Optimal. Leaf size=346 \[ -\frac {3 e^{-i d+\frac {(e+i b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {i e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-3 i d+\frac {(3 e+i b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {3 i e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 e^{i d+\frac {(e-i b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {i e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 i d-\frac {(3 i e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {3 i e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A]
time = 0.32, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4561, 2325,
2266, 2235} \begin {gather*} -\frac {3 \sqrt {\pi } f^a e^{\frac {(e+i b \log (f))^2}{4 c \log (f)}-i d} \text {Erfi}\left (\frac {-b \log (f)-2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{\frac {(3 e+i b \log (f))^2}{4 c \log (f)}-3 i d} \text {Erfi}\left (\frac {-b \log (f)-2 c x \log (f)+3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 \sqrt {\pi } f^a e^{\frac {(e-i b \log (f))^2}{4 c \log (f)}+i d} \text {Erfi}\left (\frac {b \log (f)+2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{3 i d-\frac {(b \log (f)+3 i e)^2}{4 c \log (f)}} \text {Erfi}\left (\frac {b \log (f)+2 c x \log (f)+3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \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+b x+c x^2} \cos ^3(d+e x) \, dx &=\int \left (\frac {3}{8} e^{-i d-i e x} f^{a+b x+c x^2}+\frac {3}{8} e^{i d+i e x} f^{a+b x+c x^2}+\frac {1}{8} e^{-3 i d-3 i e x} f^{a+b x+c x^2}+\frac {1}{8} e^{3 i d+3 i e x} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 i d-3 i e x} f^{a+b x+c x^2} \, dx+\frac {1}{8} \int e^{3 i d+3 i e x} f^{a+b x+c x^2} \, dx+\frac {3}{8} \int e^{-i d-i e x} f^{a+b x+c x^2} \, dx+\frac {3}{8} \int e^{i d+i e x} f^{a+b x+c x^2} \, dx\\ &=\frac {1}{8} \int \exp \left (-3 i d+a \log (f)+c x^2 \log (f)-x (3 i e-b \log (f))\right ) \, dx+\frac {1}{8} \int \exp \left (3 i d+a \log (f)+c x^2 \log (f)+x (3 i e+b \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (-i d+a \log (f)+c x^2 \log (f)-x (i e-b \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (i d+a \log (f)+c x^2 \log (f)+x (i e+b \log (f))\right ) \, dx\\ &=\frac {1}{8} \left (3 e^{i d+\frac {(e-i b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac {1}{8} \left (3 e^{-i d+\frac {(e+i b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac {1}{8} \left (\exp \left (-3 i d+\frac {(3 e+i b \log (f))^2}{4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac {(-3 i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac {1}{8} \left (e^{3 i d-\frac {(3 i e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(3 i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx\\ &=-\frac {3 e^{-i d+\frac {(e+i b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {\exp \left (-3 i d+\frac {(3 e+i b \log (f))^2}{4 c \log (f)}\right ) f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 e^{i d+\frac {(e-i b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 i d-\frac {(3 i e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A]
time = 0.97, size = 386, normalized size = 1.12 \begin {gather*} \frac {e^{\frac {e (e-6 i b \log (f))}{4 c \log (f)}} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \left (e^{\frac {e (2 e+3 i b \log (f))}{c \log (f)}} \cos (3 d) \text {Erfi}\left (\frac {-3 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )+e^{\frac {2 e^2}{c \log (f)}} \cos (3 d) \text {Erfi}\left (\frac {3 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )+3 e^{\frac {2 i b e}{c}} \text {Erfi}\left (\frac {-i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (d)-i \sin (d))+3 e^{\frac {i b e}{c}} \text {Erfi}\left (\frac {i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (d)+i \sin (d))-i e^{\frac {e (2 e+3 i b \log (f))}{c \log (f)}} \text {Erfi}\left (\frac {-3 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) \sin (3 d)+i e^{\frac {2 e^2}{c \log (f)}} \text {Erfi}\left (\frac {3 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) \sin (3 d)\right )}{16 \sqrt {c} \sqrt {\log (f)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.62, size = 334, normalized size = 0.97
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}-6 i \ln \left (f \right ) b e +12 i d \ln \left (f \right ) c -9 e^{2}}{4 \ln \left (f \right ) c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {-3 i e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}-2 i \ln \left (f \right ) b e +4 i d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )-i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}+2 i \ln \left (f \right ) b e -4 i d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {i e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}+6 i \ln \left (f \right ) b e -12 i d \ln \left (f \right ) c -9 e^{2}}{4 \ln \left (f \right ) c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {3 i e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}\) | \(334\) |
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 = 696, normalized size = 2.01 \begin {gather*} -\frac {\sqrt {\pi } {\left (f^{a} {\left (\cos \left (\frac {3 \, {\left (2 \, c d - b e\right )}}{2 \, c}\right ) + i \, \sin \left (\frac {3 \, {\left (2 \, c d - b e\right )}}{2 \, c}\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} \, {\left (b \log \left (f\right ) + 3 i \, e\right )} \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (\frac {3 \, {\left (2 \, c d - b e\right )}}{2 \, c}\right ) - i \, \sin \left (\frac {3 \, {\left (2 \, c d - b e\right )}}{2 \, c}\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} \, {\left (b \log \left (f\right ) - 3 i \, e\right )} \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (\frac {3 \, {\left (2 \, c d - b e\right )}}{2 \, c}\right ) + i \, \sin \left (\frac {3 \, {\left (2 \, c d - b e\right )}}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right ) + 3 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (\frac {3 \, {\left (2 \, c d - b e\right )}}{2 \, c}\right ) - i \, \sin \left (\frac {3 \, {\left (2 \, c d - b e\right )}}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right ) - 3 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, f^{a} {\left (\cos \left (\frac {2 \, c d - b e}{2 \, c}\right ) + i \, \sin \left (\frac {2 \, c d - b e}{2 \, c}\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} \, {\left (b \log \left (f\right ) + i \, e\right )} \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, f^{a} {\left (\cos \left (\frac {2 \, c d - b e}{2 \, c}\right ) - i \, \sin \left (\frac {2 \, c d - b e}{2 \, c}\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} \, {\left (b \log \left (f\right ) - i \, e\right )} \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, f^{a} {\left (\cos \left (\frac {2 \, c d - b e}{2 \, c}\right ) + i \, \sin \left (\frac {2 \, c d - b e}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right ) + i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, f^{a} {\left (\cos \left (\frac {2 \, c d - b e}{2 \, c}\right ) - i \, \sin \left (\frac {2 \, c d - b e}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right ) - i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )}\right )} \sqrt {-c \log \left (f\right )}}{32 \, c f^{\frac {b^{2}}{4 \, c}} \log \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.66, size = 348, normalized size = 1.01 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - 3 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 6 \, {\left (2 i \, c d - i \, b e\right )} \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 2 \, {\left (2 i \, c d - i \, b e\right )} \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )} + \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + 3 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 6 \, {\left (-2 i \, c d + i \, b e\right )} \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 2 \, {\left (-2 i \, c d + i \, b e\right )} \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, 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 + b x + c x^{2}} \cos ^{3}{\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 = -0.00 \begin {gather*} \int f^{c\,x^2+b\,x+a}\,{\cos \left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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