Optimal. Leaf size=123 \[ \frac{\sqrt{\pi } \sqrt{d} e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}+\frac{\sqrt{c+d x} \sinh (a+b x)}{b} \]
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Rubi [A] time = 0.177666, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3296, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \sqrt{d} e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}+\frac{\sqrt{c+d x} \sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \sqrt{c+d x} \cosh (a+b x) \, dx &=\frac{\sqrt{c+d x} \sinh (a+b x)}{b}-\frac{d \int \frac{\sinh (a+b x)}{\sqrt{c+d x}} \, dx}{2 b}\\ &=\frac{\sqrt{c+d x} \sinh (a+b x)}{b}-\frac{d \int \frac{e^{-i (i a+i b x)}}{\sqrt{c+d x}} \, dx}{4 b}+\frac{d \int \frac{e^{i (i a+i b x)}}{\sqrt{c+d x}} \, dx}{4 b}\\ &=\frac{\sqrt{c+d x} \sinh (a+b x)}{b}+\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 )}{2 b}-\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 )}{2 b}\\ &=\frac{\sqrt{d} e^{-a+\frac{b c}{d}} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}-\frac{\sqrt{d} e^{a-\frac{b c}{d}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 b^{3/2}}+\frac{\sqrt{c+d x} \sinh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0920261, size = 105, normalized size = 0.85 \[ \frac{\sqrt{c+d x} e^{-a-\frac{b c}{d}} \left (\frac{e^{2 a} \text{Gamma}\left (\frac{3}{2},-\frac{b (c+d x)}{d}\right )}{\sqrt{-\frac{b (c+d x)}{d}}}-\frac{e^{\frac{2 b c}{d}} \text{Gamma}\left (\frac{3}{2},\frac{b (c+d x)}{d}\right )}{\sqrt{\frac{b (c+d x)}{d}}}\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \cosh \left ( bx+a \right ) \sqrt{dx+c}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11679, size = 311, normalized size = 2.53 \begin{align*} \frac{8 \,{\left (d x + c\right )}^{\frac{3}{2}} \cosh \left (b x + a\right ) - \frac{{\left (\frac{3 \, \sqrt{\pi } d^{2} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (a - \frac{b c}{d}\right )}}{b^{2} \sqrt{-\frac{b}{d}}} - \frac{3 \, \sqrt{\pi } d^{2} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-a + \frac{b c}{d}\right )}}{b^{2} \sqrt{\frac{b}{d}}} + \frac{2 \,{\left (2 \,{\left (d x + c\right )}^{\frac{3}{2}} b d e^{\left (\frac{b c}{d}\right )} + 3 \, \sqrt{d x + c} d^{2} e^{\left (\frac{b c}{d}\right )}\right )} e^{\left (-a - \frac{{\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac{2 \,{\left (2 \,{\left (d x + c\right )}^{\frac{3}{2}} b d e^{a} - 3 \, \sqrt{d x + c} d^{2} e^{a}\right )} e^{\left (\frac{{\left (d x + c\right )} b}{d} - \frac{b c}{d}\right )}}{b^{2}}\right )} b}{d}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.08264, size = 717, normalized size = 5.83 \begin{align*} \frac{\sqrt{\pi }{\left (d \cosh \left (b x + a\right ) \cosh \left (-\frac{b c - a d}{d}\right ) - d \cosh \left (b x + a\right ) \sinh \left (-\frac{b c - a d}{d}\right ) +{\left (d \cosh \left (-\frac{b c - a d}{d}\right ) - d \sinh \left (-\frac{b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt{\frac{b}{d}} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) + \sqrt{\pi }{\left (d \cosh \left (b x + a\right ) \cosh \left (-\frac{b c - a d}{d}\right ) + d \cosh \left (b x + a\right ) \sinh \left (-\frac{b c - a d}{d}\right ) +{\left (d \cosh \left (-\frac{b c - a d}{d}\right ) + d \sinh \left (-\frac{b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt{-\frac{b}{d}} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) + 2 \,{\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b\right )} \sqrt{d x + c}}{4 \,{\left (b^{2} \cosh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )\right )}} \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 \sqrt{c + d x} \cosh{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.44222, size = 228, normalized size = 1.85 \begin{align*} -\frac{\frac{\sqrt{\pi } d^{2} \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} b} - \frac{\sqrt{\pi } d^{2} \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} b} - \frac{2 \, \sqrt{d x + c} d e^{\left (\frac{{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b} + \frac{2 \, \sqrt{d x + c} d e^{\left (-\frac{{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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