Optimal. Leaf size=42 \[ -\frac {\cosh \left (a+b x^2\right )}{2 x^2}+\frac {1}{2} b \text {Chi}\left (b x^2\right ) \sinh (a)+\frac {1}{2} b \cosh (a) \text {Shi}\left (b x^2\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 42, 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 = {5429, 3378,
3384, 3379, 3382} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5429
Rubi steps
\begin {align*} \int \frac {\cosh \left (a+b x^2\right )}{x^3} \, dx &=\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*}
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Mathematica [A]
time = 0.03, size = 38, normalized size = 0.90 \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 58, normalized size = 1.38
method | result | size |
risch | \(-\frac {{\mathrm e}^{-a} {\mathrm e}^{-x^{2} b}}{4 x^{2}}+\frac {{\mathrm e}^{-a} b \expIntegral \left (1, x^{2} b \right )}{4}-\frac {{\mathrm e}^{a} {\mathrm e}^{x^{2} b}}{4 x^{2}}-\frac {{\mathrm e}^{a} b \expIntegral \left (1, -x^{2} b \right )}{4}\) | \(58\) |
meijerg | \(\frac {i \cosh \left (a \right ) \sqrt {\pi }\, b \left (\frac {4 i \cosh \left (x^{2} b \right )}{b \,x^{2} \sqrt {\pi }}-\frac {4 i \hyperbolicSineIntegral \left (x^{2} b \right )}{\sqrt {\pi }}\right )}{8}+\frac {\sinh \left (a \right ) \sqrt {\pi }\, b \left (\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (x^{2} b \right )}{\sqrt {\pi }\, x^{2} b}+\frac {4 \hyperbolicCosineIntegral \left (x^{2} b \right )-4 \ln \left (x^{2} b \right )-4 \gamma }{\sqrt {\pi }}+\frac {4 \gamma -4+8 \ln \left (x \right )+4 \ln \left (i b \right )}{\sqrt {\pi }}\right )}{8}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 40, normalized size = 0.95 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 71, normalized size = 1.69 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh {\left (a + b x^{2} \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs.
\(2 (36) = 72\).
time = 0.40, size = 108, normalized size = 2.57 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {cosh}\left (b\,x^2+a\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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