Optimal. Leaf size=161 \[ -\frac {\sqrt {\pi } f^a e^{-\frac {e^2}{c \log (f)}-2 d} \text {erfi}\left (\frac {e-c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {e^2}{c \log (f)}} \text {erfi}\left (\frac {c x \log (f)+e}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.22, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5513, 2204, 2287, 2234} \[ -\frac {\sqrt {\pi } f^a e^{-\frac {e^2}{c \log (f)}-2 d} \text {Erfi}\left (\frac {e-c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {e^2}{c \log (f)}} \text {Erfi}\left (\frac {c x \log (f)+e}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2287
Rule 5513
Rubi steps
\begin {align*} \int f^{a+c x^2} \cosh ^2(d+e x) \, dx &=\int \left (\frac {1}{2} f^{a+c x^2}+\frac {1}{4} e^{-2 d-2 e x} f^{a+c x^2}+\frac {1}{4} e^{2 d+2 e x} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 d-2 e x} f^{a+c x^2} \, dx+\frac {1}{4} \int e^{2 d+2 e x} 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 e^{-2 d-2 e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {1}{4} \int e^{2 d+2 e x+a \log (f)+c x^2 \log (f)} \, 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}{c \log (f)}} f^a\right ) \int e^{\frac {(-2 e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{4} \left (e^{2 d-\frac {e^2}{c \log (f)}} f^a\right ) \int e^{\frac {(2 e+2 c x \log (f))^2}{4 c \log (f)}} \, 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}{c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{2 d-\frac {e^2}{c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 131, normalized size = 0.81 \[ \frac {\sqrt {\pi } f^a e^{-\frac {e^2}{c \log (f)}} \left ((\cosh (2 d)-\sinh (2 d)) \text {erfi}\left (\frac {c x \log (f)-e}{\sqrt {c} \sqrt {\log (f)}}\right )+(\sinh (2 d)+\cosh (2 d)) \text {erfi}\left (\frac {c x \log (f)+e}{\sqrt {c} \sqrt {\log (f)}}\right )+2 e^{\frac {e^2}{c \log (f)}} \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 242, normalized size = 1.50 \[ -\frac {2 \, \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (a \log \relax (f)\right ) + \sqrt {\pi } \sinh \left (a \log \relax (f)\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x\right ) + \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (\frac {a c \log \relax (f)^{2} + 2 \, c d \log \relax (f) - e^{2}}{c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (\frac {a c \log \relax (f)^{2} + 2 \, c d \log \relax (f) - e^{2}}{c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left (c x \log \relax (f) + e\right )} \sqrt {-c \log \relax (f)}}{c \log \relax (f)}\right ) + \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (\frac {a c \log \relax (f)^{2} - 2 \, c d \log \relax (f) - e^{2}}{c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (\frac {a c \log \relax (f)^{2} - 2 \, c d \log \relax (f) - e^{2}}{c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left (c x \log \relax (f) - e\right )} \sqrt {-c \log \relax (f)}}{c \log \relax (f)}\right )}{8 \, c \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 150, normalized size = 0.93 \[ -\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-c \log \relax (f)} x\right )}{4 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f)} {\left (x + \frac {e}{c \log \relax (f)}\right )}\right ) e^{\left (\frac {a c \log \relax (f)^{2} + 2 \, c d \log \relax (f) - e^{2}}{c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f)} {\left (x - \frac {e}{c \log \relax (f)}\right )}\right ) e^{\left (\frac {a c \log \relax (f)^{2} - 2 \, c d \log \relax (f) - e^{2}}{c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 139, normalized size = 0.86 \[ \frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 d \ln \relax (f ) c +e^{2}}{\ln \relax (f ) c}} \erf \left (\sqrt {-c \ln \relax (f )}\, x +\frac {e}{\sqrt {-c \ln \relax (f )}}\right )}{8 \sqrt {-c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 d \ln \relax (f ) c -e^{2}}{\ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {e}{\sqrt {-c \ln \relax (f )}}\right )}{8 \sqrt {-c \ln \relax (f )}}+\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 131, normalized size = 0.81 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {e}{\sqrt {-c \log \relax (f)}}\right ) e^{\left (2 \, d - \frac {e^{2}}{c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x + \frac {e}{\sqrt {-c \log \relax (f)}}\right ) e^{\left (-2 \, d - \frac {e^{2}}{c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x\right )}{4 \, \sqrt {-c \log \relax (f)}} \]
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 \[ \int f^{c\,x^2+a}\,{\mathrm {cosh}\left (d+e\,x\right )}^2 \,d x \]
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 \[ \int f^{a + c x^{2}} \cosh ^{2}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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