Optimal. Leaf size=77 \[ -\frac {\sqrt {\pi } \text {Erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}+\frac {\sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}} \]
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Rubi [A]
time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, 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 = {3389, 2211,
2235, 2236} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
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 {\text {Subst}\left (\int e^{-\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{d}+\frac {\text {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*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 0.61 \begin {gather*} \frac {\sqrt {-f x} \Gamma \left (\frac {1}{2},-f x\right )+\sqrt {f x} \Gamma \left (\frac {1}{2},f x\right )}{2 f \sqrt {d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.20, size = 71, normalized size = 0.92
method | result | size |
meijerg | \(-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {i f}\, \left (-\frac {\left (i f \right )^{\frac {3}{2}} \sqrt {2}\, \erf \left (\sqrt {x}\, \sqrt {f}\right )}{2 f^{\frac {3}{2}}}+\frac {\left (i f \right )^{\frac {3}{2}} \sqrt {2}\, \erfi \left (\sqrt {x}\, \sqrt {f}\right )}{2 f^{\frac {3}{2}}}\right )}{2 \sqrt {d x}\, f}\) | \(71\) |
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. 116 vs.
\(2 (49) = 98\).
time = 0.27, size = 116, normalized size = 1.51 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 58, normalized size = 0.75 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.56, size = 70, normalized size = 0.91 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 61, normalized size = 0.79 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {sinh}\left (f\,x\right )}{\sqrt {d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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