Optimal. Leaf size=340 \[ -\frac {3}{16} \sqrt [4]{-1} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {1}{4} i \left (4 d+\frac {(b \log (f)+i e)^2}{f}\right )} \text {erf}\left (\frac {\sqrt [4]{-1} (b \log (f)+i e+2 i f x)}{2 \sqrt {f}}\right )-\left (\frac {1}{16}+\frac {i}{16}\right ) \sqrt {\frac {\pi }{6}} f^{a-\frac {1}{2}} e^{\frac {i (b \log (f)+3 i e)^2}{12 f}+3 i d} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b \log (f)+3 i e+6 i f x)}{\sqrt {6} \sqrt {f}}\right )-\frac {3}{16} \sqrt [4]{-1} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {i (e+i b \log (f))^2}{4 f}-i d} \text {erfi}\left (\frac {\sqrt [4]{-1} (-b \log (f)+i e+2 i f x)}{2 \sqrt {f}}\right )-\left (\frac {1}{16}+\frac {i}{16}\right ) \sqrt {\frac {\pi }{6}} f^{a-\frac {1}{2}} e^{\frac {i (3 e+i b \log (f))^2}{12 f}-3 i d} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (-b \log (f)+3 i e+6 i f x)}{\sqrt {6} \sqrt {f}}\right ) \]
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Rubi [A] time = 0.52, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4473, 2287, 2234, 2204, 2205} \[ -\frac {3}{16} \sqrt [4]{-1} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {1}{4} i \left (4 d+\frac {(b \log (f)+i e)^2}{f}\right )} \text {Erf}\left (\frac {\sqrt [4]{-1} (b \log (f)+i e+2 i f x)}{2 \sqrt {f}}\right )-\left (\frac {1}{16}+\frac {i}{16}\right ) \sqrt {\frac {\pi }{6}} f^{a-\frac {1}{2}} e^{\frac {i (b \log (f)+3 i e)^2}{12 f}+3 i d} \text {Erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b \log (f)+3 i e+6 i f x)}{\sqrt {6} \sqrt {f}}\right )-\frac {3}{16} \sqrt [4]{-1} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {i (e+i b \log (f))^2}{4 f}-i d} \text {Erfi}\left (\frac {\sqrt [4]{-1} (-b \log (f)+i e+2 i f x)}{2 \sqrt {f}}\right )-\left (\frac {1}{16}+\frac {i}{16}\right ) \sqrt {\frac {\pi }{6}} f^{a-\frac {1}{2}} e^{\frac {i (3 e+i b \log (f))^2}{12 f}-3 i d} \text {Erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (-b \log (f)+3 i e+6 i f x)}{\sqrt {6} \sqrt {f}}\right ) \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2234
Rule 2287
Rule 4473
Rubi steps
\begin {align*} \int f^{a+b x} \cos ^3\left (d+e x+f x^2\right ) \, dx &=\int \left (\frac {1}{8} e^{-3 i \left (d+e x+f x^2\right )} f^{a+b x}+\frac {3}{8} \exp \left (2 i d+2 i e x+2 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x}+\frac {3}{8} \exp \left (4 i d+4 i e x+4 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x}+\frac {1}{8} \exp \left (6 i d+6 i e x+6 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x}\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 i \left (d+e x+f x^2\right )} f^{a+b x} \, dx+\frac {1}{8} \int \exp \left (6 i d+6 i e x+6 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx+\frac {3}{8} \int \exp \left (2 i d+2 i e x+2 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx+\frac {3}{8} \int \exp \left (4 i d+4 i e x+4 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx\\ &=\frac {1}{8} \int \exp \left (-3 i d-3 i f x^2+a \log (f)-x (3 i e-b \log (f))\right ) \, dx+\frac {1}{8} \int \exp \left (3 i d+3 i f x^2+a \log (f)+x (3 i e+b \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (-i d-i f x^2+a \log (f)-x (i e-b \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (i d+i f x^2+a \log (f)+x (i e+b \log (f))\right ) \, dx\\ &=\frac {1}{8} \exp \left (-3 i d+a \log (f)-\frac {i (-3 i e+b \log (f))^2}{12 f}\right ) \int e^{\frac {i (-3 i e-6 i f x+b \log (f))^2}{12 f}} \, dx+\frac {1}{8} \left (3 e^{-i d+\frac {i (e+i b \log (f))^2}{4 f}} f^a\right ) \int e^{\frac {i (-i e-2 i f x+b \log (f))^2}{4 f}} \, dx+\frac {1}{8} \left (3 e^{\frac {1}{4} i \left (4 d+\frac {(i e+b \log (f))^2}{f}\right )} f^a\right ) \int e^{-\frac {i (i e+2 i f x+b \log (f))^2}{4 f}} \, dx+\frac {1}{8} \left (e^{3 i d+\frac {i (3 i e+b \log (f))^2}{12 f}} f^a\right ) \int e^{-\frac {i (3 i e+6 i f x+b \log (f))^2}{12 f}} \, dx\\ &=-\frac {3}{16} \sqrt [4]{-1} e^{\frac {1}{4} i \left (4 d+\frac {(i e+b \log (f))^2}{f}\right )} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt [4]{-1} (i e+2 i f x+b \log (f))}{2 \sqrt {f}}\right )-\left (\frac {1}{16}+\frac {i}{16}\right ) e^{3 i d+\frac {i (3 i e+b \log (f))^2}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{6}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (3 i e+6 i f x+b \log (f))}{\sqrt {6} \sqrt {f}}\right )-\frac {3}{16} \sqrt [4]{-1} e^{-i d+\frac {i (e+i b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt [4]{-1} (i e+2 i f x-b \log (f))}{2 \sqrt {f}}\right )-\left (\frac {1}{16}+\frac {i}{16}\right ) \exp \left (-\frac {1}{12} i \left (36 d+\frac {(3 i e-b \log (f))^2}{f}\right )\right ) f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{6}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (3 i e+6 i f x-b \log (f))}{\sqrt {6} \sqrt {f}}\right )\\ \end {align*}
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Mathematica [A] time = 1.57, size = 322, normalized size = 0.95 \[ \frac {1}{48} \sqrt [4]{-1} \sqrt {\pi } f^{a-\frac {b e+f}{2 f}} e^{-\frac {i \left (b^2 \log ^2(f)+3 e^2\right )}{4 f}} \left (9 (\sin (d)-i \cos (d)) e^{\frac {i \left (b^2 \log ^2(f)+e^2\right )}{2 f}} \text {erfi}\left (\frac {\sqrt [4]{-1} (-i b \log (f)+e+2 f x)}{2 \sqrt {f}}\right )+e^{\frac {i e^2}{f}} \left (-\sqrt {3} (\cos (3 d)-i \sin (3 d)) e^{\frac {i \left (b^2 \log ^2(f)+3 e^2\right )}{6 f}} \text {erfi}\left (\frac {(-1)^{3/4} (i b \log (f)+3 e+6 f x)}{2 \sqrt {3} \sqrt {f}}\right )-9 (\cos (d)-i \sin (d)) \text {erfi}\left (\frac {(-1)^{3/4} (i b \log (f)+e+2 f x)}{2 \sqrt {f}}\right )\right )+\sqrt {3} e^{\frac {i b^2 \log ^2(f)}{3 f}} (\sin (3 d)-i \cos (3 d)) \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (-i b \log (f)+3 e+6 f x)}{\sqrt {6} \sqrt {f}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.56, size = 629, normalized size = 1.85 \[ \frac {\sqrt {6} \pi \sqrt {\frac {f}{\pi }} e^{\left (\frac {-i \, b^{2} \log \relax (f)^{2} + 9 i \, e^{2} - 36 i \, d f - 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right )} \operatorname {C}\left (\frac {\sqrt {6} {\left (6 \, f x + i \, b \log \relax (f) + 3 \, e\right )} \sqrt {\frac {f}{\pi }}}{6 \, f}\right ) - \sqrt {6} \pi \sqrt {\frac {f}{\pi }} e^{\left (\frac {i \, b^{2} \log \relax (f)^{2} - 9 i \, e^{2} + 36 i \, d f - 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right )} \operatorname {C}\left (-\frac {\sqrt {6} {\left (6 \, f x - i \, b \log \relax (f) + 3 \, e\right )} \sqrt {\frac {f}{\pi }}}{6 \, f}\right ) + 9 \, \sqrt {2} \pi \sqrt {\frac {f}{\pi }} e^{\left (\frac {-i \, b^{2} \log \relax (f)^{2} + i \, e^{2} - 4 i \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right )} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, f x + i \, b \log \relax (f) + e\right )} \sqrt {\frac {f}{\pi }}}{2 \, f}\right ) - 9 \, \sqrt {2} \pi \sqrt {\frac {f}{\pi }} e^{\left (\frac {i \, b^{2} \log \relax (f)^{2} - i \, e^{2} + 4 i \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right )} \operatorname {C}\left (-\frac {\sqrt {2} {\left (2 \, f x - i \, b \log \relax (f) + e\right )} \sqrt {\frac {f}{\pi }}}{2 \, f}\right ) - i \, \sqrt {6} \pi \sqrt {\frac {f}{\pi }} e^{\left (\frac {-i \, b^{2} \log \relax (f)^{2} + 9 i \, e^{2} - 36 i \, d f - 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right )} \operatorname {S}\left (\frac {\sqrt {6} {\left (6 \, f x + i \, b \log \relax (f) + 3 \, e\right )} \sqrt {\frac {f}{\pi }}}{6 \, f}\right ) - i \, \sqrt {6} \pi \sqrt {\frac {f}{\pi }} e^{\left (\frac {i \, b^{2} \log \relax (f)^{2} - 9 i \, e^{2} + 36 i \, d f - 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right )} \operatorname {S}\left (-\frac {\sqrt {6} {\left (6 \, f x - i \, b \log \relax (f) + 3 \, e\right )} \sqrt {\frac {f}{\pi }}}{6 \, f}\right ) - 9 i \, \sqrt {2} \pi \sqrt {\frac {f}{\pi }} e^{\left (\frac {-i \, b^{2} \log \relax (f)^{2} + i \, e^{2} - 4 i \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right )} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, f x + i \, b \log \relax (f) + e\right )} \sqrt {\frac {f}{\pi }}}{2 \, f}\right ) - 9 i \, \sqrt {2} \pi \sqrt {\frac {f}{\pi }} e^{\left (\frac {i \, b^{2} \log \relax (f)^{2} - i \, e^{2} + 4 i \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right )} \operatorname {S}\left (-\frac {\sqrt {2} {\left (2 \, f x - i \, b \log \relax (f) + e\right )} \sqrt {\frac {f}{\pi }}}{2 \, f}\right )}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 763, normalized size = 2.24 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 307, normalized size = 0.90 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {i \left (-9 e^{2}-6 i \ln \relax (f ) b e +\ln \relax (f )^{2} b^{2}+36 d f \right )}{12 f}} \sqrt {3}\, \erf \left (-\sqrt {3}\, \sqrt {i f}\, x +\frac {\left (b \ln \relax (f )-3 i e \right ) \sqrt {3}}{6 \sqrt {i f}}\right )}{48 \sqrt {i f}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {i \left (-e^{2}-2 i \ln \relax (f ) b e +\ln \relax (f )^{2} b^{2}+4 d f \right )}{4 f}} \erf \left (-\sqrt {i f}\, x +\frac {-i e +b \ln \relax (f )}{2 \sqrt {i f}}\right )}{16 \sqrt {i f}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {i \left (-e^{2}+2 i \ln \relax (f ) b e +\ln \relax (f )^{2} b^{2}+4 d f \right )}{4 f}} \erf \left (-\sqrt {-i f}\, x +\frac {i e +b \ln \relax (f )}{2 \sqrt {-i f}}\right )}{16 \sqrt {-i f}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {i \left (-9 e^{2}+6 i \ln \relax (f ) b e +\ln \relax (f )^{2} b^{2}+36 d f \right )}{12 f}} \erf \left (-\sqrt {-3 i f}\, x +\frac {3 i e +b \ln \relax (f )}{2 \sqrt {-3 i f}}\right )}{16 \sqrt {-3 i f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 377, normalized size = 1.11 \[ -\frac {3 \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i - 1\right ) \, f^{a} \cos \left (\frac {b^{2} \log \relax (f)^{2} - 9 \, e^{2} + 36 \, d f}{12 \, f}\right ) - \left (i + 1\right ) \, f^{a} \sin \left (\frac {b^{2} \log \relax (f)^{2} - 9 \, e^{2} + 36 \, d f}{12 \, f}\right )\right )} \operatorname {erf}\left (\frac {i \, {\left (6 i \, f x - b \log \relax (f) + 3 i \, e\right )} \sqrt {3 i \, f}}{6 \, f}\right ) + {\left (\left (i + 1\right ) \, f^{a} \cos \left (\frac {b^{2} \log \relax (f)^{2} - 9 \, e^{2} + 36 \, d f}{12 \, f}\right ) + \left (i - 1\right ) \, f^{a} \sin \left (\frac {b^{2} \log \relax (f)^{2} - 9 \, e^{2} + 36 \, d f}{12 \, f}\right )\right )} \operatorname {erf}\left (\frac {i \, {\left (6 i \, f x + b \log \relax (f) + 3 i \, e\right )} \sqrt {-3 i \, f}}{6 \, f}\right )\right )} f^{\frac {3}{2}} + \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (27 i - 27\right ) \, f^{a} \cos \left (\frac {b^{2} \log \relax (f)^{2} - e^{2} + 4 \, d f}{4 \, f}\right ) - \left (27 i + 27\right ) \, f^{a} \sin \left (\frac {b^{2} \log \relax (f)^{2} - e^{2} + 4 \, d f}{4 \, f}\right )\right )} \operatorname {erf}\left (\frac {i \, {\left (2 i \, f x - b \log \relax (f) + i \, e\right )} \sqrt {i \, f}}{2 \, f}\right ) + {\left (\left (27 i + 27\right ) \, f^{a} \cos \left (\frac {b^{2} \log \relax (f)^{2} - e^{2} + 4 \, d f}{4 \, f}\right ) + \left (27 i - 27\right ) \, f^{a} \sin \left (\frac {b^{2} \log \relax (f)^{2} - e^{2} + 4 \, d f}{4 \, f}\right )\right )} \operatorname {erf}\left (\frac {i \, {\left (2 i \, f x + b \log \relax (f) + i \, e\right )} \sqrt {-i \, f}}{2 \, f}\right )\right )} f^{\frac {3}{2}}}{288 \, f^{2} f^{\frac {b e}{2 \, f}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{a+b\,x}\,{\cos \left (f\,x^2+e\,x+d\right )}^3 \,d x \]
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 \[ \int f^{a + b x} \cos ^{3}{\left (d + e x + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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