Optimal. Leaf size=211 \[ \frac {f^a \sqrt {\pi } \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 i d-\frac {e^2}{2 i f-c \log (f)}} f^a \sqrt {\pi } \text {Erf}\left (\frac {i e+x (2 i f-c \log (f))}{\sqrt {2 i f-c \log (f)}}\right )}{8 \sqrt {2 i f-c \log (f)}}+\frac {e^{2 i d+\frac {e^2}{2 i f+c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {i e+x (2 i f+c \log (f))}{\sqrt {2 i f+c \log (f)}}\right )}{8 \sqrt {2 i f+c \log (f)}} \]
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Rubi [A]
time = 0.28, 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 = {4561, 2235,
2325, 2266, 2236} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 4561
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 = 2.44, size = 252, normalized size = 1.19 \begin {gather*} \frac {1}{8} f^a \sqrt {\pi } \left (\frac {2 \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{\sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt [4]{-1} \left (-e^{\frac {e^2}{-2 i f+c \log (f)}} \text {Erfi}\left (\frac {(-1)^{3/4} (e+2 f x+i c x \log (f))}{\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}{2 i f+c \log (f)}} \text {Erfi}\left (\frac {\sqrt [4]{-1} (e+2 f x-i c x \log (f))}{\sqrt {2 f-i c \log (f)}}\right ) \sqrt {2 f-i c \log (f)} (2 f+i c \log (f)) (-i \cos (2 d)+\sin (2 d))\right )}{4 f^2+c^2 \log ^2(f)}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.35, size = 191, normalized size = 0.91
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 i d \ln \left (f \right ) c +4 d f -e^{2}}{-2 i f +c \ln \left (f \right )}} \erf \left (x \sqrt {2 i f -c \ln \left (f \right )}+\frac {i e}{\sqrt {2 i f -c \ln \left (f \right )}}\right )}{8 \sqrt {2 i f -c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 i d \ln \left (f \right ) c -4 d f +e^{2}}{2 i f +c \ln \left (f \right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-2 i f}\, x +\frac {i e}{\sqrt {-c \ln \left (f \right )-2 i f}}\right )}{8 \sqrt {-c \ln \left (f \right )-2 i f}}+\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.29, size = 851, normalized size = 4.03 \begin {gather*} \frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 8 \, f^{2}} {\left ({\left (i \, f^{a} f^{\frac {c e^{2}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}} \cos \left (\frac {2 \, {\left (c^{2} d \log \left (f\right )^{2} + 4 \, d f^{2} - f e^{2}\right )}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right ) + f^{a} f^{\frac {c e^{2}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}} \sin \left (\frac {2 \, {\left (c^{2} d \log \left (f\right )^{2} + 4 \, d f^{2} - f e^{2}\right )}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right )\right )} \operatorname {erf}\left (\frac {{\left (c \log \left (f\right ) - 2 i \, f\right )} x - i \, e}{\sqrt {-c \log \left (f\right ) + 2 i \, f}}\right ) + {\left (-i \, f^{a} f^{\frac {c e^{2}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}} \cos \left (\frac {2 \, {\left (c^{2} d \log \left (f\right )^{2} + 4 \, d f^{2} - f e^{2}\right )}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right ) + f^{a} f^{\frac {c e^{2}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}} \sin \left (\frac {2 \, {\left (c^{2} d \log \left (f\right )^{2} + 4 \, d f^{2} - f e^{2}\right )}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right )\right )} \operatorname {erf}\left (\frac {{\left (c \log \left (f\right ) + 2 i \, f\right )} x + i \, e}{\sqrt {-c \log \left (f\right ) - 2 i \, f}}\right )\right )} \sqrt {c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}} \sqrt {-c \log \left (f\right )} - \sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 8 \, f^{2}} {\left ({\left (f^{a} f^{\frac {c e^{2}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}} \cos \left (\frac {2 \, {\left (c^{2} d \log \left (f\right )^{2} + 4 \, d f^{2} - f e^{2}\right )}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right ) - i \, f^{a} f^{\frac {c e^{2}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}} \sin \left (\frac {2 \, {\left (c^{2} d \log \left (f\right )^{2} + 4 \, d f^{2} - f e^{2}\right )}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right )\right )} \operatorname {erf}\left (\frac {{\left (c \log \left (f\right ) - 2 i \, f\right )} x - i \, e}{\sqrt {-c \log \left (f\right ) + 2 i \, f}}\right ) + {\left (f^{a} f^{\frac {c e^{2}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}} \cos \left (\frac {2 \, {\left (c^{2} d \log \left (f\right )^{2} + 4 \, d f^{2} - f e^{2}\right )}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right ) + i \, f^{a} f^{\frac {c e^{2}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}} \sin \left (\frac {2 \, {\left (c^{2} d \log \left (f\right )^{2} + 4 \, d f^{2} - f e^{2}\right )}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right )\right )} \operatorname {erf}\left (\frac {{\left (c \log \left (f\right ) + 2 i \, f\right )} x + i \, e}{\sqrt {-c \log \left (f\right ) - 2 i \, f}}\right )\right )} \sqrt {-c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}} \sqrt {-c \log \left (f\right )} + 2 \, \sqrt {\pi } {\left ({\left (c^{2} f^{a} \log \left (f\right )^{2} + 4 \, f^{a + 2}\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}}\right ) + {\left (c^{2} f^{a} \log \left (f\right )^{2} + 4 \, f^{a + 2}\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )\right )}}{16 \, {\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )} \sqrt {-c \log \left (f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 361 vs. \(2 (155) = 310\).
time = 2.33, size = 361, normalized size = 1.71 \begin {gather*} -\frac {2 \, \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )} \sqrt {-c \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 i \, c f \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) - 2 i \, f} \operatorname {erf}\left (\frac {{\left (c^{2} x \log \left (f\right )^{2} + 4 \, f^{2} x + i \, c e \log \left (f\right ) + 2 \, f e\right )} \sqrt {-c \log \left (f\right ) - 2 i \, f}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right ) e^{\left (\frac {a c^{2} \log \left (f\right )^{3} + 2 i \, c^{2} d \log \left (f\right )^{2} + 8 i \, d f^{2} - 2 i \, f e^{2} + {\left (4 \, a f^{2} + c e^{2}\right )} \log \left (f\right )}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right )} + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 i \, c f \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) + 2 i \, f} \operatorname {erf}\left (\frac {{\left (c^{2} x \log \left (f\right )^{2} + 4 \, f^{2} x - i \, c e \log \left (f\right ) + 2 \, f e\right )} \sqrt {-c \log \left (f\right ) + 2 i \, f}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right ) e^{\left (\frac {a c^{2} \log \left (f\right )^{3} - 2 i \, c^{2} d \log \left (f\right )^{2} - 8 i \, d f^{2} + 2 i \, f e^{2} + {\left (4 \, a f^{2} + c e^{2}\right )} \log \left (f\right )}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right )}}{8 \, {\left (c^{3} \log \left (f\right )^{3} + 4 \, c f^{2} \log \left (f\right )\right )}} \end {gather*}
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 \begin {gather*} \int f^{a + c x^{2}} \cos ^{2}{\left (d + e x + f x^{2} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int f^{c\,x^2+a}\,{\cos \left (f\,x^2+e\,x+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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