Integrand size = 16, antiderivative size = 133 \[ \int f^{a+c x^2} \cosh (d+e x) \, dx=-\frac {e^{-d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5624, 2325, 2266, 2235} \[ \int f^{a+c x^2} \cosh (d+e x) \, dx=\frac {\sqrt {\pi } f^a e^{d-\frac {e^2}{4 c \log (f)}} \text {erfi}\left (\frac {2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{-\frac {e^2}{4 c \log (f)}-d} \text {erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rule 2235
Rule 2266
Rule 2325
Rule 5624
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} e^{-d-e x} f^{a+c x^2}+\frac {1}{2} e^{d+e x} f^{a+c x^2}\right ) \, dx \\ & = \frac {1}{2} \int e^{-d-e x} f^{a+c x^2} \, dx+\frac {1}{2} \int e^{d+e x} f^{a+c x^2} \, dx \\ & = \frac {1}{2} \int e^{-d-e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {1}{2} \int e^{d+e x+a \log (f)+c x^2 \log (f)} \, dx \\ & = \frac {1}{2} \left (e^{-d-\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{2} \left (e^{d-\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(e+2 c x \log (f))^2}{4 c \log (f)}} \, dx \\ & = -\frac {e^{-d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.78 \[ \int f^{a+c x^2} \cosh (d+e x) \, dx=\frac {e^{-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \left (\text {erfi}\left (\frac {-e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (d)-\sinh (d))+\text {erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (d)+\sinh (d))\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {\operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d \ln \left (f \right ) c +e^{2}}{4 \ln \left (f \right ) c}}}{4 \sqrt {-c \ln \left (f \right )}}-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{2 \sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}}}{4 \sqrt {-c \ln \left (f \right )}}\) | \(117\) |
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Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (101) = 202\).
Time = 0.28 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.62 \[ \int f^{a+c x^2} \cosh (d+e x) \, dx=-\frac {\sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) + \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right )}{4 \, c \log \left (f\right )} \]
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\[ \int f^{a+c x^2} \cosh (d+e x) \, dx=\int f^{a + c x^{2}} \cosh {\left (d + e x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.79 \[ \int f^{a+c x^2} \cosh (d+e x) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (d - \frac {e^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, \sqrt {-c \log \left (f\right )}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x + \frac {e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (-d - \frac {e^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, \sqrt {-c \log \left (f\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.99 \[ \int f^{a+c x^2} \cosh (d+e x) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x - \frac {e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, \sqrt {-c \log \left (f\right )}} \]
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Timed out. \[ \int f^{a+c x^2} \cosh (d+e x) \, dx=\int f^{c\,x^2+a}\,\mathrm {cosh}\left (d+e\,x\right ) \,d x \]
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