Optimal. Leaf size=245 \[ \frac {\sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{-4 c \log (f)+8 i f}-2 i d} \text {erf}\left (\frac {b \log (f)-2 x (-c \log (f)+2 i f)}{2 \sqrt {-c \log (f)+2 i f}}\right )}{8 \sqrt {-c \log (f)+2 i f}}-\frac {\sqrt {\pi } f^a e^{2 i d-\frac {b^2 \log ^2(f)}{4 c \log (f)+8 i f}} \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+2 i f)}{2 \sqrt {c \log (f)+2 i f}}\right )}{8 \sqrt {c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.46, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4472, 2234, 2204, 2287, 2205} \[ \frac {\sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{-4 c \log (f)+8 i f}-2 i d} \text {Erf}\left (\frac {b \log (f)-2 x (-c \log (f)+2 i f)}{2 \sqrt {-c \log (f)+2 i f}}\right )}{8 \sqrt {-c \log (f)+2 i f}}-\frac {\sqrt {\pi } f^a e^{2 i d-\frac {b^2 \log ^2(f)}{4 c \log (f)+8 i f}} \text {Erfi}\left (\frac {b \log (f)+2 x (c \log (f)+2 i f)}{2 \sqrt {c \log (f)+2 i f}}\right )}{8 \sqrt {c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2234
Rule 2287
Rule 4472
Rubi steps
\begin {align*} \int f^{a+b x+c x^2} \sin ^2\left (d+f x^2\right ) \, dx &=\int \left (\frac {1}{2} f^{a+b x+c x^2}-\frac {1}{4} e^{-2 i d-2 i f x^2} f^{a+b x+c x^2}-\frac {1}{4} e^{2 i d+2 i f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=-\left (\frac {1}{4} \int e^{-2 i d-2 i f x^2} f^{a+b x+c x^2} \, dx\right )-\frac {1}{4} \int e^{2 i d+2 i f x^2} f^{a+b x+c x^2} \, dx+\frac {1}{2} \int f^{a+b x+c x^2} \, dx\\ &=-\left (\frac {1}{4} \int \exp \left (-2 i d+a \log (f)+b x \log (f)-x^2 (2 i f-c \log (f))\right ) \, dx\right )-\frac {1}{4} \int \exp \left (2 i d+a \log (f)+b x \log (f)+x^2 (2 i f+c \log (f))\right ) \, dx+\frac {1}{2} f^{a-\frac {b^2}{4 c}} \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx\\ &=\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {1}{4} \left (e^{-2 i d+\frac {b^2 \log ^2(f)}{8 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+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 {b^2 \log ^2(f)}{8 i f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (2 i f+c \log (f)))^2}{4 (2 i f+c \log (f))}\right ) \, dx\\ &=\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 i d+\frac {b^2 \log ^2(f)}{8 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (2 i f-c \log (f))}{2 \sqrt {2 i f-c \log (f)}}\right )}{8 \sqrt {2 i f-c \log (f)}}-\frac {e^{2 i d-\frac {b^2 \log ^2(f)}{8 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (2 i f+c \log (f))}{2 \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 = 3.09, size = 299, normalized size = 1.22 \[ \frac {1}{8} \sqrt {\pi } f^a \left (\frac {2 f^{-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{\sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt [4]{-1} e^{\frac {b^2 \log ^2(f)}{-4 c \log (f)+8 i f}} \left (\sqrt {2 f-i c \log (f)} (2 f+i c \log (f)) (\cos (2 d)+i \sin (2 d)) e^{\frac {i b^2 f \log ^2(f)}{c^2 \log ^2(f)+4 f^2}} \text {erf}\left (\frac {(-1)^{3/4} (4 f x-i \log (f) (b+2 c x))}{2 \sqrt {2 f-i c \log (f)}}\right )+\sqrt {2 f+i c \log (f)} (c \log (f)+2 i f) (\cos (2 d)-i \sin (2 d)) \text {erf}\left (\frac {\sqrt [4]{-1} (4 f x+i \log (f) (b+2 c x))}{2 \sqrt {2 f+i c \log (f)}}\right )\right )}{c^2 \log ^2(f)+4 f^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.96, size = 400, normalized size = 1.63 \[ \frac {\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 2 i \, c f \log \relax (f)\right )} \sqrt {-c \log \relax (f) - 2 i \, f} \operatorname {erf}\left (\frac {{\left (8 \, f^{2} x - 2 i \, b f \log \relax (f) + {\left (2 \, c^{2} x + b c\right )} \log \relax (f)^{2}\right )} \sqrt {-c \log \relax (f) - 2 i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right ) e^{\left (\frac {16 \, a f^{2} \log \relax (f) - {\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)^{3} + 32 i \, d f^{2} + {\left (8 i \, c^{2} d + 2 i \, b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right )} + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 2 i \, c f \log \relax (f)\right )} \sqrt {-c \log \relax (f) + 2 i \, f} \operatorname {erf}\left (\frac {{\left (8 \, f^{2} x + 2 i \, b f \log \relax (f) + {\left (2 \, c^{2} x + b c\right )} \log \relax (f)^{2}\right )} \sqrt {-c \log \relax (f) + 2 i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right ) e^{\left (\frac {16 \, a f^{2} \log \relax (f) - {\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)^{3} - 32 i \, d f^{2} + {\left (-8 i \, c^{2} d - 2 i \, b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right )} - \frac {2 \, \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )} \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{8 \, {\left (c^{3} \log \relax (f)^{3} + 4 \, c f^{2} \log \relax (f)\right )}} \]
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 \[ \int f^{c x^{2} + b x + a} \sin \left (f x^{2} + d\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 227, normalized size = 0.93 \[ \frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+8 i d \ln \relax (f ) c +16 d f}{4 \left (-2 i f +c \ln \relax (f )\right )}} \erf \left (-x \sqrt {2 i f -c \ln \relax (f )}+\frac {\ln \relax (f ) b}{2 \sqrt {2 i f -c \ln \relax (f )}}\right )}{8 \sqrt {2 i f -c \ln \relax (f )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-8 i d \ln \relax (f ) c +16 d f}{4 \left (2 i f +c \ln \relax (f )\right )}} \erf \left (-\sqrt {-2 i f -c \ln \relax (f )}\, x +\frac {\ln \relax (f ) b}{2 \sqrt {-2 i f -c \ln \relax (f )}}\right )}{8 \sqrt {-2 i f -c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.38, size = 997, normalized size = 4.07 \[ \text {result too large to display} \]
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 \[ \int f^{c\,x^2+b\,x+a}\,{\sin \left (f\,x^2+d\right )}^2 \,d x \]
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 \[ \int f^{a + b x + c x^{2}} \sin ^{2}{\left (d + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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