Optimal. Leaf size=301 \[ -\frac {3 i \sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)}-i d} \text {erfi}\left (\frac {-2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {i \sqrt {\pi } f^a e^{\frac {9 e^2}{4 c \log (f)}-3 i d} \text {erfi}\left (\frac {-2 c x \log (f)+3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 i \sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)}+i d} \text {erfi}\left (\frac {2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {i \sqrt {\pi } f^a e^{\frac {9 e^2}{4 c \log (f)}+3 i d} \text {erfi}\left (\frac {2 c x \log (f)+3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.35, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4472, 2287, 2234, 2204} \[ -\frac {3 i \sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)}-i d} \text {Erfi}\left (\frac {-2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {i \sqrt {\pi } f^a e^{\frac {9 e^2}{4 c \log (f)}-3 i d} \text {Erfi}\left (\frac {-2 c x \log (f)+3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 i \sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)}+i d} \text {Erfi}\left (\frac {2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {i \sqrt {\pi } f^a e^{\frac {9 e^2}{4 c \log (f)}+3 i d} \text {Erfi}\left (\frac {2 c x \log (f)+3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2287
Rule 4472
Rubi steps
\begin {align*} \int f^{a+c x^2} \sin ^3(d+e x) \, dx &=\int \left (\frac {3}{8} i e^{-i d-i e x} f^{a+c x^2}-\frac {3}{8} i e^{i d+i e x} f^{a+c x^2}-\frac {1}{8} i e^{-3 i d-3 i e x} f^{a+c x^2}+\frac {1}{8} i e^{3 i d+3 i e x} f^{a+c x^2}\right ) \, dx\\ &=-\left (\frac {1}{8} i \int e^{-3 i d-3 i e x} f^{a+c x^2} \, dx\right )+\frac {1}{8} i \int e^{3 i d+3 i e x} f^{a+c x^2} \, dx+\frac {3}{8} i \int e^{-i d-i e x} f^{a+c x^2} \, dx-\frac {3}{8} i \int e^{i d+i e x} f^{a+c x^2} \, dx\\ &=-\left (\frac {1}{8} i \int e^{-3 i d-3 i e x+a \log (f)+c x^2 \log (f)} \, dx\right )+\frac {1}{8} i \int e^{3 i d+3 i e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {3}{8} i \int e^{-i d-i e x+a \log (f)+c x^2 \log (f)} \, dx-\frac {3}{8} i \int e^{i d+i e x+a \log (f)+c x^2 \log (f)} \, dx\\ &=\frac {1}{8} \left (3 i e^{-i d+\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx-\frac {1}{8} \left (3 i e^{i d+\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx-\frac {1}{8} \left (i e^{-3 i d+\frac {9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-3 i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{8} \left (i e^{3 i d+\frac {9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(3 i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx\\ &=-\frac {3 i e^{-i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e-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 {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e-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^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+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 {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e+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 = 0.42, size = 224, normalized size = 0.74 \[ \frac {\sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)}} \left (-i e^{\frac {2 e^2}{c \log (f)}} \left ((\cos (3 d)-i \sin (3 d)) \text {erfi}\left (\frac {2 c x \log (f)-3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )-(\cos (3 d)+i \sin (3 d)) \text {erfi}\left (\frac {2 c x \log (f)+3 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )\right )+3 i (\cos (d)+i \sin (d)) \text {erfi}\left (\frac {-2 c x \log (f)-i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )+3 (\sin (d)+i \cos (d)) \text {erfi}\left (\frac {2 c x \log (f)-i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 282, normalized size = 0.94 \[ \frac {-i \, \sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) + 3 i \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} + 12 i \, c d \log \relax (f) + 9 \, e^{2}}{4 \, c \log \relax (f)}\right )} + 3 i \, \sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) + i \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} + 4 i \, c d \log \relax (f) + e^{2}}{4 \, c \log \relax (f)}\right )} - 3 i \, \sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) - i \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} - 4 i \, c d \log \relax (f) + e^{2}}{4 \, c \log \relax (f)}\right )} + i \, \sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) - 3 i \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} - 12 i \, c d \log \relax (f) + 9 \, e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, c \log \relax (f)} \]
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 \[ \int f^{c x^{2} + a} \sin \left (e x + d\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.84, size = 246, normalized size = 0.82 \[ -\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {3 i d \ln \relax (f ) c +\frac {9 e^{2}}{4}}{c \ln \relax (f )}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {3 i e}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}}-\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {3 \left (4 i d \ln \relax (f ) c -3 e^{2}\right )}{4 \ln \relax (f ) c}} \erf \left (\sqrt {-c \ln \relax (f )}\, x +\frac {3 i e}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}}+\frac {3 i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 i d \ln \relax (f ) c -e^{2}}{4 \ln \relax (f ) c}} \erf \left (\sqrt {-c \ln \relax (f )}\, x +\frac {i e}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}}+\frac {3 i \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 i d \ln \relax (f ) c +e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {i e}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.42, size = 412, normalized size = 1.37 \[ \frac {\sqrt {\pi } {\left (f^{a} {\left (i \, \cos \left (3 \, d\right ) + \sin \left (3 \, d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}} + \frac {3}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \relax (f)}}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \relax (f)}\right )} + f^{a} {\left (-i \, \cos \left (3 \, d\right ) + \sin \left (3 \, d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}} - \frac {3}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \relax (f)}}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \relax (f)}\right )} + f^{a} {\left (i \, \cos \left (3 \, d\right ) - \sin \left (3 \, d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \relax (f) + 3 i \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \relax (f)}\right )} + f^{a} {\left (-i \, \cos \left (3 \, d\right ) - \sin \left (3 \, d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \relax (f) - 3 i \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \relax (f)}\right )} + f^{a} {\left (-3 i \, \cos \relax (d) - 3 \, \sin \relax (d)\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}} + \frac {1}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \relax (f)}}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \relax (f)}\right )} + f^{a} {\left (3 i \, \cos \relax (d) - 3 \, \sin \relax (d)\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}} - \frac {1}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \relax (f)}}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \relax (f)}\right )} + f^{a} {\left (-3 i \, \cos \relax (d) + 3 \, \sin \relax (d)\right )} \operatorname {erf}\left (\frac {2 \, c x \log \relax (f) + i \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \relax (f)}\right )} + f^{a} {\left (3 i \, \cos \relax (d) + 3 \, \sin \relax (d)\right )} \operatorname {erf}\left (\frac {2 \, c x \log \relax (f) - i \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \relax (f)}\right )}\right )} \sqrt {-c \log \relax (f)}}{32 \, c \log \relax (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 \[ \int f^{c\,x^2+a}\,{\sin \left (d+e\,x\right )}^3 \,d x \]
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 \[ \int f^{a + c x^{2}} \sin ^{3}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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