Optimal. Leaf size=104 \[ \frac{\sqrt{\pi } e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\pi } e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}} \]
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Rubi [A] time = 0.129774, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\pi } e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{\cosh (a+b x)}{\sqrt{c+d x}} \, dx &=\frac{1}{2} \int \frac{e^{-i (i a+i b x)}}{\sqrt{c+d x}} \, dx+\frac{1}{2} \int \frac{e^{i (i a+i b x)}}{\sqrt{c+d x}} \, dx\\ &=\frac{\operatorname{Subst}\left (\int e^{i \left (i a-\frac{i b c}{d}\right )-\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{d}+\frac{\operatorname{Subst}\left (\int e^{-i \left (i a-\frac{i b c}{d}\right )+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{e^{-a+\frac{b c}{d}} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}}+\frac{e^{a-\frac{b c}{d}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.0428036, size = 105, normalized size = 1.01 \[ \frac{e^{-a-\frac{b c}{d}} \left (e^{2 a} \sqrt{-\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{1}{2},-\frac{b (c+d x)}{d}\right )-e^{\frac{2 b c}{d}} \sqrt{\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{1}{2},\frac{b (c+d x)}{d}\right )\right )}{2 b \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\cosh \left ( bx+a \right ){\frac{1}{\sqrt{dx+c}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0842, size = 243, normalized size = 2.34 \begin{align*} \frac{4 \, \sqrt{d x + c} \cosh \left (b x + a\right ) + \frac{{\left (\frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (a - \frac{b c}{d}\right )}}{b \sqrt{-\frac{b}{d}}} + \frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-a + \frac{b c}{d}\right )}}{b \sqrt{\frac{b}{d}}} - \frac{2 \, \sqrt{d x + c} d e^{\left (a + \frac{{\left (d x + c\right )} b}{d} - \frac{b c}{d}\right )}}{b} - \frac{2 \, \sqrt{d x + c} d e^{\left (-a - \frac{{\left (d x + c\right )} b}{d} + \frac{b c}{d}\right )}}{b}\right )} b}{d}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0964, size = 271, normalized size = 2.61 \begin{align*} \frac{\sqrt{\pi } \sqrt{\frac{b}{d}}{\left (\cosh \left (-\frac{b c - a d}{d}\right ) - \sinh \left (-\frac{b c - a d}{d}\right )\right )} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) - \sqrt{\pi } \sqrt{-\frac{b}{d}}{\left (\cosh \left (-\frac{b c - a d}{d}\right ) + \sinh \left (-\frac{b c - a d}{d}\right )\right )} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (a + b x \right )}}{\sqrt{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39583, size = 123, normalized size = 1.18 \begin{align*} -\frac{\frac{\sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{b d} \sqrt{d x + c}}{d}\right ) e^{\left (\frac{b c - a d}{d}\right )}}{\sqrt{b d}} + \frac{\sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{-b d} \sqrt{d x + c}}{d}\right ) e^{\left (-\frac{b c - a d}{d}\right )}}{\sqrt{-b d}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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