\(\int \frac {\cosh (a+b x^2)}{x} \, dx\) [5]

Optimal result

Integrand size = 12, antiderivative size = 25 \[ \int \frac {\cosh \left (a+b x^2\right )}{x} \, dx=\frac {1}{2} \cosh (a) \text {Chi}\left (b x^2\right )+\frac {1}{2} \sinh (a) \text {Shi}\left (b x^2\right ) \]

[Out]

1/2*Chi(b*x^2)*cosh(a)+1/2*Shi(b*x^2)*sinh(a)

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} \[ \int \frac {\cosh \left (a+b x^2\right )}{x} \, dx=\frac {1}{2} \cosh (a) \text {Chi}\left (b x^2\right )+\frac {1}{2} \sinh (a) \text {Shi}\left (b x^2\right ) \]

[In]

Int[Cosh[a + b*x^2]/x,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh \left (a+b x^2\right )}{x} \, dx=\frac {1}{2} \left (\cosh (a) \text {Chi}\left (b x^2\right )+\sinh (a) \text {Shi}\left (b x^2\right )\right ) \]

[In]

Integrate[Cosh[a + b*x^2]/x,x]

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32

method	result	size
risch	\(-\frac {{\mathrm e}^{2 a} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (-b \,x^{2}\right )}{4}-\frac {{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b \,x^{2}\right )}{4}\)	\(33\)
meijerg	\(\frac {\cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +4 \ln \left (x \right )+2 \ln \left (i b \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (b \,x^{2}\right )-2 \ln \left (b \,x^{2}\right )-2 \gamma }{\sqrt {\pi }}\right )}{4}+\frac {\operatorname {Shi}\left (b \,x^{2}\right ) \sinh \left (a \right )}{2}\)	\(62\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {\cosh \left (a+b x^2\right )}{x} \, dx=\frac {1}{4} \, {\left ({\rm Ei}\left (b x^{2}\right ) + {\rm Ei}\left (-b x^{2}\right )\right )} \cosh \left (a\right ) + \frac {1}{4} \, {\left ({\rm Ei}\left (b x^{2}\right ) - {\rm Ei}\left (-b x^{2}\right )\right )} \sinh \left (a\right ) \]

[In]

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

[Out]

1/4*(Ei(b*x^2) + Ei(-b*x^2))*cosh(a) + 1/4*(Ei(b*x^2) - Ei(-b*x^2))*sinh(a)

Sympy [F]

\[ \int \frac {\cosh \left (a+b x^2\right )}{x} \, dx=\int \frac {\cosh {\left (a + b x^{2} \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(cosh(a + b*x**2)/x, x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\cosh \left (a+b x^2\right )}{x} \, dx=\frac {1}{4} \, {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + \frac {1}{4} \, {\rm Ei}\left (b x^{2}\right ) e^{a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\cosh \left (a+b x^2\right )}{x} \, dx=\frac {1}{4} \, {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + \frac {1}{4} \, {\rm Ei}\left (b x^{2}\right ) e^{a} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh \left (a+b x^2\right )}{x} \, dx=\frac {\mathrm {cosh}\left (a\right )\,\mathrm {coshint}\left (b\,x^2\right )}{2}+\frac {\mathrm {sinh}\left (a\right )\,\mathrm {sinhint}\left (b\,x^2\right )}{2} \]

[In]

int(cosh(a + b*x^2)/x,x)

[Out]

(cosh(a)*coshint(b*x^2))/2 + (sinh(a)*sinhint(b*x^2))/2