Optimal. Leaf size=92 \[ \frac {\sqrt {d} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}-\frac {\sqrt {d} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}+\frac {\sqrt {d x} \sinh (f x)}{f} \]
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Rubi [A]
time = 0.07, 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 = {3377, 3389,
2211, 2235, 2236} \begin {gather*} \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} \sinh (f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3377
Rule 3389
Rubi steps
\begin {align*} \int \sqrt {d x} \cosh (f x) \, dx &=\frac {\sqrt {d x} \sinh (f x)}{f}-\frac {d \int \frac {\sinh (f x)}{\sqrt {d x}} \, dx}{2 f}\\ &=\frac {\sqrt {d x} \sinh (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} \sinh (f x)}{f}+\frac {\text {Subst}\left (\int e^{-\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{2 f}-\frac {\text {Subst}\left (\int e^{\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{2 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}}+\frac {\sqrt {d x} \sinh (f x)}{f}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 48, normalized size = 0.52 \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.72, size = 121, normalized size = 1.32
method | result | size |
meijerg | \(-\frac {i \sqrt {\pi }\, \sqrt {d x}\, \sqrt {2}\, \left (\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {3}{2}} {\mathrm e}^{f x}}{4 \sqrt {\pi }\, f}-\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {3}{2}} {\mathrm e}^{-f x}}{4 \sqrt {\pi }\, f}+\frac {\left (i f \right )^{\frac {3}{2}} \sqrt {2}\, \erf \left (\sqrt {x}\, \sqrt {f}\right )}{8 f^{\frac {3}{2}}}-\frac {\left (i f \right )^{\frac {3}{2}} \sqrt {2}\, \erfi \left (\sqrt {x}\, \sqrt {f}\right )}{8 f^{\frac {3}{2}}}\right )}{\sqrt {x}\, \sqrt {i f}\, f}\) | \(121\) |
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. 148 vs.
\(2 (62) = 124\).
time = 0.28, size = 148, normalized size = 1.61 \begin {gather*} \frac {8 \, \left (d x\right )^{\frac {3}{2}} \cosh \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 138 vs.
\(2 (62) = 124\).
time = 0.38, size = 138, normalized size = 1.50 \begin {gather*} \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 )}} \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 = 0.86, size = 100, normalized size = 1.09 \begin {gather*} \frac {3 \sqrt {d} \sqrt {x} \sinh {\left (f x \right )} \Gamma \left (\frac {3}{4}\right )}{4 f \Gamma \left (\frac {7}{4}\right )} - \frac {3 \sqrt {2} \sqrt {\pi } \sqrt {d} 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 )}{8 f^{\frac {3}{2}} \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 = 0.43, size = 108, normalized size = 1.17 \begin {gather*} -\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} \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 \mathrm {cosh}\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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