3.40 \(\int \frac{\cosh ^3(a+b x^n)}{x} \, dx\)

Optimal. Leaf size=67 \[ \frac{3 \cosh (a) \text{Chi}\left (b x^n\right )}{4 n}+\frac{\cosh (3 a) \text{Chi}\left (3 b x^n\right )}{4 n}+\frac{3 \sinh (a) \text{Shi}\left (b x^n\right )}{4 n}+\frac{\sinh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n} \]

[Out]

(3*Cosh[a]*CoshIntegral[b*x^n])/(4*n) + (Cosh[3*a]*CoshIntegral[3*b*x^n])/(4*n) + (3*Sinh[a]*SinhIntegral[b*x^
n])/(4*n) + (Sinh[3*a]*SinhIntegral[3*b*x^n])/(4*n)

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Rubi [A]  time = 0.100218, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5363, 5319, 5317, 5316} \[ \frac{3 \cosh (a) \text{Chi}\left (b x^n\right )}{4 n}+\frac{\cosh (3 a) \text{Chi}\left (3 b x^n\right )}{4 n}+\frac{3 \sinh (a) \text{Shi}\left (b x^n\right )}{4 n}+\frac{\sinh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x^n]^3/x,x]

[Out]

(3*Cosh[a]*CoshIntegral[b*x^n])/(4*n) + (Cosh[3*a]*CoshIntegral[3*b*x^n])/(4*n) + (3*Sinh[a]*SinhIntegral[b*x^
n])/(4*n) + (Sinh[3*a]*SinhIntegral[3*b*x^n])/(4*n)

Rule 5363

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandTrigReduce[(
e*x)^m, (a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

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 ^3\left (a+b x^n\right )}{x} \, dx &=\int \left (\frac{3 \cosh \left (a+b x^n\right )}{4 x}+\frac{\cosh \left (3 a+3 b x^n\right )}{4 x}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\cosh \left (3 a+3 b x^n\right )}{x} \, dx+\frac{3}{4} \int \frac{\cosh \left (a+b x^n\right )}{x} \, dx\\ &=\frac{1}{4} (3 \cosh (a)) \int \frac{\cosh \left (b x^n\right )}{x} \, dx+\frac{1}{4} \cosh (3 a) \int \frac{\cosh \left (3 b x^n\right )}{x} \, dx+\frac{1}{4} (3 \sinh (a)) \int \frac{\sinh \left (b x^n\right )}{x} \, dx+\frac{1}{4} \sinh (3 a) \int \frac{\sinh \left (3 b x^n\right )}{x} \, dx\\ &=\frac{3 \cosh (a) \text{Chi}\left (b x^n\right )}{4 n}+\frac{\cosh (3 a) \text{Chi}\left (3 b x^n\right )}{4 n}+\frac{3 \sinh (a) \text{Shi}\left (b x^n\right )}{4 n}+\frac{\sinh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n}\\ \end{align*}

Mathematica [A]  time = 0.0480971, size = 52, normalized size = 0.78 \[ \frac{3 \cosh (a) \text{Chi}\left (b x^n\right )+\cosh (3 a) \text{Chi}\left (3 b x^n\right )+3 \sinh (a) \text{Shi}\left (b x^n\right )+\sinh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x^n]^3/x,x]

[Out]

(3*Cosh[a]*CoshIntegral[b*x^n] + Cosh[3*a]*CoshIntegral[3*b*x^n] + 3*Sinh[a]*SinhIntegral[b*x^n] + Sinh[3*a]*S
inhIntegral[3*b*x^n])/(4*n)

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Maple [A]  time = 0.106, size = 67, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,b{x}^{n} \right ) }{8\,n}}-{\frac{3\,{{\rm e}^{-a}}{\it Ei} \left ( 1,b{x}^{n} \right ) }{8\,n}}-{\frac{{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,b{x}^{n} \right ) }{8\,n}}-{\frac{3\,{{\rm e}^{a}}{\it Ei} \left ( 1,-b{x}^{n} \right ) }{8\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a+b*x^n)^3/x,x)

[Out]

-1/8/n*exp(-3*a)*Ei(1,3*b*x^n)-3/8/n*exp(-a)*Ei(1,b*x^n)-1/8/n*exp(3*a)*Ei(1,-3*b*x^n)-3/8/n*exp(a)*Ei(1,-b*x^
n)

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Maxima [A]  time = 1.27732, size = 84, normalized size = 1.25 \begin{align*} \frac{{\rm Ei}\left (3 \, b x^{n}\right ) e^{\left (3 \, a\right )}}{8 \, n} + \frac{3 \,{\rm Ei}\left (-b x^{n}\right ) e^{\left (-a\right )}}{8 \, n} + \frac{{\rm Ei}\left (-3 \, b x^{n}\right ) e^{\left (-3 \, a\right )}}{8 \, n} + \frac{3 \,{\rm Ei}\left (b x^{n}\right ) e^{a}}{8 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*x^n)^3/x,x, algorithm="maxima")

[Out]

1/8*Ei(3*b*x^n)*e^(3*a)/n + 3/8*Ei(-b*x^n)*e^(-a)/n + 1/8*Ei(-3*b*x^n)*e^(-3*a)/n + 3/8*Ei(b*x^n)*e^a/n

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Fricas [A]  time = 1.76263, size = 374, normalized size = 5.58 \begin{align*} \frac{{\left (\cosh \left (3 \, a\right ) + \sinh \left (3 \, a\right )\right )}{\rm Ei}\left (3 \, b \cosh \left (n \log \left (x\right )\right ) + 3 \, b \sinh \left (n \log \left (x\right )\right )\right ) + 3 \,{\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 ) + 3 \,{\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 (3 \, a\right ) - \sinh \left (3 \, a\right )\right )}{\rm Ei}\left (-3 \, b \cosh \left (n \log \left (x\right )\right ) - 3 \, b \sinh \left (n \log \left (x\right )\right )\right )}{8 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*x^n)^3/x,x, algorithm="fricas")

[Out]

1/8*((cosh(3*a) + sinh(3*a))*Ei(3*b*cosh(n*log(x)) + 3*b*sinh(n*log(x))) + 3*(cosh(a) + sinh(a))*Ei(b*cosh(n*l
og(x)) + b*sinh(n*log(x))) + 3*(cosh(a) - sinh(a))*Ei(-b*cosh(n*log(x)) - b*sinh(n*log(x))) + (cosh(3*a) - sin
h(3*a))*Ei(-3*b*cosh(n*log(x)) - 3*b*sinh(n*log(x))))/n

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh ^{3}{\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)**3/x,x)

[Out]

Integral(cosh(a + b*x**n)**3/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 )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b*x^n)^3/x,x, algorithm="giac")

[Out]

integrate(cosh(b*x^n + a)^3/x, x)