Optimal. Leaf size=161 \[ \frac{\sqrt{\pi } f^a e^{d-\frac{(b \log (f)+e)^2}{4 (c \log (f)+f)}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+f)+e}{2 \sqrt{c \log (f)+f}}\right )}{4 \sqrt{c \log (f)+f}}-\frac{\sqrt{\pi } f^a e^{\frac{(e-b \log (f))^2}{4 (f-c \log (f))}-d} \text{Erf}\left (\frac{-b \log (f)+2 x (f-c \log (f))+e}{2 \sqrt{f-c \log (f)}}\right )}{4 \sqrt{f-c \log (f)}} \]
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Rubi [A] time = 0.449089, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {5512, 2287, 2234, 2205, 2204} \[ \frac{\sqrt{\pi } f^a e^{d-\frac{(b \log (f)+e)^2}{4 (c \log (f)+f)}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+f)+e}{2 \sqrt{c \log (f)+f}}\right )}{4 \sqrt{c \log (f)+f}}-\frac{\sqrt{\pi } f^a e^{\frac{(e-b \log (f))^2}{4 (f-c \log (f))}-d} \text{Erf}\left (\frac{-b \log (f)+2 x (f-c \log (f))+e}{2 \sqrt{f-c \log (f)}}\right )}{4 \sqrt{f-c \log (f)}} \]
Antiderivative was successfully verified.
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Rule 5512
Rule 2287
Rule 2234
Rule 2205
Rule 2204
Rubi steps
\begin{align*} \int f^{a+b x+c x^2} \sinh \left (d+e x+f x^2\right ) \, dx &=\int \left (-\frac{1}{2} e^{-d-e x-f x^2} f^{a+b x+c x^2}+\frac{1}{2} e^{d+e x+f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=-\left (\frac{1}{2} \int e^{-d-e x-f x^2} f^{a+b x+c x^2} \, dx\right )+\frac{1}{2} \int e^{d+e x+f x^2} f^{a+b x+c x^2} \, dx\\ &=-\left (\frac{1}{2} \int \exp \left (-d+a \log (f)-x (e-b \log (f))-x^2 (f-c \log (f))\right ) \, dx\right )+\frac{1}{2} \int \exp \left (d+a \log (f)+x (e+b \log (f))+x^2 (f+c \log (f))\right ) \, dx\\ &=-\left (\frac{1}{2} \left (e^{-d+\frac{(e-b \log (f))^2}{4 (f-c \log (f))}} f^a\right ) \int \exp \left (\frac{(-e+b \log (f)+2 x (-f+c \log (f)))^2}{4 (-f+c \log (f))}\right ) \, dx\right )+\frac{1}{2} \left (e^{d-\frac{(e+b \log (f))^2}{4 (f+c \log (f))}} f^a\right ) \int \exp \left (\frac{(e+b \log (f)+2 x (f+c \log (f)))^2}{4 (f+c \log (f))}\right ) \, dx\\ &=-\frac{e^{-d+\frac{(e-b \log (f))^2}{4 (f-c \log (f))}} f^a \sqrt{\pi } \text{erf}\left (\frac{e-b \log (f)+2 x (f-c \log (f))}{2 \sqrt{f-c \log (f)}}\right )}{4 \sqrt{f-c \log (f)}}+\frac{e^{d-\frac{(e+b \log (f))^2}{4 (f+c \log (f))}} f^a \sqrt{\pi } \text{erfi}\left (\frac{e+b \log (f)+2 x (f+c \log (f))}{2 \sqrt{f+c \log (f)}}\right )}{4 \sqrt{f+c \log (f)}}\\ \end{align*}
Mathematica [A] time = 1.52473, size = 252, normalized size = 1.57 \[ \frac{\sqrt{\pi } e^{-\frac{b^2 \log ^2(f)+e^2}{4 (c \log (f)+f)}} f^{a+\frac{b e f}{c^2 \log ^2(f)-f^2}} \left ((f-c \log (f)) \sqrt{c \log (f)+f} (\sinh (d)+\cosh (d)) f^{\frac{b e}{2 f-2 c \log (f)}} \text{Erfi}\left (\frac{\log (f) (b+2 c x)+e+2 f x}{2 \sqrt{c \log (f)+f}}\right )-\sqrt{f-c \log (f)} (c \log (f)+f) (\cosh (d)-\sinh (d)) f^{\frac{b e}{2 (c \log (f)+f)}} \exp \left (\frac{f \left (b^2 \log ^2(f)+e^2\right )}{2 \left (f^2-c^2 \log ^2(f)\right )}\right ) \text{Erf}\left (\frac{-\log (f) (b+2 c x)+e+2 f x}{2 \sqrt{f-c \log (f)}}\right )\right )}{4 \left (f^2-c^2 \log ^2(f)\right )} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.136, size = 186, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{\pi }{f}^{a}}{4}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+2\,\ln \left ( f \right ) be-4\,d\ln \left ( f \right ) c-4\,df+{e}^{2}}{4\,c\ln \left ( f \right ) +4\,f}}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -f}x+{\frac{e+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -f}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -f}}}}+{\frac{\sqrt{\pi }{f}^{a}}{4}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-2\,\ln \left ( f \right ) be+4\,d\ln \left ( f \right ) c-4\,df+{e}^{2}}{4\,c\ln \left ( f \right ) -4\,f}}}}{\it Erf} \left ( -x\sqrt{f-c\ln \left ( f \right ) }+{\frac{b\ln \left ( f \right ) -e}{2}{\frac{1}{\sqrt{f-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{f-c\ln \left ( f \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06733, size = 204, normalized size = 1.27 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - f} x - \frac{b \log \left (f\right ) + e}{2 \, \sqrt{-c \log \left (f\right ) - f}}\right ) e^{\left (-\frac{{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \,{\left (c \log \left (f\right ) + f\right )}} + d\right )}}{4 \, \sqrt{-c \log \left (f\right ) - f}} - \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + f} x - \frac{b \log \left (f\right ) - e}{2 \, \sqrt{-c \log \left (f\right ) + f}}\right ) e^{\left (-\frac{{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \,{\left (c \log \left (f\right ) - f\right )}} - d\right )}}{4 \, \sqrt{-c \log \left (f\right ) + f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89436, size = 973, normalized size = 6.04 \begin{align*} \frac{{\left (\sqrt{\pi }{\left (c \log \left (f\right ) + f\right )} \cosh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} - 4 \, d f + 2 \,{\left (2 \, c d - b e + 2 \, a f\right )} \log \left (f\right )}{4 \,{\left (c \log \left (f\right ) - f\right )}}\right ) + \sqrt{\pi }{\left (c \log \left (f\right ) + f\right )} \sinh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} - 4 \, d f + 2 \,{\left (2 \, c d - b e + 2 \, a f\right )} \log \left (f\right )}{4 \,{\left (c \log \left (f\right ) - f\right )}}\right )\right )} \sqrt{-c \log \left (f\right ) + f} \operatorname{erf}\left (-\frac{{\left (2 \, f x -{\left (2 \, c x + b\right )} \log \left (f\right ) + e\right )} \sqrt{-c \log \left (f\right ) + f}}{2 \,{\left (c \log \left (f\right ) - f\right )}}\right ) -{\left (\sqrt{\pi }{\left (c \log \left (f\right ) - f\right )} \cosh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} - 4 \, d f - 2 \,{\left (2 \, c d - b e + 2 \, a f\right )} \log \left (f\right )}{4 \,{\left (c \log \left (f\right ) + f\right )}}\right ) + \sqrt{\pi }{\left (c \log \left (f\right ) - f\right )} \sinh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} - 4 \, d f - 2 \,{\left (2 \, c d - b e + 2 \, a f\right )} \log \left (f\right )}{4 \,{\left (c \log \left (f\right ) + f\right )}}\right )\right )} \sqrt{-c \log \left (f\right ) - f} \operatorname{erf}\left (\frac{{\left (2 \, f x +{\left (2 \, c x + b\right )} \log \left (f\right ) + e\right )} \sqrt{-c \log \left (f\right ) - f}}{2 \,{\left (c \log \left (f\right ) + f\right )}}\right )}{4 \,{\left (c^{2} \log \left (f\right )^{2} - f^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x + c x^{2}} \sinh{\left (d + e x + f x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23494, size = 282, normalized size = 1.75 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) - f}{\left (2 \, x + \frac{b \log \left (f\right ) + e}{c \log \left (f\right ) + f}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - 4 \, a f \log \left (f\right ) + 2 \, b e \log \left (f\right ) - 4 \, d f + e^{2}}{4 \,{\left (c \log \left (f\right ) + f\right )}}\right )}}{4 \, \sqrt{-c \log \left (f\right ) - f}} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) + f}{\left (2 \, x + \frac{b \log \left (f\right ) - e}{c \log \left (f\right ) - f}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) + 4 \, a f \log \left (f\right ) - 2 \, b e \log \left (f\right ) - 4 \, d f + e^{2}}{4 \,{\left (c \log \left (f\right ) - f\right )}}\right )}}{4 \, \sqrt{-c \log \left (f\right ) + f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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