\(\int \frac {\cosh (a+b x^2)}{x^2} \, dx\) [6]

Optimal result

Integrand size = 12, antiderivative size = 66 \[ \int \frac {\cosh \left (a+b x^2\right )}{x^2} \, dx=-\frac {\cosh \left (a+b x^2\right )}{x}-\frac {1}{2} \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )+\frac {1}{2} \sqrt {b} e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5435, 5406, 2235, 2236} \[ \int \frac {\cosh \left (a+b x^2\right )}{x^2} \, dx=-\frac {1}{2} \sqrt {\pi } e^{-a} \sqrt {b} \text {erf}\left (\sqrt {b} x\right )+\frac {1}{2} \sqrt {\pi } e^a \sqrt {b} \text {erfi}\left (\sqrt {b} x\right )-\frac {\cosh \left (a+b x^2\right )}{x} \]

[In]

Int[Cosh[a + b*x^2]/x^2,x]

[Out]

-(Cosh[a + b*x^2]/x) - (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 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}& = -\frac {\cosh \left (a+b x^2\right )}{x}+(2 b) \int \sinh \left (a+b x^2\right ) \, dx \\ & = -\frac {\cosh \left (a+b x^2\right )}{x}-b \int e^{-a-b x^2} \, dx+b \int e^{a+b x^2} \, dx \\ & = -\frac {\cosh \left (a+b x^2\right )}{x}-\frac {1}{2} \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )+\frac {1}{2} \sqrt {b} e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh \left (a+b x^2\right )}{x^2} \, dx=\frac {-2 \cosh \left (a+b x^2\right )+\sqrt {b} \sqrt {\pi } x \text {erf}\left (\sqrt {b} x\right ) (-\cosh (a)+\sinh (a))+\sqrt {b} \sqrt {\pi } x \text {erfi}\left (\sqrt {b} x\right ) (\cosh (a)+\sinh (a))}{2 x} \]

[In]

Integrate[Cosh[a + b*x^2]/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06

method	result	size
risch	\(-\frac {{\mathrm e}^{-a} {\mathrm e}^{-b \,x^{2}}}{2 x}-\frac {\operatorname {erf}\left (x \sqrt {b}\right ) \sqrt {b}\, \sqrt {\pi }\, {\mathrm e}^{-a}}{2}-\frac {{\mathrm e}^{a} {\mathrm e}^{b \,x^{2}}}{2 x}+\frac {{\mathrm e}^{a} b \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b}\, x \right )}{2 \sqrt {-b}}\)	\(70\)
meijerg	\(\frac {i \cosh \left (a \right ) \sqrt {\pi }\, b \sqrt {2}\, \left (-\frac {2 \sqrt {2}\, {\mathrm e}^{b \,x^{2}}}{\sqrt {\pi }\, x \sqrt {i b}}-\frac {2 \sqrt {2}\, {\mathrm e}^{-b \,x^{2}}}{\sqrt {\pi }\, x \sqrt {i b}}-\frac {2 \sqrt {2}\, \sqrt {b}\, \operatorname {erf}\left (x \sqrt {b}\right )}{\sqrt {i b}}+\frac {2 \sqrt {2}\, \sqrt {b}\, \operatorname {erfi}\left (x \sqrt {b}\right )}{\sqrt {i b}}\right )}{8 \sqrt {i b}}+\frac {\sinh \left (a \right ) \sqrt {\pi }\, b \sqrt {2}\, \left (\frac {2 \sqrt {2}\, \sqrt {i b}\, {\mathrm e}^{-b \,x^{2}}}{\sqrt {\pi }\, x b}-\frac {2 \sqrt {2}\, \sqrt {i b}\, {\mathrm e}^{b \,x^{2}}}{\sqrt {\pi }\, x b}+\frac {2 \sqrt {i b}\, \sqrt {2}\, \operatorname {erf}\left (x \sqrt {b}\right )}{\sqrt {b}}+\frac {2 \sqrt {i b}\, \sqrt {2}\, \operatorname {erfi}\left (x \sqrt {b}\right )}{\sqrt {b}}\right )}{8 \sqrt {i b}}\)	\(219\)

[In]

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

[Out]

-1/2/exp(a)/x*exp(-b*x^2)-1/2*erf(x*b^(1/2))*b^(1/2)*Pi^(1/2)/exp(a)-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. 183 vs. \(2 (48) = 96\).

Time = 0.25 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.77 \[ \int \frac {\cosh \left (a+b x^2\right )}{x^2} \, dx=-\frac {\sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right ) \cosh \left (a\right ) + x \cosh \left (b x^{2} + a\right ) \sinh \left (a\right ) + {\left (x \cosh \left (a\right ) + x \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {-b} \operatorname {erf}\left (\sqrt {-b} x\right ) + \sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right ) \cosh \left (a\right ) - x \cosh \left (b x^{2} + a\right ) \sinh \left (a\right ) + {\left (x \cosh \left (a\right ) - x \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {b} x\right ) + \cosh \left (b x^{2} + a\right )^{2} + 2 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right ) + \sinh \left (b x^{2} + a\right )^{2} + 1}{2 \, {\left (x \cosh \left (b x^{2} + a\right ) + x \sinh \left (b x^{2} + a\right )\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(cosh(a + b*x**2)/x**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh \left (a+b x^2\right )}{x^2} \, dx=-\frac {1}{2} \, {\left (\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{\sqrt {b}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{\sqrt {-b}}\right )} b - \frac {\cosh \left (b x^{2} + a\right )}{x} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(cosh(b*x^2 + a)/x^2, x)

Mupad [F(-1)]

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

[In]

int(cosh(a + b*x^2)/x^2,x)

[Out]

int(cosh(a + b*x^2)/x^2, x)