Optimal. Leaf size=140 \[ \frac {\sqrt {\pi } e^{-2 i d} f^a \text {erf}\left (x \sqrt {-c \log (f)+2 i f}\right )}{8 \sqrt {-c \log (f)+2 i f}}+\frac {\sqrt {\pi } e^{2 i d} f^a \text {erfi}\left (x \sqrt {c \log (f)+2 i f}\right )}{8 \sqrt {c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.19, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4473, 2204, 2287, 2205} \[ \frac {\sqrt {\pi } e^{-2 i d} f^a \text {Erf}\left (x \sqrt {-c \log (f)+2 i f}\right )}{8 \sqrt {-c \log (f)+2 i f}}+\frac {\sqrt {\pi } e^{2 i d} f^a \text {Erfi}\left (x \sqrt {c \log (f)+2 i f}\right )}{8 \sqrt {c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2287
Rule 4473
Rubi steps
\begin {align*} \int f^{a+c x^2} \cos ^2\left (d+f x^2\right ) \, dx &=\int \left (\frac {1}{2} f^{a+c x^2}+\frac {1}{4} e^{-2 i d-2 i f x^2} f^{a+c x^2}+\frac {1}{4} e^{2 i d+2 i f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 i d-2 i f x^2} f^{a+c x^2} \, dx+\frac {1}{4} \int e^{2 i d+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+a \log (f)-x^2 (2 i f-c \log (f))\right ) \, dx+\frac {1}{4} \int \exp \left (2 i d+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 {e^{-2 i d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {2 i f-c \log (f)}\right )}{8 \sqrt {2 i f-c \log (f)}}+\frac {e^{2 i d} f^a \sqrt {\pi } \text {erfi}\left (x \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 = 0.83, size = 189, normalized size = 1.35 \[ \frac {1}{8} \sqrt {\pi } f^a \left (\frac {2 \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{\sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt [4]{-1} \left (\sqrt {2 f-i c \log (f)} (c \log (f)-2 i f) (\cos (2 d)+i \sin (2 d)) \text {erfi}\left (\sqrt [4]{-1} x \sqrt {2 f-i c \log (f)}\right )-(2 f-i c \log (f)) \sqrt {2 f+i c \log (f)} (\cos (2 d)-i \sin (2 d)) \text {erfi}\left ((-1)^{3/4} x \sqrt {2 f+i c \log (f)}\right )\right )}{c^2 \log ^2(f)+4 f^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.71, size = 167, normalized size = 1.19 \[ -\frac {2 \, \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )} \sqrt {-c \log \relax (f)} f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x\right ) + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 2 i \, c f \log \relax (f)\right )} \sqrt {-c \log \relax (f) - 2 i \, f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - 2 i \, f} x\right ) e^{\left (a \log \relax (f) + 2 i \, d\right )} + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 2 i \, c f \log \relax (f)\right )} \sqrt {-c \log \relax (f) + 2 i \, f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + 2 i \, f} x\right ) e^{\left (a \log \relax (f) - 2 i \, d\right )}}{8 \, {\left (c^{3} \log \relax (f)^{3} + 4 \, c f^{2} \log \relax (f)\right )}} \]
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 \[ \int f^{c x^{2} + a} \cos \left (f x^{2} + d\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 107, normalized size = 0.76 \[ \frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-2 i d} \erf \left (x \sqrt {2 i f -c \ln \relax (f )}\right )}{8 \sqrt {2 i f -c \ln \relax (f )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{2 i d} \erf \left (\sqrt {-2 i f -c \ln \relax (f )}\, x \right )}{8 \sqrt {-2 i f -c \ln \relax (f )}}+\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.35, size = 315, normalized size = 2.25 \[ -\frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \relax (f)^{2} + 8 \, f^{2}} {\left (f^{a} {\left (i \, \cos \left (2 \, d\right ) + \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + 2 i \, f} x\right ) + f^{a} {\left (-i \, \cos \left (2 \, d\right ) + \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - 2 i \, f} x\right )\right )} \sqrt {c \log \relax (f) + \sqrt {c^{2} \log \relax (f)^{2} + 4 \, f^{2}}} \sqrt {-c \log \relax (f)} - \sqrt {\pi } \sqrt {2 \, c^{2} \log \relax (f)^{2} + 8 \, f^{2}} {\left (f^{a} {\left (\cos \left (2 \, d\right ) - i \, \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + 2 i \, f} x\right ) + f^{a} {\left (\cos \left (2 \, d\right ) + i \, \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - 2 i \, f} x\right )\right )} \sqrt {-c \log \relax (f) + \sqrt {c^{2} \log \relax (f)^{2} + 4 \, f^{2}}} \sqrt {-c \log \relax (f)} - 2 \, \sqrt {\pi } {\left ({\left (c^{2} f^{a} \log \relax (f)^{2} + 4 \, f^{a + 2}\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}}\right ) + {\left (c^{2} f^{a} \log \relax (f)^{2} + 4 \, f^{a + 2}\right )} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x\right )\right )}}{16 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )} \sqrt {-c \log \relax (f)}} \]
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 \[ \int f^{c\,x^2+a}\,{\cos \left (f\,x^2+d\right )}^2 \,d x \]
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 \[ \int f^{a + c x^{2}} \cos ^{2}{\left (d + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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