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 = 0.149009, antiderivative size = 114, normalized size of antiderivative = 1., 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 3297
Rule 3308
Rule 2180
Rule 2204
Rule 2205
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*}
Mathematica [A] time = 0.0830307, size = 84, normalized size = 0.74 \[ -\frac{x e^{-f x} \left (2 e^{f x} (-f x)^{3/2} \text{Gamma}\left (\frac{1}{2},-f x\right )-2 e^{f x} (f x)^{3/2} \text{Gamma}\left (\frac{1}{2},f x\right )+e^{2 f x}+2 f x e^{2 f x}+2 f x-1\right )}{3 (d x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 132, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{\pi }\sqrt{2}}{8\,f}{x}^{{\frac{5}{2}}} \left ( if \right ) ^{{\frac{5}{2}}} \left ( -{\frac{4\,\sqrt{2} \left ( 2\,fx+1 \right ){{\rm e}^{fx}}}{3\,\sqrt{\pi }f}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{if}}}}+{\frac{4\,\sqrt{2} \left ( -2\,fx+1 \right ){{\rm e}^{-fx}}}{3\,\sqrt{\pi }f}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{if}}}}-{\frac{8\,\sqrt{2}}{3}\sqrt{f}{\it Erf} \left ( \sqrt{x}\sqrt{f} \right ){\frac{1}{\sqrt{if}}}}+{\frac{8\,\sqrt{2}}{3}\sqrt{f}{\it erfi} \left ( \sqrt{x}\sqrt{f} \right ){\frac{1}{\sqrt{if}}}} \right ) \left ( dx \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18749, size = 77, normalized size = 0.68 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.64832, size = 454, normalized size = 3.98 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 175.807, size = 129, normalized size = 1.13 \begin{align*} - \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (f x\right )}{\left (d x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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