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

Optimal. Leaf size=18 \[ \frac{\sinh \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

Sinh[a + b*Log[c*x^n]]/(b*n)

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Rubi [A]  time = 0.0160305, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2637} \[ \frac{\sinh \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*Log[c*x^n]]/x,x]

[Out]

Sinh[a + b*Log[c*x^n]]/(b*n)

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [B]  time = 0.0105704, size = 37, normalized size = 2.06 \[ \frac{\sinh (a) \cosh \left (b \log \left (c x^n\right )\right )}{b n}+\frac{\cosh (a) \sinh \left (b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*Log[c*x^n]]/x,x]

[Out]

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

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Maple [A]  time = 0.007, size = 19, normalized size = 1.1 \begin{align*}{\frac{\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0494, size = 24, normalized size = 1.33 \begin{align*} \frac{\sinh \left (b \log \left (c x^{n}\right ) + a\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sinh(b*log(c*x^n) + a)/(b*n)

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Fricas [A]  time = 1.91868, size = 53, normalized size = 2.94 \begin{align*} \frac{\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sinh(b*n*log(x) + b*log(c) + a)/(b*n)

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Sympy [A]  time = 1.1883, size = 41, normalized size = 2.28 \begin{align*} \begin{cases} \log{\left (x \right )} \cosh{\left (a \right )} & \text{for}\: b = 0 \wedge n = 0 \\\log{\left (x \right )} \cosh{\left (a + b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\log{\left (x \right )} \cosh{\left (a \right )} & \text{for}\: b = 0 \\\frac{\sinh{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(x)*cosh(a), Eq(b, 0) & Eq(n, 0)), (log(x)*cosh(a + b*log(c)), Eq(n, 0)), (log(x)*cosh(a), Eq(b,
0)), (sinh(a + b*n*log(x) + b*log(c))/(b*n), True))

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Giac [B]  time = 1.13579, size = 57, normalized size = 3.17 \begin{align*} \frac{{\left (c^{2 \, b} x^{b n} e^{\left (2 \, a\right )} - \frac{1}{x^{b n}}\right )} e^{\left (-a\right )}}{2 \, b c^{b} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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