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 = 0.13, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2180
Rule 2204
Rule 2205
Rule 3307
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*}
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Mathematica [A] time = 0.04, size = 105, normalized size = 1.01 \[ \frac {e^{-a-\frac {b c}{d}} \left (e^{2 a} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )-e^{\frac {2 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )\right )}{2 b \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 123, normalized size = 1.18 \[ \frac {\sqrt {\pi } \sqrt {\frac {b}{d}} {\left (\cosh \left (-\frac {b c - a d}{d}\right ) - \sinh \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - \sqrt {\pi } \sqrt {-\frac {b}{d}} {\left (\cosh \left (-\frac {b c - a d}{d}\right ) + \sinh \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 89, normalized size = 0.86 \[ -\frac {{\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {b d} \sqrt {d x + c}}{d}\right ) e^{\left (\frac {b c}{d}\right )}}{\sqrt {b d}} + \frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {-b d} \sqrt {d x + c}}{d}\right ) e^{\left (-\frac {b c - 2 \, a d}{d}\right )}}{\sqrt {-b d}}\right )} e^{\left (-a\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (b x +a \right )}{\sqrt {d x +c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 180, normalized size = 1.73 \[ \frac {4 \, \sqrt {d x + c} \cosh \left (b x + a\right ) + \frac {{\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b \sqrt {-\frac {b}{d}}} + \frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b \sqrt {\frac {b}{d}}} - \frac {2 \, \sqrt {d x + c} d e^{\left (a + \frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b} - \frac {2 \, \sqrt {d x + c} d e^{\left (-a - \frac {{\left (d x + c\right )} b}{d} + \frac {b c}{d}\right )}}{b}\right )} b}{d}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (a+b\,x\right )}{\sqrt {c+d\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (a + b x \right )}}{\sqrt {c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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