Optimal. Leaf size=183 \[ \frac {\sqrt {\pi } f^a e^{\frac {e^2}{2 f-c \log (f)}-2 d} \text {erf}\left (\frac {x (2 f-c \log (f))+e}{\sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {e^2}{c \log (f)+2 f}} \text {erfi}\left (\frac {x (c \log (f)+2 f)+e}{\sqrt {c \log (f)+2 f}}\right )}{8 \sqrt {c \log (f)+2 f}}+\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.33, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5513, 2204, 2287, 2234, 2205} \[ \frac {\sqrt {\pi } f^a e^{\frac {e^2}{2 f-c \log (f)}-2 d} \text {Erf}\left (\frac {x (2 f-c \log (f))+e}{\sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {e^2}{c \log (f)+2 f}} \text {Erfi}\left (\frac {x (c \log (f)+2 f)+e}{\sqrt {c \log (f)+2 f}}\right )}{8 \sqrt {c \log (f)+2 f}}+\frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2234
Rule 2287
Rule 5513
Rubi steps
\begin {align*} \int f^{a+c x^2} \cosh ^2\left (d+e x+f x^2\right ) \, dx &=\int \left (\frac {1}{2} f^{a+c x^2}+\frac {1}{4} e^{-2 d-2 e x-2 f x^2} f^{a+c x^2}+\frac {1}{4} e^{2 d+2 e x+2 f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 d-2 e x-2 f x^2} f^{a+c x^2} \, dx+\frac {1}{4} \int e^{2 d+2 e x+2 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 d-2 e x+a \log (f)-x^2 (2 f-c \log (f))\right ) \, dx+\frac {1}{4} \int \exp \left (2 d+2 e x+a \log (f)+x^2 (2 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 d+\frac {e^2}{2 f-c \log (f)}} f^a\right ) \int \exp \left (\frac {(-2 e+2 x (-2 f+c \log (f)))^2}{4 (-2 f+c \log (f))}\right ) \, dx+\frac {1}{4} \left (e^{2 d-\frac {e^2}{2 f+c \log (f)}} f^a\right ) \int \exp \left (\frac {(2 e+2 x (2 f+c \log (f)))^2}{4 (2 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 d+\frac {e^2}{2 f-c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {e+x (2 f-c \log (f))}{\sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d-\frac {e^2}{2 f+c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+x (2 f+c \log (f))}{\sqrt {2 f+c \log (f)}}\right )}{8 \sqrt {2 f+c \log (f)}}\\ \end {align*}
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Mathematica [A] time = 1.49, size = 258, normalized size = 1.41 \[ \frac {\sqrt {\pi } f^a e^{\frac {e^2}{2 f-c \log (f)}} \left (-\sqrt {c} \sqrt {\log (f)} \left ((2 f-c \log (f)) \sqrt {c \log (f)+2 f} (\sinh (2 d)+\cosh (2 d)) e^{\frac {4 e^2 f}{c^2 \log ^2(f)-4 f^2}} \text {erfi}\left (\frac {c x \log (f)+e+2 f x}{\sqrt {c \log (f)+2 f}}\right )+\sqrt {2 f-c \log (f)} (c \log (f)+2 f) (\cosh (2 d)-\sinh (2 d)) \text {erf}\left (\frac {-c x \log (f)+e+2 f x}{\sqrt {2 f-c \log (f)}}\right )\right )-2 \left (4 f^2-c^2 \log ^2(f)\right ) e^{\frac {e^2}{c \log (f)-2 f}} \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )\right )}{8 \sqrt {c} \sqrt {\log (f)} \left (c^2 \log ^2(f)-4 f^2\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 420, normalized size = 2.30 \[ -\frac {2 \, {\left (\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 4 \, f^{2}\right )} \cosh \left (a \log \relax (f)\right ) + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 4 \, f^{2}\right )} \sinh \left (a \log \relax (f)\right )\right )} \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 2 \, c f \log \relax (f)\right )} \cosh \left (\frac {a c \log \relax (f)^{2} - e^{2} + 4 \, d f - 2 \, {\left (c d + a f\right )} \log \relax (f)}{c \log \relax (f) - 2 \, f}\right ) + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 2 \, c f \log \relax (f)\right )} \sinh \left (\frac {a c \log \relax (f)^{2} - e^{2} + 4 \, d f - 2 \, {\left (c d + a f\right )} \log \relax (f)}{c \log \relax (f) - 2 \, f}\right )\right )} \sqrt {-c \log \relax (f) + 2 \, f} \operatorname {erf}\left (\frac {{\left (c x \log \relax (f) - 2 \, f x - e\right )} \sqrt {-c \log \relax (f) + 2 \, f}}{c \log \relax (f) - 2 \, f}\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 2 \, c f \log \relax (f)\right )} \cosh \left (\frac {a c \log \relax (f)^{2} - e^{2} + 4 \, d f + 2 \, {\left (c d + a f\right )} \log \relax (f)}{c \log \relax (f) + 2 \, f}\right ) + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 2 \, c f \log \relax (f)\right )} \sinh \left (\frac {a c \log \relax (f)^{2} - e^{2} + 4 \, d f + 2 \, {\left (c d + a f\right )} \log \relax (f)}{c \log \relax (f) + 2 \, f}\right )\right )} \sqrt {-c \log \relax (f) - 2 \, f} \operatorname {erf}\left (\frac {{\left (c x \log \relax (f) + 2 \, f x + e\right )} \sqrt {-c \log \relax (f) - 2 \, f}}{c \log \relax (f) + 2 \, f}\right )}{8 \, {\left (c^{3} \log \relax (f)^{3} - 4 \, c f^{2} \log \relax (f)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 198, normalized size = 1.08 \[ -\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-c \log \relax (f)} x\right )}{4 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f) - 2 \, f} {\left (x + \frac {e}{c \log \relax (f) + 2 \, f}\right )}\right ) e^{\left (\frac {a c \log \relax (f)^{2} + 2 \, c d \log \relax (f) + 2 \, a f \log \relax (f) + 4 \, d f - e^{2}}{c \log \relax (f) + 2 \, f}\right )}}{8 \, \sqrt {-c \log \relax (f) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f) + 2 \, f} {\left (x - \frac {e}{c \log \relax (f) - 2 \, f}\right )}\right ) e^{\left (\frac {a c \log \relax (f)^{2} - 2 \, c d \log \relax (f) - 2 \, a f \log \relax (f) + 4 \, d f - e^{2}}{c \log \relax (f) - 2 \, f}\right )}}{8 \, \sqrt {-c \log \relax (f) + 2 \, f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 177, normalized size = 0.97 \[ \frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 d \ln \relax (f ) c -4 d f +e^{2}}{-2 f +c \ln \relax (f )}} \erf \left (x \sqrt {2 f -c \ln \relax (f )}+\frac {e}{\sqrt {2 f -c \ln \relax (f )}}\right )}{8 \sqrt {2 f -c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 d \ln \relax (f ) c +4 d f -e^{2}}{2 f +c \ln \relax (f )}} \erf \left (-\sqrt {-c \ln \relax (f )-2 f}\, x +\frac {e}{\sqrt {-c \ln \relax (f )-2 f}}\right )}{8 \sqrt {-c \ln \relax (f )-2 f}}+\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 161, normalized size = 0.88 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - 2 \, f} x - \frac {e}{\sqrt {-c \log \relax (f) - 2 \, f}}\right ) e^{\left (2 \, d - \frac {e^{2}}{c \log \relax (f) + 2 \, f}\right )}}{8 \, \sqrt {-c \log \relax (f) - 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + 2 \, f} x + \frac {e}{\sqrt {-c \log \relax (f) + 2 \, f}}\right ) e^{\left (-2 \, d - \frac {e^{2}}{c \log \relax (f) - 2 \, f}\right )}}{8 \, \sqrt {-c \log \relax (f) + 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x\right )}{4 \, \sqrt {-c \log \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int f^{c\,x^2+a}\,{\mathrm {cosh}\left (f\,x^2+e\,x+d\right )}^2 \,d x \]
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 \[ \int f^{a + c x^{2}} \cosh ^{2}{\left (d + e x + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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