Optimal. Leaf size=66 \[ \frac {1}{2} \sqrt {\pi } e^{-a} \sqrt {b} \text {erf}\left (\sqrt {b} x\right )+\frac {1}{2} \sqrt {\pi } e^a \sqrt {b} \text {erfi}\left (\sqrt {b} x\right )-\frac {\sinh \left (a+b x^2\right )}{x} \]
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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 = {5326, 5299, 2204, 2205} \[ \frac {1}{2} \sqrt {\pi } e^{-a} \sqrt {b} \text {Erf}\left (\sqrt {b} x\right )+\frac {1}{2} \sqrt {\pi } e^a \sqrt {b} \text {Erfi}\left (\sqrt {b} x\right )-\frac {\sinh \left (a+b x^2\right )}{x} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5299
Rule 5326
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 = 0.06, size = 70, normalized size = 1.06 \[ \frac {\sqrt {\pi } \sqrt {b} x (\cosh (a)-\sinh (a)) \text {erf}\left (\sqrt {b} x\right )+\sqrt {\pi } \sqrt {b} x (\sinh (a)+\cosh (a)) \text {erfi}\left (\sqrt {b} x\right )-2 \sinh \left (a+b x^2\right )}{2 x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 184, normalized size = 2.79 \[ -\frac {\sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right ) \cosh \relax (a) + x \cosh \left (b x^{2} + a\right ) \sinh \relax (a) + {\left (x \cosh \relax (a) + x \sinh \relax (a)\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {-b} \operatorname {erf}\left (\sqrt {-b} x\right ) - \sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right ) \cosh \relax (a) - x \cosh \left (b x^{2} + a\right ) \sinh \relax (a) + {\left (x \cosh \relax (a) - x \sinh \relax (a)\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {b} x\right ) + \cosh \left (b x^{2} + a\right )^{2} + 2 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right ) + \sinh \left (b x^{2} + a\right )^{2} - 1}{2 \, {\left (x \cosh \left (b x^{2} + a\right ) + x \sinh \left (b x^{2} + a\right )\right )}} \]
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 \[ \int \frac {\sinh \left (b x^{2} + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 70, normalized size = 1.06 \[ \frac {{\mathrm e}^{-a} {\mathrm e}^{-b \,x^{2}}}{2 x}+\frac {{\mathrm e}^{-a} \sqrt {b}\, \sqrt {\pi }\, \erf \left (x \sqrt {b}\right )}{2}-\frac {{\mathrm e}^{a} {\mathrm e}^{b \,x^{2}}}{2 x}+\frac {{\mathrm e}^{a} b \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x \right )}{2 \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 54, normalized size = 0.82 \[ \frac {1}{2} \, {\left (\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{\sqrt {b}} + \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{\sqrt {-b}}\right )} b - \frac {\sinh \left (b x^{2} + a\right )}{x} \]
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 \[ \int \frac {\mathrm {sinh}\left (b\,x^2+a\right )}{x^2} \,d x \]
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 \[ \int \frac {\sinh {\left (a + b x^{2} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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