3.248 \(\int \frac {\cosh ^2(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=39 \[ \frac {\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log (x)}{2} \]

[Out]

1/2*ln(x)+1/2*cosh(a+b*ln(c*x^n))*sinh(a+b*ln(c*x^n))/b/n

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2635, 8} \[ \frac {\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log (x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*Log[c*x^n]]^2/x,x]

[Out]

Log[x]/2 + (Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(2*b*n)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rubi steps

\begin {align*} \int \frac {\cosh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \cosh ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\operatorname {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{2 n}\\ &=\frac {\log (x)}{2}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 36, normalized size = 0.92 \[ \frac {2 \left (a+b \log \left (c x^n\right )\right )+\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}{4 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*Log[c*x^n]]^2/x,x]

[Out]

(2*(a + b*Log[c*x^n]) + Sinh[2*(a + b*Log[c*x^n])])/(4*b*n)

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 39, normalized size = 1.00 \[ \frac {b n \log \relax (x) + \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{2 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*n*log(x) + cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a))/(b*n)

________________________________________________________________________________________

giac [B]  time = 0.13, size = 80, normalized size = 2.05 \[ \frac {{\left (4 \, b c^{2 \, b} n e^{\left (2 \, a\right )} \log \relax (x) + c^{4 \, b} x^{2 \, b n} e^{\left (4 \, a\right )} - \frac {2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1}{x^{2 \, b n}}\right )} e^{\left (-2 \, a\right )}}{8 \, b c^{2 \, b} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*b*c^(2*b)*n*e^(2*a)*log(x) + c^(4*b)*x^(2*b*n)*e^(4*a) - (2*c^(2*b)*x^(2*b*n)*e^(2*a) + 1)/x^(2*b*n))*e
^(-2*a)/(b*c^(2*b)*n)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 52, normalized size = 1.33 \[ \frac {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}+\frac {\ln \left (c \,x^{n}\right )}{2 n}+\frac {a}{2 b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cosh(a+b*ln(c*x^n))*sinh(a+b*ln(c*x^n))/b/n+1/2*ln(c*x^n)/n+1/2/b/n*a

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 49, normalized size = 1.26 \[ \frac {e^{\left (2 \, b \log \left (c x^{n}\right ) + 2 \, a\right )}}{8 \, b n} - \frac {e^{\left (-2 \, b \log \left (c x^{n}\right ) - 2 \, a\right )}}{8 \, b n} + \frac {1}{2} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*e^(2*b*log(c*x^n) + 2*a)/(b*n) - 1/8*e^(-2*b*log(c*x^n) - 2*a)/(b*n) + 1/2*log(x)

________________________________________________________________________________________

mupad [B]  time = 1.05, size = 32, normalized size = 0.82 \[ \frac {\ln \left (x^n\right )}{2\,n}+\frac {\mathrm {sinh}\left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4\,b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*log(c*x^n))^2/x,x)

[Out]

log(x^n)/(2*n) + sinh(2*a + 2*b*log(c*x^n))/(4*b*n)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________