Optimal. Leaf size=171 \[ \frac {f^a \sqrt {\pi } \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 i d+\frac {e^2}{c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {i e-c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{2 i d+\frac {e^2}{c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {i e+c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A]
time = 0.15, 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 = {4561, 2235,
2325, 2266} \begin {gather*} -\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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 2325
Rule 4561
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 = 0.27, size = 131, normalized size = 0.77 \begin {gather*} \frac {f^a \sqrt {\pi } \left (2 \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )+e^{\frac {e^2}{c \log (f)}} \left (\text {Erfi}\left (\frac {-i e+c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right ) (\cos (2 d)-i \sin (2 d))+\text {Erfi}\left (\frac {i e+c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right ) (\cos (2 d)+i \sin (2 d))\right )\right )}{8 \sqrt {c} \sqrt {\log (f)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 145, normalized size = 0.85
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 i d \ln \left (f \right ) c -e^{2}}{\ln \left (f \right ) c}} \erf \left (\sqrt {-c \ln \left (f \right )}\, x +\frac {i e}{\sqrt {-c \ln \left (f \right )}}\right )}{8 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 i d \ln \left (f \right ) c +e^{2}}{\ln \left (f \right ) c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {i e}{\sqrt {-c \ln \left (f \right )}}\right )}{8 \sqrt {-c \ln \left (f \right )}}+\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(145\) |
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 = 0.31, size = 236, normalized size = 1.38 \begin {gather*} \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 \left (f\right )}} + i \, \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}} e\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (2 \, d\right ) + i \, \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - i \, \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}} e\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} - f^{a} {\left (\cos \left (2 \, d\right ) + i \, \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (\frac {c x \log \left (f\right ) + i \, e}{\sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} - f^{a} {\left (\cos \left (2 \, d\right ) - i \, \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (\frac {c x \log \left (f\right ) - i \, e}{\sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} + 2 \, f^{a} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}}\right ) + 2 \, f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )\right )}}{16 \, \sqrt {-c \log \left (f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.13, size = 159, normalized size = 0.93 \begin {gather*} -\frac {2 \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right ) + \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (c x \log \left (f\right ) + i \, e\right )} \sqrt {-c \log \left (f\right )}}{c \log \left (f\right )}\right ) e^{\left (\frac {a c \log \left (f\right )^{2} + 2 i \, c d \log \left (f\right ) + e^{2}}{c \log \left (f\right )}\right )} + \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (c x \log \left (f\right ) - i \, e\right )} \sqrt {-c \log \left (f\right )}}{c \log \left (f\right )}\right ) e^{\left (\frac {a c \log \left (f\right )^{2} - 2 i \, c d \log \left (f\right ) + e^{2}}{c \log \left (f\right )}\right )}}{8 \, c \log \left (f\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 \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 = -0.01 \begin {gather*} \int f^{c\,x^2+a}\,{\cos \left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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