Optimal. Leaf size=75 \[ -\frac{\sqrt{\pi } e^{-a} \text{Erf}\left (\frac{\sqrt{b}}{x}\right )}{8 b^{3/2}}+\frac{\sqrt{\pi } e^a \text{Erfi}\left (\frac{\sqrt{b}}{x}\right )}{8 b^{3/2}}-\frac{\sinh \left (a+\frac{b}{x^2}\right )}{2 b x} \]
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Rubi [A] time = 0.0493617, antiderivative size = 75, normalized size of antiderivative = 1., 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.
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Rule 5347
Rule 5325
Rule 5298
Rule 2204
Rule 2205
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 = 0.0711987, 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.
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Maple [A] time = 0.033, size = 82, normalized size = 1.1 \begin{align*}{\frac{{{\rm e}^{-a}}}{4\,bx}{{\rm e}^{-{\frac{b}{{x}^{2}}}}}}-{\frac{{{\rm e}^{-a}}\sqrt{\pi }}{8}{\it Erf} \left ({\frac{1}{x}\sqrt{b}} \right ){b}^{-{\frac{3}{2}}}}-{\frac{{{\rm e}^{a}}}{4\,bx}{{\rm e}^{{\frac{b}{{x}^{2}}}}}}+{\frac{{{\rm e}^{a}}\sqrt{\pi }}{8\,b}{\it Erf} \left ({\frac{1}{x}\sqrt{-b}} \right ){\frac{1}{\sqrt{-b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16182, size = 85, normalized size = 1.13 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88464, size = 653, normalized size = 8.71 \begin{align*} -\frac{2 \, b \cosh \left (\frac{a x^{2} + b}{x^{2}}\right )^{2} + \sqrt{\pi }{\left (x \cosh \left (a\right ) \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) + x \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) +{\left (x \cosh \left (a\right ) + x \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 (x \cosh \left (a\right ) \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) - x \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) +{\left (x \cosh \left (a\right ) - x \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 ) + 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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (a + \frac{b}{x^{2}} \right )}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (a + \frac{b}{x^{2}}\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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