Optimal. Leaf size=171 \[ -\frac {\sqrt {\pi } f^a e^{\frac {e^2}{c \log (f)}-2 i d} \text {erfi}\left (\frac {-c x \log (f)+i e}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{\frac {e^2}{c \log (f)}+2 i d} \text {erfi}\left (\frac {c x \log (f)+i e}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.20, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4473, 2204, 2287, 2234} \[ -\frac {\sqrt {\pi } f^a e^{\frac {e^2}{c \log (f)}-2 i d} \text {Erfi}\left (\frac {-c x \log (f)+i e}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{\frac {e^2}{c \log (f)}+2 i d} \text {Erfi}\left (\frac {c x \log (f)+i e}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2287
Rule 4473
Rubi steps
\begin {align*} \int f^{a+c x^2} \cos ^2(d+e x) \, dx &=\int \left (\frac {1}{2} f^{a+c x^2}+\frac {1}{4} e^{-2 i d-2 i e x} f^{a+c x^2}+\frac {1}{4} e^{2 i d+2 i e x} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 i d-2 i e x} f^{a+c x^2} \, dx+\frac {1}{4} \int e^{2 i d+2 i e x} f^{a+c x^2} \, dx+\frac {1}{2} \int f^{a+c x^2} \, dx\\ &=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {1}{4} \int e^{-2 i d-2 i e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {1}{4} \int e^{2 i d+2 i e x+a \log (f)+c x^2 \log (f)} \, dx\\ &=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {1}{4} \left (e^{-2 i d+\frac {e^2}{c \log (f)}} f^a\right ) \int e^{\frac {(-2 i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{4} \left (e^{2 i d+\frac {e^2}{c \log (f)}} f^a\right ) \int e^{\frac {(2 i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx\\ &=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 i d+\frac {e^2}{c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e-c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{2 i d+\frac {e^2}{c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 131, normalized size = 0.77 \[ \frac {\sqrt {\pi } f^a \left (2 \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )+e^{\frac {e^2}{c \log (f)}} \left ((\cos (2 d)-i \sin (2 d)) \text {erfi}\left (\frac {c x \log (f)-i e}{\sqrt {c} \sqrt {\log (f)}}\right )+(\cos (2 d)+i \sin (2 d)) \text {erfi}\left (\frac {c x \log (f)+i e}{\sqrt {c} \sqrt {\log (f)}}\right )\right )\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 159, normalized size = 0.93 \[ -\frac {2 \, \sqrt {\pi } \sqrt {-c \log \relax (f)} f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x\right ) + \sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (c x \log \relax (f) + i \, e\right )} \sqrt {-c \log \relax (f)}}{c \log \relax (f)}\right ) e^{\left (\frac {a c \log \relax (f)^{2} + 2 i \, c d \log \relax (f) + e^{2}}{c \log \relax (f)}\right )} + \sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (c x \log \relax (f) - i \, e\right )} \sqrt {-c \log \relax (f)}}{c \log \relax (f)}\right ) e^{\left (\frac {a c \log \relax (f)^{2} - 2 i \, c d \log \relax (f) + e^{2}}{c \log \relax (f)}\right )}}{8 \, c \log \relax (f)} \]
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 \[ \int f^{c x^{2} + a} \cos \left (e x + d\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 145, normalized size = 0.85 \[ \frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 i d \ln \relax (f ) c -e^{2}}{\ln \relax (f ) c}} \erf \left (\sqrt {-c \ln \relax (f )}\, x +\frac {i e}{\sqrt {-c \ln \relax (f )}}\right )}{8 \sqrt {-c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 i d \ln \relax (f ) c +e^{2}}{\ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {i e}{\sqrt {-c \ln \relax (f )}}\right )}{8 \sqrt {-c \ln \relax (f )}}+\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.36, size = 236, normalized size = 1.38 \[ \frac {\sqrt {\pi } {\left (f^{a} {\left (\cos \left (2 \, d\right ) - i \, \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}} + i \, e \overline {\frac {1}{\sqrt {-c \log \relax (f)}}}\right ) e^{\left (\frac {e^{2}}{c \log \relax (f)}\right )} + f^{a} {\left (\cos \left (2 \, d\right ) + i \, \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}} - i \, e \overline {\frac {1}{\sqrt {-c \log \relax (f)}}}\right ) e^{\left (\frac {e^{2}}{c \log \relax (f)}\right )} - f^{a} {\left (\cos \left (2 \, d\right ) + i \, \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (\frac {c x \log \relax (f) + i \, e}{\sqrt {-c \log \relax (f)}}\right ) e^{\left (\frac {e^{2}}{c \log \relax (f)}\right )} - f^{a} {\left (\cos \left (2 \, d\right ) - i \, \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (\frac {c x \log \relax (f) - i \, e}{\sqrt {-c \log \relax (f)}}\right ) e^{\left (\frac {e^{2}}{c \log \relax (f)}\right )} + 2 \, f^{a} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}}\right ) + 2 \, f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x\right )\right )}}{16 \, \sqrt {-c \log \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int f^{c\,x^2+a}\,{\cos \left (d+e\,x\right )}^2 \,d x \]
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 \[ \int f^{a + c x^{2}} \cos ^{2}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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