Optimal. Leaf size=301 \[ -\frac {3 i e^{-i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {i e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {i e^{-3 i d+\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {3 i e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 i e^{i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {i e^{3 i d+\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {3 i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A]
time = 0.24, 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 = {4560, 2325,
2266, 2235} \begin {gather*} -\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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 2325
Rule 4560
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 = 0.46, size = 224, normalized size = 0.74 \begin {gather*} \frac {e^{\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \left (3 i \text {Erfi}\left (\frac {-i e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (d)+i \sin (d))+3 \text {Erfi}\left (\frac {-i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (i \cos (d)+\sin (d))-i e^{\frac {2 e^2}{c \log (f)}} \left (\text {Erfi}\left (\frac {-3 i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (3 d)-i \sin (3 d))-\text {Erfi}\left (\frac {3 i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (3 d)+i \sin (3 d))\right )\right )}{16 \sqrt {c} \sqrt {\log (f)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 246, normalized size = 0.82
method | result | size |
risch | \(-\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {3 i d \ln \left (f \right ) c +\frac {9 e^{2}}{4}}{c \ln \left (f \right )}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {3 i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}-\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {3 \left (4 i d \ln \left (f \right ) c -3 e^{2}\right )}{4 \ln \left (f \right ) c}} \erf \left (\sqrt {-c \ln \left (f \right )}\, x +\frac {3 i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}+\frac {3 i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 i d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}} \erf \left (\sqrt {-c \ln \left (f \right )}\, x +\frac {i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}+\frac {3 i \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 i d \ln \left (f \right ) c +e^{2}}{4 \ln \left (f \right ) c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}\) | \(246\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.30, size = 412, normalized size = 1.37 \begin {gather*} \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 \left (f\right )}} + \frac {3}{2} i \, \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}} e\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + f^{a} {\left (-i \, \cos \left (3 \, d\right ) + \sin \left (3 \, d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {3}{2} i \, \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}} e\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + f^{a} {\left (i \, \cos \left (3 \, d\right ) - \sin \left (3 \, d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) + 3 i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + f^{a} {\left (-i \, \cos \left (3 \, d\right ) - \sin \left (3 \, d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) - 3 i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} - 3 \, f^{a} {\left (i \, \cos \left (d\right ) + \sin \left (d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} + \frac {1}{2} i \, \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}} e\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} - 3 \, f^{a} {\left (-i \, \cos \left (d\right ) + \sin \left (d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} i \, \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}} e\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} - 3 \, f^{a} {\left (i \, \cos \left (d\right ) - \sin \left (d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) + i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} - 3 \, f^{a} {\left (-i \, \cos \left (d\right ) - \sin \left (d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) - i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )}\right )} \sqrt {-c \log \left (f\right )}}{32 \, c \log \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.02, size = 282, normalized size = 0.94 \begin {gather*} \frac {-i \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + 3 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 12 i \, c d \log \left (f\right ) + 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 i \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 4 i \, c d \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )} - 3 i \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 4 i \, c d \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )} + i \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - 3 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 12 i \, c d \log \left (f\right ) + 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, c \log \left (f\right )} \end {gather*}
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 \begin {gather*} \int f^{a + c x^{2}} \sin ^{3}{\left (d + e x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int f^{c\,x^2+a}\,{\sin \left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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