3.251 \(\int \frac{\cosh ^5(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=65 \[ \frac{\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n}+\frac{2 \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{\sinh \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

Sinh[a + b*Log[c*x^n]]/(b*n) + (2*Sinh[a + b*Log[c*x^n]]^3)/(3*b*n) + Sinh[a + b*Log[c*x^n]]^5/(5*b*n)

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Rubi [A]  time = 0.0372635, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2633} \[ \frac{\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n}+\frac{2 \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{\sinh \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*Log[c*x^n]]^5/x,x]

[Out]

Sinh[a + b*Log[c*x^n]]/(b*n) + (2*Sinh[a + b*Log[c*x^n]]^3)/(3*b*n) + Sinh[a + b*Log[c*x^n]]^5/(5*b*n)

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \frac{\cosh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \cosh ^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{i \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-i \sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=\frac{\sinh \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac{2 \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n}\\ \end{align*}

Mathematica [A]  time = 0.0178927, size = 65, normalized size = 1. \[ \frac{\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n}+\frac{2 \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{\sinh \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*Log[c*x^n]]^5/x,x]

[Out]

Sinh[a + b*Log[c*x^n]]/(b*n) + (2*Sinh[a + b*Log[c*x^n]]^3)/(3*b*n) + Sinh[a + b*Log[c*x^n]]^5/(5*b*n)

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Maple [A]  time = 0.017, size = 51, normalized size = 0.8 \begin{align*}{\frac{\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{nb} \left ({\frac{8}{15}}+{\frac{ \left ( \cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}}{5}}+{\frac{4\, \left ( \cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{15}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a+b*ln(c*x^n))^5/x,x)

[Out]

1/n/b*(8/15+1/5*cosh(a+b*ln(c*x^n))^4+4/15*cosh(a+b*ln(c*x^n))^2)*sinh(a+b*ln(c*x^n))

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Maxima [B]  time = 1.03966, size = 176, normalized size = 2.71 \begin{align*} \frac{e^{\left (5 \, b \log \left (c x^{n}\right ) + 5 \, a\right )}}{160 \, b n} + \frac{5 \, e^{\left (3 \, b \log \left (c x^{n}\right ) + 3 \, a\right )}}{96 \, b n} + \frac{5 \, e^{\left (b \log \left (c x^{n}\right ) + a\right )}}{16 \, b n} - \frac{5 \, e^{\left (-b \log \left (c x^{n}\right ) - a\right )}}{16 \, b n} - \frac{5 \, e^{\left (-3 \, b \log \left (c x^{n}\right ) - 3 \, a\right )}}{96 \, b n} - \frac{e^{\left (-5 \, b \log \left (c x^{n}\right ) - 5 \, a\right )}}{160 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*log(c*x^n))^5/x,x, algorithm="maxima")

[Out]

1/160*e^(5*b*log(c*x^n) + 5*a)/(b*n) + 5/96*e^(3*b*log(c*x^n) + 3*a)/(b*n) + 5/16*e^(b*log(c*x^n) + a)/(b*n) -
5/16*e^(-b*log(c*x^n) - a)/(b*n) - 5/96*e^(-3*b*log(c*x^n) - 3*a)/(b*n) - 1/160*e^(-5*b*log(c*x^n) - 5*a)/(b*
n)

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Fricas [A]  time = 1.94643, size = 333, normalized size = 5.12 \begin{align*} \frac{3 \, \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{5} + 5 \,{\left (6 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 5\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 15 \,{\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 5 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 10\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{240 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*log(c*x^n))^5/x,x, algorithm="fricas")

[Out]

1/240*(3*sinh(b*n*log(x) + b*log(c) + a)^5 + 5*(6*cosh(b*n*log(x) + b*log(c) + a)^2 + 5)*sinh(b*n*log(x) + b*l
og(c) + a)^3 + 15*(cosh(b*n*log(x) + b*log(c) + a)^4 + 5*cosh(b*n*log(x) + b*log(c) + a)^2 + 10)*sinh(b*n*log(
x) + b*log(c) + a))/(b*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*ln(c*x**n))**5/x,x)

[Out]

Timed out

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Giac [A]  time = 1.19242, size = 157, normalized size = 2.42 \begin{align*} \frac{{\left (3 \, c^{10 \, b} x^{5 \, b n} e^{\left (10 \, a\right )} + 25 \, c^{8 \, b} x^{3 \, b n} e^{\left (8 \, a\right )} + 150 \, c^{6 \, b} x^{b n} e^{\left (6 \, a\right )} - \frac{150 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 25 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 3}{x^{5 \, b n}}\right )} e^{\left (-5 \, a\right )}}{480 \, b c^{5 \, b} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*log(c*x^n))^5/x,x, algorithm="giac")

[Out]

1/480*(3*c^(10*b)*x^(5*b*n)*e^(10*a) + 25*c^(8*b)*x^(3*b*n)*e^(8*a) + 150*c^(6*b)*x^(b*n)*e^(6*a) - (150*c^(4*
b)*x^(4*b*n)*e^(4*a) + 25*c^(2*b)*x^(2*b*n)*e^(2*a) + 3)/x^(5*b*n))*e^(-5*a)/(b*c^(5*b)*n)