Integrand size = 18, antiderivative size = 293 \[ \int f^{a+c x^2} \cos ^3(d+e x) \, dx=-\frac {3 e^{-i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-3 i d+\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 e^{i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 i d+\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.35 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4561, 2325, 2266, 2235} \[ \int f^{a+c x^2} \cos ^3(d+e x) \, dx=-\frac {3 \sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)}-i d} \text {erfi}\left (\frac {-2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{\frac {9 e^2}{4 c \log (f)}-3 i d} \text {erfi}\left (\frac {-2 c x \log (f)+3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 \sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)}+i d} \text {erfi}\left (\frac {2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{\frac {9 e^2}{4 c \log (f)}+3 i d} \text {erfi}\left (\frac {2 c x \log (f)+3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Rule 2235
Rule 2266
Rule 2325
Rule 4561
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{8} e^{-i d-i e x} f^{a+c x^2}+\frac {3}{8} e^{i d+i e x} f^{a+c x^2}+\frac {1}{8} e^{-3 i d-3 i e x} f^{a+c x^2}+\frac {1}{8} e^{3 i d+3 i e x} f^{a+c x^2}\right ) \, dx \\ & = \frac {1}{8} \int e^{-3 i d-3 i e x} f^{a+c x^2} \, dx+\frac {1}{8} \int e^{3 i d+3 i e x} f^{a+c x^2} \, dx+\frac {3}{8} \int e^{-i d-i e x} f^{a+c x^2} \, dx+\frac {3}{8} \int e^{i d+i e x} f^{a+c x^2} \, dx \\ & = \frac {1}{8} \int e^{-3 i d-3 i e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {1}{8} \int e^{3 i d+3 i e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {3}{8} \int e^{-i d-i e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {3}{8} \int e^{i d+i e x+a \log (f)+c x^2 \log (f)} \, dx \\ & = \frac {1}{8} \left (3 e^{-i d+\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{8} \left (3 e^{i d+\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{8} \left (e^{-3 i d+\frac {9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-3 i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{8} \left (e^{3 i d+\frac {9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(3 i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx \\ & = -\frac {3 e^{-i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-3 i d+\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 e^{i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 i d+\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.74 \[ \int f^{a+c x^2} \cos ^3(d+e x) \, dx=\frac {e^{\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \left (3 \text {erfi}\left (\frac {-i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (d)-i \sin (d))+3 \text {erfi}\left (\frac {i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (d)+i \sin (d))+e^{\frac {2 e^2}{c \log (f)}} \left (\text {erfi}\left (\frac {-3 i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (3 d)-i \sin (3 d))+\text {erfi}\left (\frac {3 i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (3 d)+i \sin (3 d))\right )\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 1.51 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {3 \left (4 i d \ln \left (f \right ) c -3 e^{2}\right )}{4 \ln \left (f \right ) c}} \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x +\frac {3 i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 i d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}} \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x +\frac {i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 i d \ln \left (f \right ) c +e^{2}}{4 \ln \left (f \right ) c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {3 i d \ln \left (f \right ) c +\frac {9 e^{2}}{4}}{c \ln \left (f \right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {3 i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}\) | \(242\) |
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Time = 0.26 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.96 \[ \int f^{a+c x^2} \cos ^3(d+e x) \, dx=-\frac {\sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + 3 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 12 i \, c d \log \left (f\right ) + 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 4 i \, c d \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 4 i \, c d \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )} + \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - 3 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 12 i \, c d \log \left (f\right ) + 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, c \log \left (f\right )} \]
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\[ \int f^{a+c x^2} \cos ^3(d+e x) \, dx=\int f^{a + c x^{2}} \cos ^{3}{\left (d + e x \right )}\, dx \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.26 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.39 \[ \int f^{a+c x^2} \cos ^3(d+e x) \, dx=-\frac {\sqrt {\pi } {\left (f^{a} {\left (\cos \left (3 \, d\right ) - i \, \sin \left (3 \, d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} + \frac {3}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (3 \, d\right ) + i \, \sin \left (3 \, d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {3}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} - f^{a} {\left (\cos \left (3 \, d\right ) + i \, \sin \left (3 \, d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) + 3 i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} - f^{a} {\left (\cos \left (3 \, d\right ) - i \, \sin \left (3 \, d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) - 3 i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, f^{a} {\left (\cos \left (d\right ) - i \, \sin \left (d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} + \frac {1}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, f^{a} {\left (\cos \left (d\right ) + i \, \sin \left (d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} - 3 \, f^{a} {\left (\cos \left (d\right ) + i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) + i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} - 3 \, f^{a} {\left (\cos \left (d\right ) - i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) - i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )}\right )} \sqrt {-c \log \left (f\right )}}{32 \, c \log \left (f\right )} \]
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\[ \int f^{a+c x^2} \cos ^3(d+e x) \, dx=\int { f^{c x^{2} + a} \cos \left (e x + d\right )^{3} \,d x } \]
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Timed out. \[ \int f^{a+c x^2} \cos ^3(d+e x) \, dx=\int f^{c\,x^2+a}\,{\cos \left (d+e\,x\right )}^3 \,d x \]
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