Optimal. Leaf size=114 \[ -\frac {2 \sqrt {\pi } f^{3/2} \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 \sqrt {\pi } f^{3/2} \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}} \]
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Rubi [A] time = 0.15, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3297, 3308, 2180, 2204, 2205} \[ -\frac {2 \sqrt {\pi } f^{3/2} \text {Erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 \sqrt {\pi } f^{3/2} \text {Erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3297
Rule 3308
Rubi steps
\begin {align*} \int \frac {\sinh (f x)}{(d x)^{5/2}} \, dx &=-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}}+\frac {(2 f) \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx}{3 d}\\ &=-\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}}+\frac {\left (4 f^2\right ) \int \frac {\sinh (f x)}{\sqrt {d x}} \, dx}{3 d^2}\\ &=-\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}}-\frac {\left (2 f^2\right ) \int \frac {e^{-f x}}{\sqrt {d x}} \, dx}{3 d^2}+\frac {\left (2 f^2\right ) \int \frac {e^{f x}}{\sqrt {d x}} \, dx}{3 d^2}\\ &=-\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}}-\frac {\left (4 f^2\right ) \operatorname {Subst}\left (\int e^{-\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{3 d^3}+\frac {\left (4 f^2\right ) \operatorname {Subst}\left (\int e^{\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{3 d^3}\\ &=-\frac {4 f \cosh (f x)}{3 d^2 \sqrt {d x}}-\frac {2 f^{3/2} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 f^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \sinh (f x)}{3 d (d x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 84, normalized size = 0.74 \[ -\frac {x e^{-f x} \left (e^{2 f x}+2 f x e^{2 f x}+2 f x+2 e^{f x} (-f x)^{3/2} \Gamma \left (\frac {1}{2},-f x\right )-2 e^{f x} (f x)^{3/2} \Gamma \left (\frac {1}{2},f x\right )-1\right )}{3 (d x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 178, normalized size = 1.56 \[ -\frac {2 \, \sqrt {\pi } {\left (d f x^{2} \cosh \left (f x\right ) + d f x^{2} \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) + 2 \, \sqrt {\pi } {\left (d f x^{2} \cosh \left (f x\right ) + d f x^{2} \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) + {\left ({\left (2 \, f x + 1\right )} \cosh \left (f x\right )^{2} + 2 \, {\left (2 \, f x + 1\right )} \cosh \left (f x\right ) \sinh \left (f x\right ) + {\left (2 \, f x + 1\right )} \sinh \left (f x\right )^{2} + 2 \, f x - 1\right )} \sqrt {d x}}{3 \, {\left (d^{3} x^{2} \cosh \left (f x\right ) + d^{3} x^{2} \sinh \left (f x\right )\right )}} \]
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 \[ \int \frac {\sinh \left (f x\right )}{\left (d x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 132, normalized size = 1.16 \[ -\frac {\sqrt {\pi }\, x^{\frac {5}{2}} \sqrt {2}\, \left (i f \right )^{\frac {5}{2}} \left (-\frac {4 \sqrt {2}\, \left (2 f x +1\right ) {\mathrm e}^{f x}}{3 \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {i f}\, f}+\frac {4 \sqrt {2}\, \left (-2 f x +1\right ) {\mathrm e}^{-f x}}{3 \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {i f}\, f}-\frac {8 \sqrt {2}\, \sqrt {f}\, \erf \left (\sqrt {x}\, \sqrt {f}\right )}{3 \sqrt {i f}}+\frac {8 \sqrt {2}\, \sqrt {f}\, \erfi \left (\sqrt {x}\, \sqrt {f}\right )}{3 \sqrt {i f}}\right )}{8 \left (d x \right )^{\frac {5}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 57, normalized size = 0.50 \[ -\frac {\frac {f {\left (\frac {\sqrt {f x} \Gamma \left (-\frac {1}{2}, f x\right )}{\sqrt {d x}} + \frac {\sqrt {-f x} \Gamma \left (-\frac {1}{2}, -f x\right )}{\sqrt {d x}}\right )}}{d} + \frac {2 \, \sinh \left (f x\right )}{\left (d x\right )^{\frac {3}{2}}}}{3 \, d} \]
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 \[ \int \frac {\mathrm {sinh}\left (f\,x\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 19.63, size = 129, normalized size = 1.13 \[ - \frac {\sqrt {2} \sqrt {\pi } f^{\frac {3}{2}} e^{- \frac {3 i \pi }{4}} S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (- \frac {1}{4}\right )}{3 d^{\frac {5}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {f \cosh {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{3 d^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {\sinh {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{6 d^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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