Optimal. Leaf size=77 \[ \frac{\sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}}-\frac{\sqrt{\pi } \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}} \]
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Rubi [A] time = 0.0761857, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3308, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}}-\frac{\sqrt{\pi } \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
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Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{\sinh (f x)}{\sqrt{d x}} \, dx &=-\left (\frac{1}{2} \int \frac{e^{-f x}}{\sqrt{d x}} \, dx\right )+\frac{1}{2} \int \frac{e^{f x}}{\sqrt{d x}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int e^{-\frac{f x^2}{d}} \, dx,x,\sqrt{d x}\right )}{d}+\frac{\operatorname{Subst}\left (\int e^{\frac{f x^2}{d}} \, dx,x,\sqrt{d x}\right )}{d}\\ &=-\frac{\sqrt{\pi } \text{erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}}+\frac{\sqrt{\pi } \text{erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}}\\ \end{align*}
Mathematica [A] time = 0.0087227, size = 47, normalized size = 0.61 \[ \frac{\sqrt{-f x} \text{Gamma}\left (\frac{1}{2},-f x\right )+\sqrt{f x} \text{Gamma}\left (\frac{1}{2},f x\right )}{2 f \sqrt{d x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 71, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{\pi }\sqrt{2}}{2\,f}\sqrt{x}\sqrt{if} \left ( -{\frac{\sqrt{2}}{2} \left ( if \right ) ^{{\frac{3}{2}}}{\it Erf} \left ( \sqrt{x}\sqrt{f} \right ){f}^{-{\frac{3}{2}}}}+{\frac{\sqrt{2}}{2} \left ( if \right ) ^{{\frac{3}{2}}}{\it erfi} \left ( \sqrt{x}\sqrt{f} \right ){f}^{-{\frac{3}{2}}}} \right ){\frac{1}{\sqrt{dx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.21827, size = 157, normalized size = 2.04 \begin{align*} \frac{4 \, \sqrt{d x} \sinh \left (f x\right ) - \frac{{\left (\frac{2 \, \sqrt{d x} d e^{\left (f x\right )}}{f} - \frac{2 \, \sqrt{d x} d e^{\left (-f x\right )}}{f} + \frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{f}{d}}\right )}{f \sqrt{\frac{f}{d}}} - \frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{f}{d}}\right )}{f \sqrt{-\frac{f}{d}}}\right )} f}{d}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.63551, size = 138, normalized size = 1.79 \begin{align*} -\frac{\sqrt{\pi } \sqrt{\frac{f}{d}} \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{f}{d}}\right ) + \sqrt{\pi } \sqrt{-\frac{f}{d}} \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{f}{d}}\right )}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.47673, size = 70, normalized size = 0.91 \begin{align*} \frac{3 \sqrt{2} \sqrt{\pi } e^{- \frac{3 i \pi }{4}} S\left (\frac{\sqrt{2} \sqrt{f} \sqrt{x} e^{\frac{i \pi }{4}}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{3}{4}\right )}{4 \sqrt{d} \sqrt{f} \Gamma \left (\frac{7}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.314, size = 82, normalized size = 1.06 \begin{align*} \frac{\frac{\sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{d f} \sqrt{d x}}{d}\right )}{\sqrt{d f}} - \frac{\sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{-d f} \sqrt{d x}}{d}\right )}{\sqrt{-d f}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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