Integrand size = 18, antiderivative size = 138 \[ \int \frac {\cosh ^2(a+b x)}{\sqrt {c+d x}} \, dx=\frac {\sqrt {c+d x}}{d}+\frac {e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 \sqrt {b} \sqrt {d}}+\frac {e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 \sqrt {b} \sqrt {d}} \]
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Time = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3393, 3388, 2211, 2235, 2236} \[ \int \frac {\cosh ^2(a+b x)}{\sqrt {c+d x}} \, dx=\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 b c}{d}-2 a} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {2 b c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 \sqrt {b} \sqrt {d}}+\frac {\sqrt {c+d x}}{d} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2 \sqrt {c+d x}}+\frac {\cosh (2 a+2 b x)}{2 \sqrt {c+d x}}\right ) \, dx \\ & = \frac {\sqrt {c+d x}}{d}+\frac {1}{2} \int \frac {\cosh (2 a+2 b x)}{\sqrt {c+d x}} \, dx \\ & = \frac {\sqrt {c+d x}}{d}+\frac {1}{4} \int \frac {e^{-i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx+\frac {1}{4} \int \frac {e^{i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx \\ & = \frac {\sqrt {c+d x}}{d}+\frac {\text {Subst}\left (\int e^{i \left (2 i a-\frac {2 i b c}{d}\right )-\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d}+\frac {\text {Subst}\left (\int e^{-i \left (2 i a-\frac {2 i b c}{d}\right )+\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d} \\ & = \frac {\sqrt {c+d x}}{d}+\frac {e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 \sqrt {b} \sqrt {d}}+\frac {e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 \sqrt {b} \sqrt {d}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.02 \[ \int \frac {\cosh ^2(a+b x)}{\sqrt {c+d x}} \, dx=\frac {\sqrt {c+d x}}{d}+\frac {e^{2 a-\frac {2 b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {2 b (c+d x)}{d}\right )}{4 \sqrt {2} b \sqrt {c+d x}}-\frac {e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},\frac {2 b (c+d x)}{d}\right )}{4 \sqrt {2} b \sqrt {c+d x}} \]
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\[\int \frac {\cosh \left (b x +a \right )^{2}}{\sqrt {d x +c}}d x\]
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Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.12 \[ \int \frac {\cosh ^2(a+b x)}{\sqrt {c+d x}} \, dx=\frac {\sqrt {2} \sqrt {\pi } {\left (d \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - \sqrt {2} \sqrt {\pi } {\left (d \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) + 8 \, \sqrt {d x + c} b}{8 \, b d} \]
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\[ \int \frac {\cosh ^2(a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {\cosh ^{2}{\left (a + b x \right )}}{\sqrt {c + d x}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78 \[ \int \frac {\cosh ^2(a+b x)}{\sqrt {c+d x}} \, dx=\frac {\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )}}{\sqrt {-\frac {b}{d}}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )}}{\sqrt {\frac {b}{d}}} + 8 \, \sqrt {d x + c}}{8 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh ^2(a+b x)}{\sqrt {c+d x}} \, dx=-\frac {{\left (\frac {\sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c}}{d}\right ) e^{\left (\frac {2 \, b c}{d}\right )}}{\sqrt {b d}} + \frac {\sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {-b d} \sqrt {d x + c}}{d}\right ) e^{\left (-\frac {2 \, {\left (b c - 2 \, a d\right )}}{d}\right )}}{\sqrt {-b d}} - 8 \, \sqrt {d x + c} e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, a\right )}}{8 \, d} \]
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Timed out. \[ \int \frac {\cosh ^2(a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{\sqrt {c+d\,x}} \,d x \]
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