Optimal. Leaf size=75 \[ -\frac {\sqrt {\pi } e^{-a} \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{8 b^{3/2}}+\frac {\sqrt {\pi } e^a \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{8 b^{3/2}}-\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5347, 5325, 5298, 2204, 2205} \[ -\frac {\sqrt {\pi } e^{-a} \text {Erf}\left (\frac {\sqrt {b}}{x}\right )}{8 b^{3/2}}+\frac {\sqrt {\pi } e^a \text {Erfi}\left (\frac {\sqrt {b}}{x}\right )}{8 b^{3/2}}-\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2204
Rule 2205
Rule 5298
Rule 5325
Rule 5347
Rubi steps
\begin {align*} \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x^4} \, dx &=-\operatorname {Subst}\left (\int x^2 \cosh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b x}+\frac {\operatorname {Subst}\left (\int \sinh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )}{2 b}\\ &=-\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\operatorname {Subst}\left (\int e^{-a-b x^2} \, dx,x,\frac {1}{x}\right )}{4 b}+\frac {\operatorname {Subst}\left (\int e^{a+b x^2} \, dx,x,\frac {1}{x}\right )}{4 b}\\ &=-\frac {e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{8 b^{3/2}}+\frac {e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{8 b^{3/2}}-\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 74, normalized size = 0.99 \[ \frac {\sqrt {\pi } x (\sinh (a)-\cosh (a)) \text {erf}\left (\frac {\sqrt {b}}{x}\right )+\sqrt {\pi } x (\sinh (a)+\cosh (a)) \text {erfi}\left (\frac {\sqrt {b}}{x}\right )-4 \sqrt {b} \sinh \left (a+\frac {b}{x^2}\right )}{8 b^{3/2} x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 250, normalized size = 3.33 \[ -\frac {2 \, b \cosh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} + \sqrt {\pi } {\left (x \cosh \relax (a) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + x \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \relax (a) + {\left (x \cosh \relax (a) + x \sinh \relax (a)\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {-b} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) + \sqrt {\pi } {\left (x \cosh \relax (a) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - x \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \relax (a) + {\left (x \cosh \relax (a) - x \sinh \relax (a)\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) + 4 \, b \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) + 2 \, b \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} - 2 \, b}{8 \, {\left (b^{2} x \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + b^{2} x \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (a + \frac {b}{x^{2}}\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.13, size = 82, normalized size = 1.09 \[ \frac {{\mathrm e}^{-a} {\mathrm e}^{-\frac {b}{x^{2}}}}{4 b x}-\frac {{\mathrm e}^{-a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {b}}{x}\right )}{8 b^{\frac {3}{2}}}-\frac {{\mathrm e}^{a} {\mathrm e}^{\frac {b}{x^{2}}}}{4 x b}+\frac {{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-b}}{x}\right )}{8 b \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 63, normalized size = 0.84 \[ \frac {1}{6} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (\frac {5}{2}, \frac {b}{x^{2}}\right )}{x^{5} \left (\frac {b}{x^{2}}\right )^{\frac {5}{2}}} - \frac {e^{a} \Gamma \left (\frac {5}{2}, -\frac {b}{x^{2}}\right )}{x^{5} \left (-\frac {b}{x^{2}}\right )^{\frac {5}{2}}}\right )} - \frac {\cosh \left (a + \frac {b}{x^{2}}\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (a+\frac {b}{x^2}\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (a + \frac {b}{x^{2}} \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________