Optimal. Leaf size=53 \[ \frac{\sqrt{\pi } e^a \text{Erfi}\left (\sqrt{b} x\right )}{4 \sqrt{b}}-\frac{\sqrt{\pi } e^{-a} \text{Erf}\left (\sqrt{b} x\right )}{4 \sqrt{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0182759, antiderivative size = 53, normalized size of antiderivative = 1., 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 = {5298, 2204, 2205} \[ \frac{\sqrt{\pi } e^a \text{Erfi}\left (\sqrt{b} x\right )}{4 \sqrt{b}}-\frac{\sqrt{\pi } e^{-a} \text{Erf}\left (\sqrt{b} x\right )}{4 \sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5298
Rule 2204
Rule 2205
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*}
Mathematica [A] time = 0.0384149, size = 45, normalized size = 0.85 \[ \frac{\sqrt{\pi } \left ((\sinh (a)-\cosh (a)) \text{Erf}\left (\sqrt{b} x\right )+(\sinh (a)+\cosh (a)) \text{Erfi}\left (\sqrt{b} x\right )\right )}{4 \sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.017, size = 40, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{\pi }{{\rm e}^{-a}}}{4}{\it Erf} \left ( x\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}+{\frac{{{\rm e}^{a}}\sqrt{\pi }}{4}{\it Erf} \left ( \sqrt{-b}x \right ){\frac{1}{\sqrt{-b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.14263, size = 116, normalized size = 2.19 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.91211, size = 159, normalized size = 3. \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (a + b x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20513, size = 55, normalized size = 1.04 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]