Optimal. Leaf size=231 \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{\frac {(2 e+i b \log (f))^2}{4 c \log (f)}-2 i d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+2 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{2 i d-\frac {(b \log (f)+2 i e)^2}{4 c \log (f)}} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+2 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.28, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4473, 2234, 2204, 2287} \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{\frac {(2 e+i b \log (f))^2}{4 c \log (f)}-2 i d} \text {Erfi}\left (\frac {-b \log (f)-2 c x \log (f)+2 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{2 i d-\frac {(b \log (f)+2 i e)^2}{4 c \log (f)}} \text {Erfi}\left (\frac {b \log (f)+2 c x \log (f)+2 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2287
Rule 4473
Rubi steps
\begin {align*} \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx &=\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*}
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Mathematica [A] time = 0.61, size = 204, normalized size = 0.88 \[ \frac {\sqrt {\pi } e^{-\frac {i b e}{c}} f^{a-\frac {b^2}{4 c}} \left ((\cos (2 d)+i \sin (2 d)) e^{\frac {e^2}{c \log (f)}} \text {erfi}\left (\frac {\log (f) (b+2 c x)+2 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )+(\cos (2 d)-i \sin (2 d)) e^{\frac {e (e+2 i b \log (f))}{c \log (f)}} \text {erfi}\left (\frac {\log (f) (b+2 c x)-2 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )+2 e^{\frac {i b e}{c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 224, normalized size = 0.97 \[ -\frac {\sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \relax (f) + 2 i \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} - 4 \, e^{2} - {\left (8 i \, c d - 4 i \, b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right )} + \sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \relax (f) - 2 i \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} - 4 \, e^{2} - {\left (-8 i \, c d + 4 i \, b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right )} + \frac {2 \, \sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{8 \, c \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{c x^{2} + b x + a} \cos \left (e x + d\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 217, normalized size = 0.94 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-4 i \ln \relax (f ) b e +8 i d \ln \relax (f ) c -4 e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )-2 i e}{2 \sqrt {-c \ln \relax (f )}}\right )}{8 \sqrt {-c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+4 i \ln \relax (f ) b e -8 i d \ln \relax (f ) c -4 e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {2 i e +b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{8 \sqrt {-c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.38, size = 399, normalized size = 1.73 \[ \frac {\sqrt {\pi } {\left (f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) - i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}} - \frac {1}{2} \, {\left (b \log \relax (f) + 2 i \, e\right )} \overline {\frac {1}{\sqrt {-c \log \relax (f)}}}\right ) e^{\left (\frac {e^{2}}{c \log \relax (f)}\right )} + f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) + i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}} - \frac {1}{2} \, {\left (b \log \relax (f) - 2 i \, e\right )} \overline {\frac {1}{\sqrt {-c \log \relax (f)}}}\right ) e^{\left (\frac {e^{2}}{c \log \relax (f)}\right )} + f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) - i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) + b \log \relax (f) + 2 i \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) e^{\left (\frac {e^{2}}{c \log \relax (f)}\right )} + f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) + i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) + b \log \relax (f) - 2 i \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) e^{\left (\frac {e^{2}}{c \log \relax (f)}\right )} + 2 \, f^{a} \operatorname {erf}\left (-\frac {1}{2} \, b \overline {\frac {1}{\sqrt {-c \log \relax (f)}}} \log \relax (f) + x \overline {\sqrt {-c \log \relax (f)}}\right ) - 2 \, f^{a} \operatorname {erf}\left (\frac {2 \, c x \log \relax (f) + b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f)}}\right )\right )}}{16 \, \sqrt {-c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{c\,x^2+b\,x+a}\,{\cos \left (d+e\,x\right )}^2 \,d x \]
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 \[ \int f^{a + b x + c x^{2}} \cos ^{2}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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