Integrand size = 21, antiderivative size = 231 \[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 i d+\frac {(2 e+i b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {2 i e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{2 i d-\frac {(2 i e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {2 i e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.39 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4561, 2266, 2235, 2325} \[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{\frac {(2 e+i b \log (f))^2}{4 c \log (f)}-2 i d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+2 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{2 i d-\frac {(b \log (f)+2 i e)^2}{4 c \log (f)}} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+2 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
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Rule 2235
Rule 2266
Rule 2325
Rule 4561
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} f^{a+b x+c x^2}+\frac {1}{4} e^{-2 i d-2 i e x} f^{a+b x+c x^2}+\frac {1}{4} e^{2 i d+2 i e x} f^{a+b x+c x^2}\right ) \, dx \\ & = \frac {1}{4} \int e^{-2 i d-2 i e x} f^{a+b x+c x^2} \, dx+\frac {1}{4} \int e^{2 i d+2 i e x} f^{a+b x+c x^2} \, dx+\frac {1}{2} \int f^{a+b x+c x^2} \, dx \\ & = \frac {1}{4} \int \exp \left (-2 i d+a \log (f)+c x^2 \log (f)-x (2 i e-b \log (f))\right ) \, dx+\frac {1}{4} \int \exp \left (2 i d+a \log (f)+c x^2 \log (f)+x (2 i e+b \log (f))\right ) \, dx+\frac {1}{2} f^{a-\frac {b^2}{4 c}} \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx \\ & = \frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {1}{4} \left (\exp \left (-2 i d+\frac {(2 e+i b \log (f))^2}{4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac {(-2 i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac {1}{4} \left (e^{2 i d-\frac {(2 i e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(2 i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx \\ & = \frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\exp \left (-2 i d+\frac {(2 e+i b \log (f))^2}{4 c \log (f)}\right ) f^a \sqrt {\pi } \text {erfi}\left (\frac {2 i e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{2 i d-\frac {(2 i e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {2 i e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.88 \[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\frac {e^{-\frac {i b e}{c}} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \left (2 e^{\frac {i b e}{c}} \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )+e^{\frac {e (e+2 i b \log (f))}{c \log (f)}} \text {erfi}\left (\frac {-2 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (2 d)-i \sin (2 d))+e^{\frac {e^2}{c \log (f)}} \text {erfi}\left (\frac {2 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (2 d)+i \sin (2 d))\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.74 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.94
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{\frac {i \ln \left (f \right ) b e -2 i d \ln \left (f \right ) c +e^{2}}{\ln \left (f \right ) c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )-2 i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{8 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{-\frac {i \ln \left (f \right ) b e -2 i d \ln \left (f \right ) c -e^{2}}{\ln \left (f \right ) c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {2 i e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{8 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} f^{a} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(218\) |
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Time = 0.25 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.97 \[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=-\frac {\sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - 2 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, e^{2} + 4 \, {\left (2 i \, c d - i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + 2 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, e^{2} + 4 \, {\left (-2 i \, c d + i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + \frac {2 \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{8 \, c \log \left (f\right )} \]
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\[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\int f^{a + b x + c x^{2}} \cos ^{2}{\left (d + e x \right )}\, dx \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.26 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.73 \[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\frac {\sqrt {\pi } {\left (f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) - i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} \, {\left (b \log \left (f\right ) + 2 i \, e\right )} \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) + i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} \, {\left (b \log \left (f\right ) - 2 i \, e\right )} \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) - i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right ) + 2 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) + i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right ) - 2 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} + 2 \, f^{a} \operatorname {erf}\left (-\frac {1}{2} \, b \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}} \log \left (f\right ) + x \overline {\sqrt {-c \log \left (f\right )}}\right ) - 2 \, f^{a} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) + b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right )}}\right )\right )}}{16 \, \sqrt {-c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} \]
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\[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\int { f^{c x^{2} + b x + a} \cos \left (e x + d\right )^{2} \,d x } \]
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Timed out. \[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\int f^{c\,x^2+b\,x+a}\,{\cos \left (d+e\,x\right )}^2 \,d x \]
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