Integrand size = 23, antiderivative size = 211 \[ \int f^{a+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 i d-\frac {e^2}{2 i f-c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {i e+x (2 i f-c \log (f))}{\sqrt {2 i f-c \log (f)}}\right )}{8 \sqrt {2 i f-c \log (f)}}-\frac {e^{2 i d+\frac {e^2}{2 i f+c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+x (2 i f+c \log (f))}{\sqrt {2 i f+c \log (f)}}\right )}{8 \sqrt {2 i f+c \log (f)}} \]
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Time = 0.48 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4560, 2235, 2325, 2266, 2236} \[ \int f^{a+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {\pi } f^a e^{-\frac {e^2}{-c \log (f)+2 i f}-2 i d} \text {erf}\left (\frac {x (-c \log (f)+2 i f)+i e}{\sqrt {-c \log (f)+2 i f}}\right )}{8 \sqrt {-c \log (f)+2 i f}}-\frac {\sqrt {\pi } f^a e^{\frac {e^2}{c \log (f)+2 i f}+2 i d} \text {erfi}\left (\frac {x (c \log (f)+2 i f)+i e}{\sqrt {c \log (f)+2 i f}}\right )}{8 \sqrt {c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 4560
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} f^{a+c x^2}-\frac {1}{4} e^{-2 i d-2 i e x-2 i f x^2} f^{a+c x^2}-\frac {1}{4} e^{2 i d+2 i e x+2 i f x^2} f^{a+c x^2}\right ) \, dx \\ & = -\left (\frac {1}{4} \int e^{-2 i d-2 i e x-2 i f x^2} f^{a+c x^2} \, dx\right )-\frac {1}{4} \int e^{2 i d+2 i e x+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-2 i e x+a \log (f)-x^2 (2 i f-c \log (f))\right ) \, dx-\frac {1}{4} \int \exp \left (2 i d+2 i e x+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 {1}{4} \left (e^{-2 i d-\frac {e^2}{2 i f-c \log (f)}} f^a\right ) \int \exp \left (\frac {(-2 i e+2 x (-2 i f+c \log (f)))^2}{4 (-2 i f+c \log (f))}\right ) \, dx-\frac {1}{4} \left (e^{2 i d+\frac {e^2}{2 i f+c \log (f)}} f^a\right ) \int \exp \left (\frac {(2 i e+2 x (2 i f+c \log (f)))^2}{4 (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-\frac {e^2}{2 i f-c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {i e+x (2 i f-c \log (f))}{\sqrt {2 i f-c \log (f)}}\right )}{8 \sqrt {2 i f-c \log (f)}}-\frac {e^{2 i d+\frac {e^2}{2 i f+c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+x (2 i f+c \log (f))}{\sqrt {2 i f+c \log (f)}}\right )}{8 \sqrt {2 i f+c \log (f)}} \\ \end{align*}
Time = 1.76 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.19 \[ \int f^{a+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\frac {1}{8} f^a \sqrt {\pi } \left (\frac {2 \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{\sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt [4]{-1} \left (e^{\frac {e^2}{2 i f+c \log (f)}} \text {erf}\left (\frac {(-1)^{3/4} (e+2 f x-i c x \log (f))}{\sqrt {2 f-i c \log (f)}}\right ) \sqrt {2 f-i c \log (f)} (2 f+i c \log (f)) (\cos (2 d)+i \sin (2 d))+e^{\frac {e^2}{-2 i f+c \log (f)}} \text {erf}\left (\frac {\sqrt [4]{-1} (e+2 f x+i c x \log (f))}{\sqrt {2 f+i c \log (f)}}\right ) (2 f-i c \log (f)) \sqrt {2 f+i c \log (f)} (i \cos (2 d)+\sin (2 d))\right )}{4 f^2+c^2 \log ^2(f)}\right ) \]
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Time = 0.69 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.91
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 i d \ln \left (f \right ) c +4 d f -e^{2}}{c \ln \left (f \right )-2 i f}} \operatorname {erf}\left (x \sqrt {2 i f -c \ln \left (f \right )}+\frac {i e}{\sqrt {2 i f -c \ln \left (f \right )}}\right )}{8 \sqrt {2 i f -c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 i d \ln \left (f \right ) c -4 d f +e^{2}}{2 i f +c \ln \left (f \right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-2 i f}\, x +\frac {i e}{\sqrt {-c \ln \left (f \right )-2 i f}}\right )}{8 \sqrt {-c \ln \left (f \right )-2 i f}}+\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(191\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (155) = 310\).
Time = 0.26 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.72 \[ \int f^{a+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=-\frac {2 \, \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )} \sqrt {-c \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right ) - \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 i \, c f \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) - 2 i \, f} \operatorname {erf}\left (\frac {{\left (c^{2} x \log \left (f\right )^{2} + 4 \, f^{2} x + i \, c e \log \left (f\right ) + 2 \, e f\right )} \sqrt {-c \log \left (f\right ) - 2 i \, f}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right ) e^{\left (\frac {a c^{2} \log \left (f\right )^{3} + 2 i \, c^{2} d \log \left (f\right )^{2} - 2 i \, e^{2} f + 8 i \, d f^{2} + {\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right )} - \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 i \, c f \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) + 2 i \, f} \operatorname {erf}\left (\frac {{\left (c^{2} x \log \left (f\right )^{2} + 4 \, f^{2} x - i \, c e \log \left (f\right ) + 2 \, e f\right )} \sqrt {-c \log \left (f\right ) + 2 i \, f}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right ) e^{\left (\frac {a c^{2} \log \left (f\right )^{3} - 2 i \, c^{2} d \log \left (f\right )^{2} + 2 i \, e^{2} f - 8 i \, d f^{2} + {\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right )}}{8 \, {\left (c^{3} \log \left (f\right )^{3} + 4 \, c f^{2} \log \left (f\right )\right )}} \]
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\[ \int f^{a+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\int f^{a + c x^{2}} \sin ^{2}{\left (d + e x + f x^{2} \right )}\, dx \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.25 (sec) , antiderivative size = 863, normalized size of antiderivative = 4.09 \[ \int f^{a+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
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\[ \int f^{a+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\int { f^{c x^{2} + a} \sin \left (f x^{2} + e x + d\right )^{2} \,d x } \]
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Timed out. \[ \int f^{a+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\int f^{c\,x^2+a}\,{\sin \left (f\,x^2+e\,x+d\right )}^2 \,d x \]
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