\(\int \cosh (a+\frac {b}{x^2}) \, dx\) [30]

Optimal result

Integrand size = 8, antiderivative size = 67 \[ \int \cosh \left (a+\frac {b}{x^2}\right ) \, dx=x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{2} \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{2} \sqrt {b} e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right ) \]

[Out]

x*cosh(a+b/x^2)+1/2*erf(b^(1/2)/x)*b^(1/2)*Pi^(1/2)/exp(a)-1/2*exp(a)*erfi(b^(1/2)/x)*b^(1/2)*Pi^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5411, 5435, 5406, 2235, 2236} \[ \int \cosh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{2} \sqrt {\pi } e^{-a} \sqrt {b} \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{2} \sqrt {\pi } e^a \sqrt {b} \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+x \cosh \left (a+\frac {b}{x^2}\right ) \]

[In]

Int[Cosh[a + b/x^2],x]

[Out]

x*Cosh[a + b/x^2] + (Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]/x])/(2*E^a) - (Sqrt[b]*E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/2

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 5406

Int[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5411

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.), x_Symbol] :> -Subst[Int[(a + b*Cosh[c + d/x^n])^p/x^2
, x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n, 0] && IntegerQ[p]

Rule 5435

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*x)^(m + 1)*(Cosh[c + d*x^n]/(e*(m +
1))), x] - Dist[d*(n/(e^n*(m + 1))), Int[(e*x)^(m + n)*Sinh[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[
n, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\cosh \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = x \cosh \left (a+\frac {b}{x^2}\right )-(2 b) \text {Subst}\left (\int \sinh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right ) \\ & = x \cosh \left (a+\frac {b}{x^2}\right )+b \text {Subst}\left (\int e^{-a-b x^2} \, dx,x,\frac {1}{x}\right )-b \text {Subst}\left (\int e^{a+b x^2} \, dx,x,\frac {1}{x}\right ) \\ & = x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{2} \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{2} \sqrt {b} e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06 \[ \int \cosh \left (a+\frac {b}{x^2}\right ) \, dx=x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{2} \sqrt {b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)-\sinh (a))-\frac {1}{2} \sqrt {b} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)+\sinh (a)) \]

[In]

Integrate[Cosh[a + b/x^2],x]

[Out]

x*Cosh[a + b/x^2] + (Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]/x]*(Cosh[a] - Sinh[a]))/2 - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[b]/x
]*(Cosh[a] + Sinh[a]))/2

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04

method	result	size
risch	\(\frac {\operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) \sqrt {b}\, \sqrt {\pi }\, {\mathrm e}^{-a}}{2}+\frac {{\mathrm e}^{-a} x \,{\mathrm e}^{-\frac {b}{x^{2}}}}{2}+\frac {{\mathrm e}^{a} {\mathrm e}^{\frac {b}{x^{2}}} x}{2}-\frac {{\mathrm e}^{a} b \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right )}{2 \sqrt {-b}}\)	\(70\)
meijerg	\(-\frac {\sqrt {\pi }\, \cosh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (-\frac {2 x \sqrt {2}\, {\mathrm e}^{\frac {b}{x^{2}}}}{\sqrt {\pi }\, \sqrt {i b}}-\frac {2 x \sqrt {2}\, {\mathrm e}^{-\frac {b}{x^{2}}}}{\sqrt {\pi }\, \sqrt {i b}}-\frac {2 \sqrt {2}\, \sqrt {b}\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{\sqrt {i b}}+\frac {2 \sqrt {2}\, \sqrt {b}\, \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{\sqrt {i b}}\right )}{8}+\frac {i \sqrt {\pi }\, \sinh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (\frac {2 x \sqrt {2}\, \sqrt {i b}\, {\mathrm e}^{-\frac {b}{x^{2}}}}{\sqrt {\pi }\, b}-\frac {2 x \sqrt {2}\, \sqrt {i b}\, {\mathrm e}^{\frac {b}{x^{2}}}}{\sqrt {\pi }\, b}+\frac {2 \sqrt {i b}\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{\sqrt {b}}+\frac {2 \sqrt {i b}\, \sqrt {2}\, \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{\sqrt {b}}\right )}{8}\)	\(217\)

[In]

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

[Out]

1/2*erf(b^(1/2)/x)*b^(1/2)*Pi^(1/2)/exp(a)+1/2/exp(a)*x*exp(-b/x^2)+1/2*exp(a)*exp(b/x^2)*x-1/2*exp(a)*b*Pi^(1
/2)/(-b)^(1/2)*erf((-b)^(1/2)/x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (49) = 98\).

Time = 0.26 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.36 \[ \int \cosh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {x \cosh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} + \sqrt {\pi } {\left (\cosh \left (a\right ) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) + {\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {-b} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) + \sqrt {\pi } {\left (\cosh \left (a\right ) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) + {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) + 2 \, x \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) + x \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} + x}{2 \, {\left (\cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )}} \]

[In]

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

[Out]

1/2*(x*cosh((a*x^2 + b)/x^2)^2 + sqrt(pi)*(cosh(a)*cosh((a*x^2 + b)/x^2) + cosh((a*x^2 + b)/x^2)*sinh(a) + (co
sh(a) + sinh(a))*sinh((a*x^2 + b)/x^2))*sqrt(-b)*erf(sqrt(-b)/x) + sqrt(pi)*(cosh(a)*cosh((a*x^2 + b)/x^2) - c
osh((a*x^2 + b)/x^2)*sinh(a) + (cosh(a) - sinh(a))*sinh((a*x^2 + b)/x^2))*sqrt(b)*erf(sqrt(b)/x) + 2*x*cosh((a
*x^2 + b)/x^2)*sinh((a*x^2 + b)/x^2) + x*sinh((a*x^2 + b)/x^2)^2 + x)/(cosh((a*x^2 + b)/x^2) + sinh((a*x^2 + b
)/x^2))

Sympy [F]

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

[In]

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

[Out]

Integral(cosh(a + b/x**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.07 \[ \int \cosh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{2} \, b {\left (\frac {\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {b}{x^{2}}}\right ) - 1\right )} e^{\left (-a\right )}}{x \sqrt {\frac {b}{x^{2}}}} - \frac {\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {b}{x^{2}}}\right ) - 1\right )} e^{a}}{x \sqrt {-\frac {b}{x^{2}}}}\right )} + x \cosh \left (a + \frac {b}{x^{2}}\right ) \]

[In]

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

[Out]

1/2*b*(sqrt(pi)*(erf(sqrt(b/x^2)) - 1)*e^(-a)/(x*sqrt(b/x^2)) - sqrt(pi)*(erf(sqrt(-b/x^2)) - 1)*e^a/(x*sqrt(-
b/x^2))) + x*cosh(a + b/x^2)

Giac [F]

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

[In]

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

[Out]

integrate(cosh(a + b/x^2), x)

Mupad [F(-1)]

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

[In]

int(cosh(a + b/x^2),x)

[Out]

int(cosh(a + b/x^2), x)