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

Optimal. Leaf size=25 \[ \frac{\cosh (a) \text{Chi}\left (b x^n\right )}{n}+\frac{\sinh (a) \text{Shi}\left (b x^n\right )}{n} \]

[Out]

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

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Rubi [A]  time = 0.0382602, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5319, 5317, 5316} \[ \frac{\cosh (a) \text{Chi}\left (b x^n\right )}{n}+\frac{\sinh (a) \text{Shi}\left (b x^n\right )}{n} \]

Antiderivative was successfully verified.

[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 5319

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]

Rule 5317

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

Rule 5316

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

Rubi steps

\begin{align*} \int \frac{\cosh \left (a+b x^n\right )}{x} \, dx &=\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]  time = 0.0214314, size = 23, normalized size = 0.92 \[ \frac{\cosh (a) \text{Chi}\left (b x^n\right )+\sinh (a) \text{Shi}\left (b x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x^n]/x,x]

[Out]

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

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Maple [A]  time = 0.02, size = 33, normalized size = 1.3 \begin{align*} -{\frac{{{\rm e}^{-a}}{\it Ei} \left ( 1,b{x}^{n} \right ) }{2\,n}}-{\frac{{{\rm e}^{a}}{\it Ei} \left ( 1,-b{x}^{n} \right ) }{2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15545, size = 41, normalized size = 1.64 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.8384, size = 178, normalized size = 7.12 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (a + b x^{n} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(a + b*x**n)/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (b x^{n} + a\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cosh(b*x^n + a)/x, x)