Integrand size = 16, antiderivative size = 104 \[ \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx=\frac {e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}} \]
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Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3388, 2211, 2235, 2236} \[ \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx=\frac {\sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rubi steps \begin{align*} \text {integral}& = \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 {\text {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 {\text {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*}
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.01 \[ \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx=\frac {e^{-a-\frac {b c}{d}} \left (e^{2 a} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )-e^{\frac {2 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )\right )}{2 b \sqrt {c+d x}} \]
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\[\int \frac {\cosh \left (b x +a \right )}{\sqrt {d x +c}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.18 \[ \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx=\frac {\sqrt {\pi } \sqrt {\frac {b}{d}} {\left (\cosh \left (-\frac {b c - a d}{d}\right ) - \sinh \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - \sqrt {\pi } \sqrt {-\frac {b}{d}} {\left (\cosh \left (-\frac {b c - a d}{d}\right ) + \sinh \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right )}{2 \, b} \]
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\[ \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {\cosh {\left (a + b x \right )}}{\sqrt {c + d x}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (74) = 148\).
Time = 0.20 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.73 \[ \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx=\frac {4 \, \sqrt {d x + c} \cosh \left (b x + a\right ) + \frac {{\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b \sqrt {-\frac {b}{d}}} + \frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b \sqrt {\frac {b}{d}}} - \frac {2 \, \sqrt {d x + c} d e^{\left (a + \frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b} - \frac {2 \, \sqrt {d x + c} d e^{\left (-a - \frac {{\left (d x + c\right )} b}{d} + \frac {b c}{d}\right )}}{b}\right )} b}{d}}{2 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx=-\frac {{\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {b d} \sqrt {d x + c}}{d}\right ) e^{\left (\frac {b c}{d}\right )}}{\sqrt {b d}} + \frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {-b d} \sqrt {d x + c}}{d}\right ) e^{\left (-\frac {b c - 2 \, a d}{d}\right )}}{\sqrt {-b d}}\right )} e^{\left (-a\right )}}{2 \, d} \]
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Timed out. \[ \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x\right )}{\sqrt {c+d\,x}} \,d x \]
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