3.12 \(\int \frac{\cosh ^2(a+b x^2)}{x} \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{4} \cosh (2 a) \text{Chi}\left (2 b x^2\right )+\frac{1}{4} \sinh (2 a) \text{Shi}\left (2 b x^2\right )+\frac{\log (x)}{2} \]

[Out]

(Cosh[2*a]*CoshIntegral[2*b*x^2])/4 + Log[x]/2 + (Sinh[2*a]*SinhIntegral[2*b*x^2])/4

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Rubi [A]  time = 0.0607286, antiderivative size = 37, 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 = {5341, 5319, 5317, 5316} \[ \frac{1}{4} \cosh (2 a) \text{Chi}\left (2 b x^2\right )+\frac{1}{4} \sinh (2 a) \text{Shi}\left (2 b x^2\right )+\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x^2]^2/x,x]

[Out]

(Cosh[2*a]*CoshIntegral[2*b*x^2])/4 + Log[x]/2 + (Sinh[2*a]*SinhIntegral[2*b*x^2])/4

Rule 5341

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}, x] && IGtQ[p, 1] && IGtQ[n, 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 ^2\left (a+b x^2\right )}{x} \, dx &=\int \left (\frac{1}{2 x}+\frac{\cosh \left (2 a+2 b x^2\right )}{2 x}\right ) \, dx\\ &=\frac{\log (x)}{2}+\frac{1}{2} \int \frac{\cosh \left (2 a+2 b x^2\right )}{x} \, dx\\ &=\frac{\log (x)}{2}+\frac{1}{2} \cosh (2 a) \int \frac{\cosh \left (2 b x^2\right )}{x} \, dx+\frac{1}{2} \sinh (2 a) \int \frac{\sinh \left (2 b x^2\right )}{x} \, dx\\ &=\frac{1}{4} \cosh (2 a) \text{Chi}\left (2 b x^2\right )+\frac{\log (x)}{2}+\frac{1}{4} \sinh (2 a) \text{Shi}\left (2 b x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0196797, size = 33, normalized size = 0.89 \[ \frac{1}{4} \left (\cosh (2 a) \text{Chi}\left (2 b x^2\right )+\sinh (2 a) \text{Shi}\left (2 b x^2\right )+2 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x^2]^2/x,x]

[Out]

(Cosh[2*a]*CoshIntegral[2*b*x^2] + 2*Log[x] + Sinh[2*a]*SinhIntegral[2*b*x^2])/4

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Maple [A]  time = 0.033, size = 34, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{2}}-{\frac{{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,b{x}^{2} \right ) }{8}}-{\frac{{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,b{x}^{2} \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(x)-1/8*exp(-2*a)*Ei(1,2*b*x^2)-1/8*exp(2*a)*Ei(1,-2*b*x^2)

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Maxima [A]  time = 1.3177, size = 42, normalized size = 1.14 \begin{align*} \frac{1}{8} \,{\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} + \frac{1}{8} \,{\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} + \frac{1}{2} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*Ei(2*b*x^2)*e^(2*a) + 1/8*Ei(-2*b*x^2)*e^(-2*a) + 1/2*log(x)

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Fricas [A]  time = 1.84838, size = 138, normalized size = 3.73 \begin{align*} \frac{1}{8} \,{\left ({\rm Ei}\left (2 \, b x^{2}\right ) +{\rm Ei}\left (-2 \, b x^{2}\right )\right )} \cosh \left (2 \, a\right ) + \frac{1}{8} \,{\left ({\rm Ei}\left (2 \, b x^{2}\right ) -{\rm Ei}\left (-2 \, b x^{2}\right )\right )} \sinh \left (2 \, a\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(Ei(2*b*x^2) + Ei(-2*b*x^2))*cosh(2*a) + 1/8*(Ei(2*b*x^2) - Ei(-2*b*x^2))*sinh(2*a) + 1/2*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x**2+a)**2/x,x)

[Out]

Integral(cosh(a + b*x**2)**2/x, x)

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Giac [A]  time = 1.30248, size = 47, normalized size = 1.27 \begin{align*} \frac{1}{8} \,{\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} + \frac{1}{8} \,{\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} + \frac{1}{4} \, \log \left (b x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*Ei(2*b*x^2)*e^(2*a) + 1/8*Ei(-2*b*x^2)*e^(-2*a) + 1/4*log(b*x^2)