Optimal. Leaf size=92 \[ -\frac {\sqrt {\pi } \sqrt {d} \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}-\frac {\sqrt {\pi } \sqrt {d} \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}+\frac {\sqrt {d x} \cosh (f x)}{f} \]
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Rubi [A] time = 0.11, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3296, 3307, 2180, 2204, 2205} \[ -\frac {\sqrt {\pi } \sqrt {d} \text {Erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}-\frac {\sqrt {\pi } \sqrt {d} \text {Erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}+\frac {\sqrt {d x} \cosh (f x)}{f} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3296
Rule 3307
Rubi steps
\begin {align*} \int \sqrt {d x} \sinh (f x) \, dx &=\frac {\sqrt {d x} \cosh (f x)}{f}-\frac {d \int \frac {\cosh (f x)}{\sqrt {d x}} \, dx}{2 f}\\ &=\frac {\sqrt {d x} \cosh (f x)}{f}-\frac {d \int \frac {e^{-f x}}{\sqrt {d x}} \, dx}{4 f}-\frac {d \int \frac {e^{f x}}{\sqrt {d x}} \, dx}{4 f}\\ &=\frac {\sqrt {d x} \cosh (f x)}{f}-\frac {\operatorname {Subst}\left (\int e^{-\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{2 f}-\frac {\operatorname {Subst}\left (\int e^{\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{2 f}\\ &=\frac {\sqrt {d x} \cosh (f x)}{f}-\frac {\sqrt {d} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}-\frac {\sqrt {d} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 0.53 \[ \frac {d \left (\sqrt {f x} \Gamma \left (\frac {3}{2},f x\right )-\sqrt {-f x} \Gamma \left (\frac {3}{2},-f x\right )\right )}{2 f^2 \sqrt {d x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 137, normalized size = 1.49 \[ -\frac {\sqrt {\pi } {\left (d \cosh \left (f x\right ) + d \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) - \sqrt {\pi } {\left (d \cosh \left (f x\right ) + d \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) - 2 \, {\left (f \cosh \left (f x\right )^{2} + 2 \, f \cosh \left (f x\right ) \sinh \left (f x\right ) + f \sinh \left (f x\right )^{2} + f\right )} \sqrt {d x}}{4 \, {\left (f^{2} \cosh \left (f x\right ) + f^{2} \sinh \left (f x\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 108, normalized size = 1.17 \[ \frac {\frac {\sqrt {\pi } d^{2} \operatorname {erf}\left (-\frac {\sqrt {d f} \sqrt {d x}}{d}\right )}{\sqrt {d f} f} + \frac {2 \, \sqrt {d x} d e^{\left (-f x\right )}}{f}}{4 \, d} + \frac {\frac {\sqrt {\pi } d^{2} \operatorname {erf}\left (-\frac {\sqrt {-d f} \sqrt {d x}}{d}\right )}{\sqrt {-d f} f} + \frac {2 \, \sqrt {d x} d e^{\left (f x\right )}}{f}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 120, normalized size = 1.30 \[ -\frac {\sqrt {\pi }\, \sqrt {d x}\, \sqrt {2}\, \left (\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {5}{2}} {\mathrm e}^{-f x}}{4 \sqrt {\pi }\, f^{2}}+\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {5}{2}} {\mathrm e}^{f x}}{4 \sqrt {\pi }\, f^{2}}-\frac {\left (i f \right )^{\frac {5}{2}} \sqrt {2}\, \erf \left (\sqrt {x}\, \sqrt {f}\right )}{8 f^{\frac {5}{2}}}-\frac {\left (i f \right )^{\frac {5}{2}} \sqrt {2}\, \erfi \left (\sqrt {x}\, \sqrt {f}\right )}{8 f^{\frac {5}{2}}}\right )}{\sqrt {x}\, \sqrt {i f}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 149, normalized size = 1.62 \[ \frac {8 \, \left (d x\right )^{\frac {3}{2}} \sinh \left (f x\right ) - \frac {f {\left (\frac {3 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{f^{2} \sqrt {\frac {f}{d}}} + \frac {3 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{f^{2} \sqrt {-\frac {f}{d}}} + \frac {2 \, {\left (2 \, \left (d x\right )^{\frac {3}{2}} d f - 3 \, \sqrt {d x} d^{2}\right )} e^{\left (f x\right )}}{f^{2}} - \frac {2 \, {\left (2 \, \left (d x\right )^{\frac {3}{2}} d f + 3 \, \sqrt {d x} d^{2}\right )} e^{\left (-f x\right )}}{f^{2}}\right )}}{d}}{12 \, d} \]
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 \[ \int \mathrm {sinh}\left (f\,x\right )\,\sqrt {d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.68, size = 99, normalized size = 1.08 \[ \frac {5 \sqrt {d} \sqrt {x} \cosh {\left (f x \right )} \Gamma \left (\frac {5}{4}\right )}{4 f \Gamma \left (\frac {9}{4}\right )} - \frac {5 \sqrt {2} \sqrt {\pi } \sqrt {d} e^{- \frac {i \pi }{4}} C\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {5}{4}\right )}{8 f^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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