Optimal. Leaf size=153 \[ \frac {\sqrt {\pi } f^a e^{d-\frac {(b \log (f)+e)^2}{4 c \log (f)}} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{-\frac {(e-b \log (f))^2}{4 c \log (f)}-d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.29, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5513, 2287, 2234, 2204} \[ \frac {\sqrt {\pi } f^a e^{d-\frac {(b \log (f)+e)^2}{4 c \log (f)}} \text {Erfi}\left (\frac {b \log (f)+2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{-\frac {(e-b \log (f))^2}{4 c \log (f)}-d} \text {Erfi}\left (\frac {-b \log (f)-2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2287
Rule 5513
Rubi steps
\begin {align*} \int f^{a+b x+c x^2} \cosh (d+e x) \, dx &=\int \left (\frac {1}{2} e^{-d-e x} f^{a+b x+c x^2}+\frac {1}{2} e^{d+e x} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{2} \int e^{-d-e x} f^{a+b x+c x^2} \, dx+\frac {1}{2} \int e^{d+e x} f^{a+b x+c x^2} \, dx\\ &=\frac {1}{2} \int \exp \left (-d+a \log (f)+c x^2 \log (f)-x (e-b \log (f))\right ) \, dx+\frac {1}{2} \int \exp \left (d+a \log (f)+c x^2 \log (f)+x (e+b \log (f))\right ) \, dx\\ &=\frac {1}{2} \left (e^{-d-\frac {(e-b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac {1}{2} \left (e^{d-\frac {(e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}} \, dx\\ &=-\frac {e^{-d-\frac {(e-b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{d-\frac {(e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 134, normalized size = 0.88 \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} e^{-\frac {e (2 b \log (f)+e)}{4 c \log (f)}} \left (e^{\frac {b e}{c}} (\cosh (d)-\sinh (d)) \text {erfi}\left (\frac {\log (f) (b+2 c x)-e}{2 \sqrt {c} \sqrt {\log (f)}}\right )+(\sinh (d)+\cosh (d)) \text {erfi}\left (\frac {\log (f) (b+2 c x)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 262, normalized size = 1.71 \[ -\frac {\sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + e^{2} - 2 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + e^{2} - 2 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \relax (f) + e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) + \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + e^{2} + 2 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + e^{2} + 2 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \relax (f) - e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right )}{4 \, c \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 169, normalized size = 1.10 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b \log \relax (f) - e}{c \log \relax (f)}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} + 4 \, c d \log \relax (f) - 2 \, b e \log \relax (f) + e^{2}}{4 \, c \log \relax (f)}\right )}}{4 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b \log \relax (f) + e}{c \log \relax (f)}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} - 4 \, c d \log \relax (f) + 2 \, b e \log \relax (f) + e^{2}}{4 \, c \log \relax (f)}\right )}}{4 \, \sqrt {-c \log \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 156, normalized size = 1.02 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-2 \ln \relax (f ) b e +4 d \ln \relax (f ) c +e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )-e}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+2 \ln \relax (f ) b e -4 d \ln \relax (f ) c +e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {e +b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 129, normalized size = 0.84 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f) + e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (d - \frac {{\left (b \log \relax (f) + e\right )}^{2}}{4 \, c \log \relax (f)}\right )}}{4 \, \sqrt {-c \log \relax (f)}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f) - e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (-d - \frac {{\left (b \log \relax (f) - e\right )}^{2}}{4 \, c \log \relax (f)}\right )}}{4 \, \sqrt {-c \log \relax (f)}} \]
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 \[ \int f^{c\,x^2+b\,x+a}\,\mathrm {cosh}\left (d+e\,x\right ) \,d x \]
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 \[ \int f^{a + b x + c x^{2}} \cosh {\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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