Optimal. Leaf size=140 \[ \frac {f^a \sqrt {\pi } \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 i d} f^a \sqrt {\pi } \text {Erf}\left (x \sqrt {2 i f-c \log (f)}\right )}{8 \sqrt {2 i f-c \log (f)}}+\frac {e^{2 i d} f^a \sqrt {\pi } \text {Erfi}\left (x \sqrt {2 i f+c \log (f)}\right )}{8 \sqrt {2 i f+c \log (f)}} \]
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Rubi [A]
time = 0.15, 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 = {4561, 2235,
2325, 2236} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 2325
Rule 4561
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 = 0.89, size = 189, normalized size = 1.35 \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 (-\text {Erfi}\left ((-1)^{3/4} 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 (\sqrt [4]{-1} x \sqrt {2 f-i c \log (f)}\right ) \sqrt {2 f-i c \log (f)} (-2 i f+c \log (f)) (\cos (2 d)+i \sin (2 d))\right )}{4 f^2+c^2 \log ^2(f)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 107, normalized size = 0.76
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-2 i d} \erf \left (x \sqrt {2 i f -c \ln \left (f \right )}\right )}{8 \sqrt {2 i f -c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{2 i d} \erf \left (\sqrt {-c \ln \left (f \right )-2 i f}\, x \right )}{8 \sqrt {-c \ln \left (f \right )-2 i f}}+\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(107\) |
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 = 0.28, size = 315, normalized size = 2.25 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{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 \left (f\right ) + 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 \left (f\right ) - 2 i \, f} x\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 (f^{a} {\left (\cos \left (2 \, d\right ) - i \, \sin \left (2 \, d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 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 \left (f\right ) - 2 i \, f} x\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 [A]
time = 2.27, size = 167, normalized size = 1.19 \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 (\sqrt {-c \log \left (f\right ) - 2 i \, f} x\right ) e^{\left (a \log \left (f\right ) + 2 i \, d\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 (\sqrt {-c \log \left (f\right ) + 2 i \, f} x\right ) e^{\left (a \log \left (f\right ) - 2 i \, d\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 + 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 = -0.01 \begin {gather*} \int f^{c\,x^2+a}\,{\cos \left (f\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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