3.250 \(\int \frac {\cosh ^4(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=73 \[ \frac {\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {3 \log (x)}{8} \]

[Out]

3/8*ln(x)+3/8*cosh(a+b*ln(c*x^n))*sinh(a+b*ln(c*x^n))/b/n+1/4*cosh(a+b*ln(c*x^n))^3*sinh(a+b*ln(c*x^n))/b/n

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Rubi [A]  time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2635, 8} \[ \frac {\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {3 \log (x)}{8} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*Log[c*x^n]]^4/x,x]

[Out]

(3*Log[x])/8 + (3*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(8*b*n) + (Cosh[a + b*Log[c*x^n]]^3*Sinh[a +
b*Log[c*x^n]])/(4*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 ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \cosh ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\cosh ^3\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \operatorname {Subst}\left (\int \cosh ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n}\\ &=\frac {3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cosh ^3\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \operatorname {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{8 n}\\ &=\frac {3 \log (x)}{8}+\frac {3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cosh ^3\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 51, normalized size = 0.70 \[ \frac {12 \left (a+b \log \left (c x^n\right )\right )+8 \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*Log[c*x^n]]^4/x,x]

[Out]

(12*(a + b*Log[c*x^n]) + 8*Sinh[2*(a + b*Log[c*x^n])] + Sinh[4*(a + b*Log[c*x^n])])/(32*b*n)

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fricas [A]  time = 0.53, size = 84, normalized size = 1.15 \[ \frac {\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + 3 \, b n \log \relax (x) + {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + 4 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{8 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*log(c*x^n))^4/x,x, algorithm="fricas")

[Out]

1/8*(cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + 3*b*n*log(x) + (cosh(b*n*log(x) + b*l
og(c) + a)^3 + 4*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a))/(b*n)

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giac [A]  time = 0.15, size = 114, normalized size = 1.56 \[ \frac {{\left (24 \, b c^{4 \, b} n e^{\left (4 \, a\right )} \log \relax (x) + c^{8 \, b} x^{4 \, b n} e^{\left (8 \, a\right )} + 8 \, c^{6 \, b} x^{2 \, b n} e^{\left (6 \, a\right )} - \frac {18 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 8 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1}{x^{4 \, b n}}\right )} e^{\left (-4 \, a\right )}}{64 \, b c^{4 \, b} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*log(c*x^n))^4/x,x, algorithm="giac")

[Out]

1/64*(24*b*c^(4*b)*n*e^(4*a)*log(x) + c^(8*b)*x^(4*b*n)*e^(8*a) + 8*c^(6*b)*x^(2*b*n)*e^(6*a) - (18*c^(4*b)*x^
(4*b*n)*e^(4*a) + 8*c^(2*b)*x^(2*b*n)*e^(2*a) + 1)/x^(4*b*n))*e^(-4*a)/(b*c^(4*b)*n)

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maple [A]  time = 0.23, size = 84, normalized size = 1.15 \[ \frac {\left (\cosh ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{4 b n}+\frac {3 \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{8 b n}+\frac {3 \ln \left (c \,x^{n}\right )}{8 n}+\frac {3 a}{8 b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a+b*ln(c*x^n))^4/x,x)

[Out]

1/4*cosh(a+b*ln(c*x^n))^3*sinh(a+b*ln(c*x^n))/b/n+3/8*cosh(a+b*ln(c*x^n))*sinh(a+b*ln(c*x^n))/b/n+3/8*ln(c*x^n
)/n+3/8/b/n*a

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maxima [A]  time = 0.33, size = 93, normalized size = 1.27 \[ \frac {e^{\left (4 \, b \log \left (c x^{n}\right ) + 4 \, a\right )}}{64 \, b n} + \frac {e^{\left (2 \, b \log \left (c x^{n}\right ) + 2 \, a\right )}}{8 \, b n} - \frac {e^{\left (-2 \, b \log \left (c x^{n}\right ) - 2 \, a\right )}}{8 \, b n} - \frac {e^{\left (-4 \, b \log \left (c x^{n}\right ) - 4 \, a\right )}}{64 \, b n} + \frac {3}{8} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*log(c*x^n))^4/x,x, algorithm="maxima")

[Out]

1/64*e^(4*b*log(c*x^n) + 4*a)/(b*n) + 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)
- 1/64*e^(-4*b*log(c*x^n) - 4*a)/(b*n) + 3/8*log(x)

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mupad [B]  time = 1.11, size = 50, normalized size = 0.68 \[ \frac {3\,\ln \left (x^n\right )}{8\,n}+\frac {\frac {\mathrm {sinh}\left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4}+\frac {\mathrm {sinh}\left (4\,a+4\,b\,\ln \left (c\,x^n\right )\right )}{32}}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*log(c*x^n))^4/x,x)

[Out]

(3*log(x^n))/(8*n) + (sinh(2*a + 2*b*log(c*x^n))/4 + sinh(4*a + 4*b*log(c*x^n))/32)/(b*n)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*ln(c*x**n))**4/x,x)

[Out]

Integral(cosh(a + b*log(c*x**n))**4/x, x)

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