Optimal. Leaf size=87 \[ \frac {\sqrt {f} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {f} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sinh (f x)}{d \sqrt {d x}} \]
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Rubi [A]
time = 0.08, antiderivative size = 87, 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 = {3378, 3388,
2211, 2235, 2236} \begin {gather*} \frac {\sqrt {\pi } \sqrt {f} \text {Erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {\pi } \sqrt {f} \text {Erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sinh (f x)}{d \sqrt {d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3378
Rule 3388
Rubi steps
\begin {align*} \int \frac {\sinh (f x)}{(d x)^{3/2}} \, dx &=-\frac {2 \sinh (f x)}{d \sqrt {d x}}+\frac {(2 f) \int \frac {\cosh (f x)}{\sqrt {d x}} \, dx}{d}\\ &=-\frac {2 \sinh (f x)}{d \sqrt {d x}}+\frac {f \int \frac {e^{-f x}}{\sqrt {d x}} \, dx}{d}+\frac {f \int \frac {e^{f x}}{\sqrt {d x}} \, dx}{d}\\ &=-\frac {2 \sinh (f x)}{d \sqrt {d x}}+\frac {(2 f) \text {Subst}\left (\int e^{-\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{d^2}+\frac {(2 f) \text {Subst}\left (\int e^{\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{d^2}\\ &=\frac {\sqrt {f} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {f} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sinh (f x)}{d \sqrt {d x}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 0.56 \begin {gather*} \frac {x \left (\sqrt {-f x} \Gamma \left (\frac {1}{2},-f x\right )-\sqrt {f x} \Gamma \left (\frac {1}{2},f x\right )-2 \sinh (f x)\right )}{(d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.20, size = 120, normalized size = 1.38
method | result | size |
meijerg | \(-\frac {\sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {2}\, \left (i f \right )^{\frac {3}{2}} \left (\frac {2 \sqrt {2}\, \sqrt {i f}\, {\mathrm e}^{-f x}}{\sqrt {\pi }\, \sqrt {x}\, f}-\frac {2 \sqrt {2}\, \sqrt {i f}\, {\mathrm e}^{f x}}{\sqrt {\pi }\, \sqrt {x}\, f}+\frac {2 \sqrt {i f}\, \sqrt {2}\, \erf \left (\sqrt {x}\, \sqrt {f}\right )}{\sqrt {f}}+\frac {2 \sqrt {i f}\, \sqrt {2}\, \erfi \left (\sqrt {x}\, \sqrt {f}\right )}{\sqrt {f}}\right )}{4 \left (d x \right )^{\frac {3}{2}} f}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 74, normalized size = 0.85 \begin {gather*} \frac {\frac {f {\left (\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{\sqrt {\frac {f}{d}}} + \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{\sqrt {-\frac {f}{d}}}\right )}}{d} - \frac {2 \, \sinh \left (f x\right )}{\sqrt {d x}}}{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. 137 vs.
\(2 (61) = 122\).
time = 0.44, size = 137, normalized size = 1.57 \begin {gather*} \frac {\sqrt {\pi } {\left (d x \cosh \left (f x\right ) + d x \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 x \cosh \left (f x\right ) + d x \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) - \sqrt {d x} {\left (\cosh \left (f x\right )^{2} + 2 \, \cosh \left (f x\right ) \sinh \left (f x\right ) + \sinh \left (f x\right )^{2} - 1\right )}}{d^{2} x \cosh \left (f x\right ) + d^{2} x \sinh \left (f x\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 = 1.99, size = 94, normalized size = 1.08 \begin {gather*} \frac {\sqrt {2} \sqrt {\pi } \sqrt {f} 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 {1}{4}\right )}{2 d^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} - \frac {\sinh {\left (f x \right )} \Gamma \left (\frac {1}{4}\right )}{2 d^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {5}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 )}{{\left (d\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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