Optimal. Leaf size=53 \[ -\frac {e^{-a} \sqrt {\pi } \text {Erf}\left (\sqrt {b} x\right )}{4 \sqrt {b}}+\frac {e^a \sqrt {\pi } \text {Erfi}\left (\sqrt {b} x\right )}{4 \sqrt {b}} \]
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Rubi [A]
time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5406, 2235,
2236} \begin {gather*} \frac {\sqrt {\pi } e^a \text {Erfi}\left (\sqrt {b} x\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } e^{-a} \text {Erf}\left (\sqrt {b} x\right )}{4 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 5406
Rubi steps
\begin {align*} \int \sinh \left (a+b x^2\right ) \, dx &=-\left (\frac {1}{2} \int e^{-a-b x^2} \, dx\right )+\frac {1}{2} \int e^{a+b x^2} \, dx\\ &=-\frac {e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{4 \sqrt {b}}+\frac {e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{4 \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 45, normalized size = 0.85 \begin {gather*} \frac {\sqrt {\pi } \left (\text {Erf}\left (\sqrt {b} x\right ) (-\cosh (a)+\sinh (a))+\text {Erfi}\left (\sqrt {b} x\right ) (\cosh (a)+\sinh (a))\right )}{4 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 40, normalized size = 0.75
method | result | size |
risch | \(-\frac {\erf \left (x \sqrt {b}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{4 \sqrt {b}}+\frac {{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x \right )}{4 \sqrt {-b}}\) | \(40\) |
meijerg | \(\frac {\sinh \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, \left (\frac {\sqrt {i b}\, \sqrt {2}\, \erf \left (x \sqrt {b}\right )}{2 \sqrt {b}}+\frac {\sqrt {i b}\, \sqrt {2}\, \erfi \left (x \sqrt {b}\right )}{2 \sqrt {b}}\right )}{4 \sqrt {i b}}-\frac {i \cosh \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {\left (i b \right )^{\frac {3}{2}} \sqrt {2}\, \erf \left (x \sqrt {b}\right )}{2 b^{\frac {3}{2}}}+\frac {\left (i b \right )^{\frac {3}{2}} \sqrt {2}\, \erfi \left (x \sqrt {b}\right )}{2 b^{\frac {3}{2}}}\right )}{4 \sqrt {i b}}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs.
\(2 (35) = 70\).
time = 0.26, size = 86, normalized size = 1.62 \begin {gather*} -\frac {1}{4} \, b {\left (\frac {2 \, x e^{\left (b x^{2} + a\right )}}{b} - \frac {2 \, x e^{\left (-b x^{2} - a\right )}}{b} + \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{b^{\frac {3}{2}}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{\sqrt {-b} b}\right )} + x \sinh \left (b x^{2} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 48, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-b} {\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-b} x\right ) + \sqrt {\pi } \sqrt {b} {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {b} x\right )}{4 \, b} \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 \sinh {\left (a + b x^{2} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 41, normalized size = 0.77 \begin {gather*} \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x\right ) e^{\left (-a\right )}}{4 \, \sqrt {b}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} x\right ) e^{a}}{4 \, \sqrt {-b}} \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 \mathrm {sinh}\left (b\,x^2+a\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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