\(\int \frac {\cosh (a+b \log (c x^n))}{x} \, dx\) [247]

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sinh \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2717} \[ \int \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sinh \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[In]

Int[Cosh[a + b*Log[c*x^n]]/x,x]

[Out]

Sinh[a + b*Log[c*x^n]]/(b*n)

Rule 2717

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cosh \left (b \log \left (c x^n\right )\right ) \sinh (a)}{b n}+\frac {\cosh (a) \sinh \left (b \log \left (c x^n\right )\right )}{b n} \]

[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)

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(\frac {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\)	\(19\)
default	\(\frac {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\)	\(19\)
parallelrisch	\(\frac {\sinh \left (2 b \ln \left (\sqrt {c \,x^{n}}\right )+a \right )}{b n}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).

Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \cosh {\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cosh {\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sinh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sinh \left (b \log \left (c x^{n}\right ) + a\right )}{b n} \]

[In]

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

[Out]

sinh(b*log(c*x^n) + a)/(b*n)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (18) = 36\).

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.33 \[ \int \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (c^{2 \, b} x^{b n} e^{\left (2 \, a\right )} - \frac {1}{x^{b n}}\right )} e^{\left (-a\right )}}{2 \, b c^{b} n} \]

[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)

Mupad [B] (verification not implemented)

Time = 1.79 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}{b\,n} \]

[In]

int(cosh(a + b*log(c*x^n))/x,x)

[Out]

sinh(a + b*log(c*x^n))/(b*n)