Optimal. Leaf size=211 \[ \frac {\sqrt {\pi } f^a e^{-\frac {e^2}{-c \log (f)+2 i f}-2 i d} \text {erf}\left (\frac {x (-c \log (f)+2 i f)+i e}{\sqrt {-c \log (f)+2 i f}}\right )}{8 \sqrt {-c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^a e^{\frac {e^2}{c \log (f)+2 i f}+2 i d} \text {erfi}\left (\frac {x (c \log (f)+2 i f)+i e}{\sqrt {c \log (f)+2 i f}}\right )}{8 \sqrt {c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.36, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4473, 2204, 2287, 2234, 2205} \[ \frac {\sqrt {\pi } f^a e^{-\frac {e^2}{-c \log (f)+2 i f}-2 i d} \text {Erf}\left (\frac {x (-c \log (f)+2 i f)+i e}{\sqrt {-c \log (f)+2 i f}}\right )}{8 \sqrt {-c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^a e^{\frac {e^2}{c \log (f)+2 i f}+2 i d} \text {Erfi}\left (\frac {x (c \log (f)+2 i f)+i e}{\sqrt {c \log (f)+2 i f}}\right )}{8 \sqrt {c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2234
Rule 2287
Rule 4473
Rubi steps
\begin {align*} \int f^{a+c x^2} \cos ^2\left (d+e x+f x^2\right ) \, dx &=\int \left (\frac {1}{2} f^{a+c x^2}+\frac {1}{4} e^{-2 i d-2 i e x-2 i f x^2} f^{a+c x^2}+\frac {1}{4} e^{2 i d+2 i e x+2 i f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 i d-2 i e x-2 i f x^2} f^{a+c x^2} \, dx+\frac {1}{4} \int e^{2 i d+2 i e x+2 i f x^2} 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 \exp \left (-2 i d-2 i e x+a \log (f)-x^2 (2 i f-c \log (f))\right ) \, dx+\frac {1}{4} \int \exp \left (2 i d+2 i e x+a \log (f)+x^2 (2 i f+c \log (f))\right ) \, 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}{2 i f-c \log (f)}} f^a\right ) \int \exp \left (\frac {(-2 i e+2 x (-2 i f+c \log (f)))^2}{4 (-2 i f+c \log (f))}\right ) \, dx+\frac {1}{4} \left (e^{2 i d+\frac {e^2}{2 i f+c \log (f)}} f^a\right ) \int \exp \left (\frac {(2 i e+2 x (2 i f+c \log (f)))^2}{4 (2 i f+c \log (f))}\right ) \, 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}{2 i f-c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {i e+x (2 i f-c \log (f))}{\sqrt {2 i f-c \log (f)}}\right )}{8 \sqrt {2 i f-c \log (f)}}+\frac {e^{2 i d+\frac {e^2}{2 i f+c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+x (2 i f+c \log (f))}{\sqrt {2 i f+c \log (f)}}\right )}{8 \sqrt {2 i f+c \log (f)}}\\ \end {align*}
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Mathematica [A] time = 2.32, size = 252, normalized size = 1.19 \[ \frac {1}{8} \sqrt {\pi } f^a \left (\frac {2 \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{\sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt [4]{-1} \left (\sqrt {2 f-i c \log (f)} (2 f+i c \log (f)) (\sin (2 d)-i \cos (2 d)) e^{\frac {e^2}{c \log (f)+2 i f}} \text {erfi}\left (\frac {\sqrt [4]{-1} (-i c x \log (f)+e+2 f x)}{\sqrt {2 f-i c \log (f)}}\right )-(2 f-i c \log (f)) \sqrt {2 f+i c \log (f)} (\cos (2 d)-i \sin (2 d)) e^{\frac {e^2}{c \log (f)-2 i f}} \text {erfi}\left (\frac {(-1)^{3/4} (i c x \log (f)+e+2 f x)}{\sqrt {2 f+i c \log (f)}}\right )\right )}{c^2 \log ^2(f)+4 f^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.51, size = 361, normalized size = 1.71 \[ -\frac {2 \, \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )} \sqrt {-c \log \relax (f)} f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x\right ) + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 2 i \, c f \log \relax (f)\right )} \sqrt {-c \log \relax (f) - 2 i \, f} \operatorname {erf}\left (\frac {{\left (c^{2} x \log \relax (f)^{2} + 4 \, f^{2} x + i \, c e \log \relax (f) + 2 \, e f\right )} \sqrt {-c \log \relax (f) - 2 i \, f}}{c^{2} \log \relax (f)^{2} + 4 \, f^{2}}\right ) e^{\left (\frac {a c^{2} \log \relax (f)^{3} + 2 i \, c^{2} d \log \relax (f)^{2} - 2 i \, e^{2} f + 8 i \, d f^{2} + {\left (c e^{2} + 4 \, a f^{2}\right )} \log \relax (f)}{c^{2} \log \relax (f)^{2} + 4 \, f^{2}}\right )} + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 2 i \, c f \log \relax (f)\right )} \sqrt {-c \log \relax (f) + 2 i \, f} \operatorname {erf}\left (\frac {{\left (c^{2} x \log \relax (f)^{2} + 4 \, f^{2} x - i \, c e \log \relax (f) + 2 \, e f\right )} \sqrt {-c \log \relax (f) + 2 i \, f}}{c^{2} \log \relax (f)^{2} + 4 \, f^{2}}\right ) e^{\left (\frac {a c^{2} \log \relax (f)^{3} - 2 i \, c^{2} d \log \relax (f)^{2} + 2 i \, e^{2} f - 8 i \, d f^{2} + {\left (c e^{2} + 4 \, a f^{2}\right )} \log \relax (f)}{c^{2} \log \relax (f)^{2} + 4 \, f^{2}}\right )}}{8 \, {\left (c^{3} \log \relax (f)^{3} + 4 \, c f^{2} \log \relax (f)\right )}} \]
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 \[ \int f^{c x^{2} + a} \cos \left (f x^{2} + e x + d\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 191, normalized size = 0.91 \[ \frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 i d \ln \relax (f ) c +4 d f -e^{2}}{-2 i f +c \ln \relax (f )}} \erf \left (x \sqrt {2 i f -c \ln \relax (f )}+\frac {i e}{\sqrt {2 i f -c \ln \relax (f )}}\right )}{8 \sqrt {2 i f -c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 i d \ln \relax (f ) c -4 d f +e^{2}}{2 i f +c \ln \relax (f )}} \erf \left (-\sqrt {-2 i f -c \ln \relax (f )}\, x +\frac {i e}{\sqrt {-2 i f -c \ln \relax (f )}}\right )}{8 \sqrt {-2 i f -c \ln \relax (f )}}+\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.36, size = 863, normalized size = 4.09 \[ \text {result too large to display} \]
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 \[ \int f^{c\,x^2+a}\,{\cos \left (f\,x^2+e\,x+d\right )}^2 \,d x \]
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 \[ \int f^{a + c x^{2}} \cos ^{2}{\left (d + e x + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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