Integrand size = 21, antiderivative size = 354 \[ \int f^{a+b x+c x^2} \sin ^3(d+e x) \, dx=-\frac {3 i e^{-i d+\frac {(e+i b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {i e^{-3 i d+\frac {(3 e+i b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 i e^{i d+\frac {(e-i b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {i e^{3 i d-\frac {(3 i e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.61 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4560, 2325, 2266, 2235} \[ \int f^{a+b x+c x^2} \sin ^3(d+e x) \, dx=-\frac {3 i \sqrt {\pi } f^a e^{\frac {(e+i b \log (f))^2}{4 c \log (f)}-i d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {i \sqrt {\pi } f^a e^{\frac {(3 e+i b \log (f))^2}{4 c \log (f)}-3 i d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 i \sqrt {\pi } f^a e^{\frac {(e-i b \log (f))^2}{4 c \log (f)}+i d} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {i \sqrt {\pi } f^a e^{3 i d-\frac {(b \log (f)+3 i e)^2}{4 c \log (f)}} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Rule 2235
Rule 2266
Rule 2325
Rule 4560
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{8} i e^{-i d-i e x} f^{a+b x+c x^2}-\frac {3}{8} i e^{i d+i e x} f^{a+b x+c x^2}-\frac {1}{8} i e^{-3 i d-3 i e x} f^{a+b x+c x^2}+\frac {1}{8} i e^{3 i d+3 i e x} f^{a+b x+c x^2}\right ) \, dx \\ & = -\left (\frac {1}{8} i \int e^{-3 i d-3 i e x} f^{a+b x+c x^2} \, dx\right )+\frac {1}{8} i \int e^{3 i d+3 i e x} f^{a+b x+c x^2} \, dx+\frac {3}{8} i \int e^{-i d-i e x} f^{a+b x+c x^2} \, dx-\frac {3}{8} i \int e^{i d+i e x} f^{a+b x+c x^2} \, dx \\ & = -\left (\frac {1}{8} i \int \exp \left (-3 i d+a \log (f)+c x^2 \log (f)-x (3 i e-b \log (f))\right ) \, dx\right )+\frac {1}{8} i \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} i \int \exp \left (-i d+a \log (f)+c x^2 \log (f)-x (i e-b \log (f))\right ) \, dx-\frac {3}{8} i \int \exp \left (i d+a \log (f)+c x^2 \log (f)+x (i e+b \log (f))\right ) \, dx \\ & = -\left (\frac {1}{8} \left (3 i 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\right )+\frac {1}{8} \left (3 i 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 (i \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 (i 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 i 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 {i \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 i 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 {i 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*}
Time = 0.72 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.10 \[ \int f^{a+b x+c x^2} \sin ^3(d+e x) \, dx=\frac {e^{\frac {e (e-6 i b \log (f))}{4 c \log (f)}} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \left (-i e^{\frac {e (2 e+3 i b \log (f))}{c \log (f)}} \cos (3 d) \text {erfi}\left (\frac {-3 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )+i e^{\frac {2 e^2}{c \log (f)}} \cos (3 d) \text {erfi}\left (\frac {3 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )+3 i e^{\frac {i b e}{c}} \text {erfi}\left (\frac {-i e-(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (d)+i \sin (d))+3 e^{\frac {2 i b e}{c}} \text {erfi}\left (\frac {-i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (i \cos (d)+\sin (d))-e^{\frac {e (2 e+3 i b \log (f))}{c \log (f)}} \text {erfi}\left (\frac {-3 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) \sin (3 d)-e^{\frac {2 e^2}{c \log (f)}} \text {erfi}\left (\frac {3 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) \sin (3 d)\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 1.38 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-\frac {i \sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{-\frac {3 \left (2 i \ln \left (f \right ) b e -4 i d \ln \left (f \right ) c -3 e^{2}\right )}{4 \ln \left (f \right ) c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {3 i e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}+\frac {i \sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{\frac {\frac {3 i \ln \left (f \right ) b e}{2}-3 i d \ln \left (f \right ) c +\frac {9 e^{2}}{4}}{c \ln \left (f \right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )-3 i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}-\frac {3 i \sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{\frac {2 i \ln \left (f \right ) b e -4 i d \ln \left (f \right ) c +e^{2}}{4 \ln \left (f \right ) c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )-i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}+\frac {3 i \sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{-\frac {2 i \ln \left (f \right ) b e -4 i d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {i e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}\) | \(344\) |
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Time = 0.26 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.98 \[ \int f^{a+b x+c x^2} \sin ^3(d+e x) \, dx=\frac {-3 i \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - e^{2} + 2 \, {\left (2 i \, c d - i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + 3 i \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - e^{2} + 2 \, {\left (-2 i \, c d + i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + i \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - 3 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 9 \, e^{2} + 6 \, {\left (2 i \, c d - i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} - i \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + 3 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 9 \, e^{2} + 6 \, {\left (-2 i \, c d + i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )}}{16 \, c \log \left (f\right )} \]
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\[ \int f^{a+b x+c x^2} \sin ^3(d+e x) \, dx=\int f^{a + b x + c x^{2}} \sin ^{3}{\left (d + e x \right )}\, dx \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.27 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.92 \[ \int f^{a+b x+c x^2} \sin ^3(d+e x) \, dx=\text {Too large to display} \]
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\[ \int f^{a+b x+c x^2} \sin ^3(d+e x) \, dx=\int { f^{c x^{2} + b x + a} \sin \left (e x + d\right )^{3} \,d x } \]
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Timed out. \[ \int f^{a+b x+c x^2} \sin ^3(d+e x) \, dx=\int f^{c\,x^2+b\,x+a}\,{\sin \left (d+e\,x\right )}^3 \,d x \]
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