\(\int \frac {\cosh (a+b x^n)}{x} \, dx\) [36]

Optimal result

Integrand size = 12, antiderivative size = 25 \[ \int \frac {\cosh \left (a+b x^n\right )}{x} \, dx=\frac {\cosh (a) \text {Chi}\left (b x^n\right )}{n}+\frac {\sinh (a) \text {Shi}\left (b x^n\right )}{n} \]

[Out]

Chi(b*x^n)*cosh(a)/n+Shi(b*x^n)*sinh(a)/n

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5427, 5425, 5424} \[ \int \frac {\cosh \left (a+b x^n\right )}{x} \, dx=\frac {\cosh (a) \text {Chi}\left (b x^n\right )}{n}+\frac {\sinh (a) \text {Shi}\left (b x^n\right )}{n} \]

[In]

Int[Cosh[a + b*x^n]/x,x]

[Out]

(Cosh[a]*CoshIntegral[b*x^n])/n + (Sinh[a]*SinhIntegral[b*x^n])/n

Rule 5424

Int[Sinh[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinhIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 5425

Int[Cosh[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CoshIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 5427

Int[Cosh[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Dist[Cosh[c], Int[Cosh[d*x^n]/x, x], x] + Dist[Sinh[c], In
t[Sinh[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]

Rubi steps \begin{align*} \text {integral}& = \cosh (a) \int \frac {\cosh \left (b x^n\right )}{x} \, dx+\sinh (a) \int \frac {\sinh \left (b x^n\right )}{x} \, dx \\ & = \frac {\cosh (a) \text {Chi}\left (b x^n\right )}{n}+\frac {\sinh (a) \text {Shi}\left (b x^n\right )}{n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh \left (a+b x^n\right )}{x} \, dx=\frac {\cosh (a) \text {Chi}\left (b x^n\right )+\sinh (a) \text {Shi}\left (b x^n\right )}{n} \]

[In]

Integrate[Cosh[a + b*x^n]/x,x]

[Out]

(Cosh[a]*CoshIntegral[b*x^n] + Sinh[a]*SinhIntegral[b*x^n])/n

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32

method	result	size
risch	\(-\frac {{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b \,x^{n}\right )}{2 n}-\frac {{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b \,x^{n}\right )}{2 n}\)	\(33\)
meijerg	\(\frac {\sqrt {\pi }\, \left (\frac {2 \gamma +2 n \ln \left (x \right )+2 \ln \left (i b \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (b \,x^{n}\right )-2 \ln \left (b \,x^{n}\right )-2 \gamma }{\sqrt {\pi }}\right ) \cosh \left (a \right )}{2 n}+\frac {\operatorname {Shi}\left (b \,x^{n}\right ) \sinh \left (a \right )}{n}\)	\(68\)

[In]

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

[Out]

-1/2/n*exp(-a)*Ei(1,b*x^n)-1/2/n*exp(a)*Ei(1,-b*x^n)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).

Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {\cosh \left (a+b x^n\right )}{x} \, dx=\frac {{\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} {\rm Ei}\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right ) + {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} {\rm Ei}\left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right )\right )}{2 \, n} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\cosh \left (a+b x^n\right )}{x} \, dx=\int \frac {\cosh {\left (a + b x^{n} \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(cosh(a + b*x**n)/x, x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {\cosh \left (a+b x^n\right )}{x} \, dx=\frac {{\rm Ei}\left (-b x^{n}\right ) e^{\left (-a\right )}}{2 \, n} + \frac {{\rm Ei}\left (b x^{n}\right ) e^{a}}{2 \, n} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\cosh \left (a+b x^n\right )}{x} \, dx=\int { \frac {\cosh \left (b x^{n} + a\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(cosh(b*x^n + a)/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh \left (a+b x^n\right )}{x} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x^n\right )}{x} \,d x \]

[In]

int(cosh(a + b*x^n)/x,x)

[Out]

int(cosh(a + b*x^n)/x, x)