Optimal. Leaf size=66 \[ \frac {1}{2} \sqrt {b} e^{-a} \sqrt {\pi } \text {Erf}\left (\sqrt {b} x\right )+\frac {1}{2} \sqrt {b} e^a \sqrt {\pi } \text {Erfi}\left (\sqrt {b} x\right )-\frac {\sinh \left (a+b x^2\right )}{x} \]
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Rubi [A]
time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5434, 5407,
2235, 2236} \begin {gather*} \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 {\sinh \left (a+b x^2\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 5407
Rule 5434
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+b x^2\right )}{x^2} \, dx &=-\frac {\sinh \left (a+b x^2\right )}{x}+(2 b) \int \cosh \left (a+b x^2\right ) \, dx\\ &=-\frac {\sinh \left (a+b x^2\right )}{x}+b \int e^{-a-b x^2} \, dx+b \int e^{a+b x^2} \, dx\\ &=\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 )-\frac {\sinh \left (a+b x^2\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 70, normalized size = 1.06 \begin {gather*} \frac {\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 \sinh \left (a+b x^2\right )}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 70, normalized size = 1.06
method | result | size |
risch | \(\frac {{\mathrm e}^{-a} {\mathrm e}^{-x^{2} b}}{2 x}+\frac {{\mathrm e}^{-a} \sqrt {b}\, \sqrt {\pi }\, \erf \left (x \sqrt {b}\right )}{2}-\frac {{\mathrm e}^{a} {\mathrm e}^{x^{2} b}}{2 x}+\frac {{\mathrm e}^{a} b \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x \right )}{2 \sqrt {-b}}\) | \(70\) |
meijerg | \(\frac {i \sinh \left (a \right ) \sqrt {\pi }\, b \sqrt {2}\, \left (-\frac {2 \sqrt {2}\, {\mathrm e}^{x^{2} b}}{\sqrt {\pi }\, x \sqrt {i b}}-\frac {2 \sqrt {2}\, {\mathrm e}^{-x^{2} b}}{\sqrt {\pi }\, x \sqrt {i b}}-\frac {2 \sqrt {2}\, \sqrt {b}\, \erf \left (x \sqrt {b}\right )}{\sqrt {i b}}+\frac {2 \sqrt {2}\, \sqrt {b}\, \erfi \left (x \sqrt {b}\right )}{\sqrt {i b}}\right )}{8 \sqrt {i b}}+\frac {\cosh \left (a \right ) \sqrt {\pi }\, b \sqrt {2}\, \left (\frac {2 \sqrt {2}\, \sqrt {i b}\, {\mathrm e}^{-x^{2} b}}{\sqrt {\pi }\, x b}-\frac {2 \sqrt {2}\, \sqrt {i b}\, {\mathrm e}^{x^{2} b}}{\sqrt {\pi }\, x b}+\frac {2 \sqrt {i b}\, \sqrt {2}\, \erf \left (x \sqrt {b}\right )}{\sqrt {b}}+\frac {2 \sqrt {i b}\, \sqrt {2}\, \erfi \left (x \sqrt {b}\right )}{\sqrt {b}}\right )}{8 \sqrt {i b}}\) | \(219\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 54, normalized size = 0.82 \begin {gather*} \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 {\sinh \left (b x^{2} + a\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 184 vs.
\(2 (48) = 96\).
time = 0.37, size = 184, normalized size = 2.79 \begin {gather*} -\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 )}} \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 {\sinh {\left (a + b x^{2} \right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {sinh}\left (b\,x^2+a\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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