3.67 \(\int \frac{\sinh ^2(a+b x^n)}{x} \, dx\)

Optimal. Leaf size=43 \[ \frac{\cosh (2 a) \text{Chi}\left (2 b x^n\right )}{2 n}+\frac{\sinh (2 a) \text{Shi}\left (2 b x^n\right )}{2 n}-\frac{\log (x)}{2} \]

[Out]

(Cosh[2*a]*CoshIntegral[2*b*x^n])/(2*n) - Log[x]/2 + (Sinh[2*a]*SinhIntegral[2*b*x^n])/(2*n)

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Rubi [A]  time = 0.0639129, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5362, 5319, 5317, 5316} \[ \frac{\cosh (2 a) \text{Chi}\left (2 b x^n\right )}{2 n}+\frac{\sinh (2 a) \text{Shi}\left (2 b x^n\right )}{2 n}-\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x^n]^2/x,x]

[Out]

(Cosh[2*a]*CoshIntegral[2*b*x^n])/(2*n) - Log[x]/2 + (Sinh[2*a]*SinhIntegral[2*b*x^n])/(2*n)

Rule 5362

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(
e*x)^m, (a + b*Sinh[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{\sinh ^2\left (a+b x^n\right )}{x} \, dx &=\int \left (-\frac{1}{2 x}+\frac{\cosh \left (2 a+2 b x^n\right )}{2 x}\right ) \, dx\\ &=-\frac{\log (x)}{2}+\frac{1}{2} \int \frac{\cosh \left (2 a+2 b x^n\right )}{x} \, dx\\ &=-\frac{\log (x)}{2}+\frac{1}{2} \cosh (2 a) \int \frac{\cosh \left (2 b x^n\right )}{x} \, dx+\frac{1}{2} \sinh (2 a) \int \frac{\sinh \left (2 b x^n\right )}{x} \, dx\\ &=\frac{\cosh (2 a) \text{Chi}\left (2 b x^n\right )}{2 n}-\frac{\log (x)}{2}+\frac{\sinh (2 a) \text{Shi}\left (2 b x^n\right )}{2 n}\\ \end{align*}

Mathematica [A]  time = 0.0304173, size = 39, normalized size = 0.91 \[ \frac{\cosh (2 a) \text{Chi}\left (2 b x^n\right )+\sinh (2 a) \text{Shi}\left (2 b x^n\right )}{2 n}-\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x^n]^2/x,x]

[Out]

-Log[x]/2 + (Cosh[2*a]*CoshIntegral[2*b*x^n] + Sinh[2*a]*SinhIntegral[2*b*x^n])/(2*n)

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Maple [A]  time = 0.079, size = 40, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( x \right ) }{2}}-{\frac{{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,b{x}^{n} \right ) }{4\,n}}-{\frac{{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,b{x}^{n} \right ) }{4\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a+b*x^n)^2/x,x)

[Out]

-1/2*ln(x)-1/4/n*exp(-2*a)*Ei(1,2*b*x^n)-1/4/n*exp(2*a)*Ei(1,-2*b*x^n)

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Maxima [A]  time = 1.1796, size = 50, normalized size = 1.16 \begin{align*} \frac{{\rm Ei}\left (2 \, b x^{n}\right ) e^{\left (2 \, a\right )}}{4 \, n} + \frac{{\rm Ei}\left (-2 \, b x^{n}\right ) e^{\left (-2 \, a\right )}}{4 \, n} - \frac{1}{2} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*x^n)^2/x,x, algorithm="maxima")

[Out]

1/4*Ei(2*b*x^n)*e^(2*a)/n + 1/4*Ei(-2*b*x^n)*e^(-2*a)/n - 1/2*log(x)

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Fricas [A]  time = 1.8706, size = 217, normalized size = 5.05 \begin{align*} \frac{{\left (\cosh \left (2 \, a\right ) + \sinh \left (2 \, a\right )\right )}{\rm Ei}\left (2 \, b \cosh \left (n \log \left (x\right )\right ) + 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) +{\left (\cosh \left (2 \, a\right ) - \sinh \left (2 \, a\right )\right )}{\rm Ei}\left (-2 \, b \cosh \left (n \log \left (x\right )\right ) - 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) - 2 \, n \log \left (x\right )}{4 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*x^n)^2/x,x, algorithm="fricas")

[Out]

1/4*((cosh(2*a) + sinh(2*a))*Ei(2*b*cosh(n*log(x)) + 2*b*sinh(n*log(x))) + (cosh(2*a) - sinh(2*a))*Ei(-2*b*cos
h(n*log(x)) - 2*b*sinh(n*log(x))) - 2*n*log(x))/n

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (a + b x^{n} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*x**n)**2/x,x)

[Out]

Integral(sinh(a + b*x**n)**2/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (b x^{n} + a\right )^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*x^n)^2/x,x, algorithm="giac")

[Out]

integrate(sinh(b*x^n + a)^2/x, x)