Integrand size = 12, antiderivative size = 42 \[ \int \frac {\cosh \left (a+b x^2\right )}{x^3} \, dx=-\frac {\cosh \left (a+b x^2\right )}{2 x^2}+\frac {1}{2} b \text {Chi}\left (b x^2\right ) \sinh (a)+\frac {1}{2} b \cosh (a) \text {Shi}\left (b x^2\right ) \]
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Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5429, 3378, 3384, 3379, 3382} \[ \int \frac {\cosh \left (a+b x^2\right )}{x^3} \, dx=\frac {1}{2} b \sinh (a) \text {Chi}\left (b x^2\right )+\frac {1}{2} b \cosh (a) \text {Shi}\left (b x^2\right )-\frac {\cosh \left (a+b x^2\right )}{2 x^2} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5429
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\cosh (a+b x)}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {\cosh \left (a+b x^2\right )}{2 x^2}+\frac {1}{2} b \text {Subst}\left (\int \frac {\sinh (a+b x)}{x} \, dx,x,x^2\right ) \\ & = -\frac {\cosh \left (a+b x^2\right )}{2 x^2}+\frac {1}{2} (b \cosh (a)) \text {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,x^2\right )+\frac {1}{2} (b \sinh (a)) \text {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,x^2\right ) \\ & = -\frac {\cosh \left (a+b x^2\right )}{2 x^2}+\frac {1}{2} b \text {Chi}\left (b x^2\right ) \sinh (a)+\frac {1}{2} b \cosh (a) \text {Shi}\left (b x^2\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90 \[ \int \frac {\cosh \left (a+b x^2\right )}{x^3} \, dx=\frac {1}{2} \left (-\frac {\cosh \left (a+b x^2\right )}{x^2}+b \text {Chi}\left (b x^2\right ) \sinh (a)+b \cosh (a) \text {Shi}\left (b x^2\right )\right ) \]
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Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.38
method | result | size |
risch | \(-\frac {-{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b \,x^{2}\right ) b \,x^{2}+\operatorname {Ei}_{1}\left (-b \,x^{2}\right ) {\mathrm e}^{a} b \,x^{2}+{\mathrm e}^{-b \,x^{2}-a}+{\mathrm e}^{b \,x^{2}+a}}{4 x^{2}}\) | \(58\) |
meijerg | \(\frac {i \cosh \left (a \right ) \sqrt {\pi }\, b \left (\frac {4 i \cosh \left (b \,x^{2}\right )}{b \,x^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (b \,x^{2}\right )}{\sqrt {\pi }}\right )}{8}+\frac {\sinh \left (a \right ) \sqrt {\pi }\, b \left (\frac {4 \gamma -4+8 \ln \left (x \right )+4 \ln \left (i b \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (b \,x^{2}\right )}{\sqrt {\pi }\, b \,x^{2}}+\frac {4 \,\operatorname {Chi}\left (b \,x^{2}\right )-4 \ln \left (b \,x^{2}\right )-4 \gamma }{\sqrt {\pi }}\right )}{8}\) | \(117\) |
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Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.69 \[ \int \frac {\cosh \left (a+b x^2\right )}{x^3} \, dx=\frac {{\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) - b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \cosh \left (a\right ) + {\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) + b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \sinh \left (a\right ) - 2 \, \cosh \left (b x^{2} + a\right )}{4 \, x^{2}} \]
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\[ \int \frac {\cosh \left (a+b x^2\right )}{x^3} \, dx=\int \frac {\cosh {\left (a + b x^{2} \right )}}{x^{3}}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int \frac {\cosh \left (a+b x^2\right )}{x^3} \, dx=-\frac {1}{4} \, {\left ({\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} - {\rm Ei}\left (b x^{2}\right ) e^{a}\right )} b - \frac {\cosh \left (b x^{2} + a\right )}{2 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (36) = 72\).
Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.57 \[ \int \frac {\cosh \left (a+b x^2\right )}{x^3} \, dx=-\frac {{\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} - a b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} - {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} + a b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} + b^{2} e^{\left (b x^{2} + a\right )} + b^{2} e^{\left (-b x^{2} - a\right )}}{4 \, b^{2} x^{2}} \]
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Timed out. \[ \int \frac {\cosh \left (a+b x^2\right )}{x^3} \, dx=\int \frac {\mathrm {cosh}\left (b\,x^2+a\right )}{x^3} \,d x \]
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