Optimal. Leaf size=88 \[ -\frac{\sqrt{\pi } \sqrt{f} \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{\sqrt{\pi } \sqrt{f} \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{2 \cosh (f x)}{d \sqrt{d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.112532, antiderivative size = 88, normalized size of antiderivative = 1., 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 = {3297, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \sqrt{f} \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{\sqrt{\pi } \sqrt{f} \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{2 \cosh (f x)}{d \sqrt{d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{\cosh (f x)}{(d x)^{3/2}} \, dx &=-\frac{2 \cosh (f x)}{d \sqrt{d x}}+\frac{(2 f) \int \frac{\sinh (f x)}{\sqrt{d x}} \, dx}{d}\\ &=-\frac{2 \cosh (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 \cosh (f x)}{d \sqrt{d x}}-\frac{(2 f) \operatorname{Subst}\left (\int e^{-\frac{f x^2}{d}} \, dx,x,\sqrt{d x}\right )}{d^2}+\frac{(2 f) \operatorname{Subst}\left (\int e^{\frac{f x^2}{d}} \, dx,x,\sqrt{d x}\right )}{d^2}\\ &=-\frac{2 \cosh (f x)}{d \sqrt{d x}}-\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}}\\ \end{align*}
Mathematica [A] time = 0.0342088, size = 67, normalized size = 0.76 \[ \frac{x e^{-f x} \left (e^{f x} \sqrt{-f x} \text{Gamma}\left (\frac{1}{2},-f x\right )+e^{f x} \sqrt{f x} \text{Gamma}\left (\frac{1}{2},f x\right )-e^{2 f x}-1\right )}{(d x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.022, size = 115, normalized size = 1.3 \begin{align*}{\frac{-{\frac{i}{4}}\sqrt{\pi }\sqrt{2}}{f}{x}^{{\frac{3}{2}}} \left ( if \right ) ^{{\frac{3}{2}}} \left ( -2\,{\frac{\sqrt{2}{{\rm e}^{fx}}}{\sqrt{\pi }\sqrt{x}\sqrt{if}}}-2\,{\frac{\sqrt{2}{{\rm e}^{-fx}}}{\sqrt{\pi }\sqrt{x}\sqrt{if}}}-2\,{\frac{\sqrt{2}\sqrt{f}{\it Erf} \left ( \sqrt{x}\sqrt{f} \right ) }{\sqrt{if}}}+2\,{\frac{\sqrt{2}\sqrt{f}{\it erfi} \left ( \sqrt{x}\sqrt{f} \right ) }{\sqrt{if}}} \right ) \left ( dx \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04295, size = 103, normalized size = 1.17 \begin{align*} -\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 \, \cosh \left (f x\right )}{\sqrt{d x}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.79208, size = 356, normalized size = 4.05 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 7.4263, size = 99, normalized size = 1.12 \begin{align*} - \frac{\sqrt{2} \sqrt{\pi } \sqrt{f} 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 )}{2 d^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} + \frac{\cosh{\left (f x \right )} \Gamma \left (- \frac{1}{4}\right )}{2 d^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (f x\right )}{\left (d x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]