1 + 1/ 
(E^(2*a)*(c*x^n)^(4/n)))^(3/2))
 

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

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 6052

Int[Cosh[((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)*Cosh[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 6060

Int[Cosh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] 
:> Simp[Cosh[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]

\[\int {\cosh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )}^{\frac {3}{2}}d x\]

Input:

int(cosh(a+2*ln(c*x^n)/n)^(3/2),x)
 

Output:

int(cosh(a+2*ln(c*x^n)/n)^(3/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.04 \[ \int \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\frac {{\left (3 \, \sqrt {2} x e^{\left (\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )} \log \left (-2 \, \sqrt {2} \sqrt {\frac {1}{2}} x^{3} \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1}{x^{2}}} e^{\left (\frac {a n + 2 \, \log \left (c\right )}{n}\right )} - 2 \, x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 1\right ) + 4 \, \sqrt {\frac {1}{2}} {\left (x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 2\right )} \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1}{x^{2}}} e^{\left (-\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )}\right )} e^{\left (-\frac {a n + 2 \, \log \left (c\right )}{n}\right )}}{32 \, x} \] Input:

integrate(cosh(a+2*log(c*x^n)/n)^(3/2),x, algorithm="fricas")
 

Output:

1/32*(3*sqrt(2)*x*e^(1/2*(a*n + 2*log(c))/n)*log(-2*sqrt(2)*sqrt(1/2)*x^3* 
sqrt((x^4*e^(2*(a*n + 2*log(c))/n) + 1)/x^2)*e^((a*n + 2*log(c))/n) - 2*x^ 
4*e^(2*(a*n + 2*log(c))/n) - 1) + 4*sqrt(1/2)*(x^4*e^(2*(a*n + 2*log(c))/n 
) - 2)*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)/x
 

Sympy [F]

\[ \int \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\int \cosh ^{\frac {3}{2}}{\left (a + \frac {2 \log {\left (c x^{n} \right )}}{n} \right )}\, dx \] Input:

integrate(cosh(a+2*ln(c*x**n)/n)**(3/2),x)
 

Output:

Integral(cosh(a + 2*log(c*x**n)/n)**(3/2), x)
 

Maxima [F]

\[ \int \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\int { \cosh \left (a + \frac {2 \, \log \left (c x^{n}\right )}{n}\right )^{\frac {3}{2}} \,d x } \] Input:

integrate(cosh(a+2*log(c*x^n)/n)^(3/2),x, algorithm="maxima")
 

Output:

integrate(cosh(a + 2*log(c*x^n)/n)^(3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\text {Exception raised: NotImplementedError} \] Input:

integrate(cosh(a+2*log(c*x^n)/n)^(3/2),x, algorithm="giac")
 

Output:

Exception raised: NotImplementedError >> unable to parse Giac output: sqrt 
(2)/2*2/16*sageVARx*sqrt(sageVARx^2*exp(sageVARa)*exp(ln(sageVARc)/sageVAR 
n)^2+sageVARx^6*exp(sageVARa)^3*exp(ln(sageVARc)/sageVARn)^6)+sqrt(2)/2/ex 
p(ln(sageVARc)/sage
 

Mupad [F(-1)]

Timed out. \[ \int \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\int {\mathrm {cosh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )}^{3/2} \,d x \] Input:

int(cosh(a + (2*log(c*x^n))/n)^(3/2),x)
 

Output:

int(cosh(a + (2*log(c*x^n))/n)^(3/2), x)
 

Reduce [F]

\[ \int \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\int \sqrt {\cosh \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right )+a n}{n}\right )}\, \cosh \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right )+a n}{n}\right )d x \] Input:

int(cosh(a+2*log(c*x^n)/n)^(3/2),x)
 

Output:

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