Optimal. Leaf size=219 \[ -\frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a e^{-\frac{(2 e-b \log (f))^2}{4 c \log (f)}-2 d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+2 e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{2 d-\frac{(b \log (f)+2 e)^2}{4 c \log (f)}} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+2 e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}} \]
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Rubi [A] time = 0.363978, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5512, 2234, 2204, 2287} \[ -\frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a e^{-\frac{(2 e-b \log (f))^2}{4 c \log (f)}-2 d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+2 e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{2 d-\frac{(b \log (f)+2 e)^2}{4 c \log (f)}} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+2 e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Rule 5512
Rule 2234
Rule 2204
Rule 2287
Rubi steps
\begin{align*} \int f^{a+b x+c x^2} \sinh ^2(d+e x) \, dx &=\int \left (-\frac{1}{2} f^{a+b x+c x^2}+\frac{1}{4} e^{-2 d-2 e x} f^{a+b x+c x^2}+\frac{1}{4} e^{2 d+2 e x} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac{1}{4} \int e^{-2 d-2 e x} f^{a+b x+c x^2} \, dx+\frac{1}{4} \int e^{2 d+2 e x} f^{a+b x+c x^2} \, dx-\frac{1}{2} \int f^{a+b x+c x^2} \, dx\\ &=\frac{1}{4} \int \exp \left (-2 d+a \log (f)+c x^2 \log (f)-x (2 e-b \log (f))\right ) \, dx+\frac{1}{4} \int \exp \left (2 d+a \log (f)+c x^2 \log (f)+x (2 e+b \log (f))\right ) \, dx-\frac{1}{2} f^{a-\frac{b^2}{4 c}} \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx\\ &=-\frac{f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}+\frac{1}{4} \left (e^{-2 d-\frac{(2 e-b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(-2 e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac{1}{4} \left (e^{2 d-\frac{(2 e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(2 e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx\\ &=-\frac{f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}-\frac{e^{-2 d-\frac{(2 e-b \log (f))^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{2 e-b \log (f)-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}+\frac{e^{2 d-\frac{(2 e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{2 e+b \log (f)+2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}\\ \end{align*}
Mathematica [A] time = 0.614931, size = 183, normalized size = 0.84 \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} e^{-\frac{e (b \log (f)+e)}{c \log (f)}} \left (e^{\frac{2 b e}{c}} (\cosh (2 d)-\sinh (2 d)) \text{Erfi}\left (\frac{\log (f) (b+2 c x)-2 e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+(\sinh (2 d)+\cosh (2 d)) \text{Erfi}\left (\frac{\log (f) (b+2 c x)+2 e}{2 \sqrt{c} \sqrt{\log (f)}}\right )-2 e^{\frac{e (b \log (f)+e)}{c \log (f)}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )\right )}{8 \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 211, normalized size = 1. \begin{align*} -{\frac{\sqrt{\pi }{f}^{a}}{8}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-4\,\ln \left ( f \right ) be+8\,d\ln \left ( f \right ) c+4\,{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) -2\,e}{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{\sqrt{\pi }{f}^{a}}{8}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+4\,\ln \left ( f \right ) be-8\,d\ln \left ( f \right ) c+4\,{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{2\,e+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{\frac{\sqrt{\pi }{f}^{a}}{4}{f}^{-{\frac{{b}^{2}}{4\,c}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06067, size = 250, normalized size = 1.14 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x - \frac{b \log \left (f\right ) + 2 \, e}{2 \, \sqrt{-c \log \left (f\right )}}\right ) e^{\left (2 \, d - \frac{{\left (b \log \left (f\right ) + 2 \, e\right )}^{2}}{4 \, c \log \left (f\right )}\right )}}{8 \, \sqrt{-c \log \left (f\right )}} + \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x - \frac{b \log \left (f\right ) - 2 \, e}{2 \, \sqrt{-c \log \left (f\right )}}\right ) e^{\left (-2 \, d - \frac{{\left (b \log \left (f\right ) - 2 \, e\right )}^{2}}{4 \, c \log \left (f\right )}\right )}}{8 \, \sqrt{-c \log \left (f\right )}} - \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x - \frac{b \log \left (f\right )}{2 \, \sqrt{-c \log \left (f\right )}}\right )}{4 \, \sqrt{-c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97486, size = 933, normalized size = 4.26 \begin{align*} \frac{2 \, \sqrt{-c \log \left (f\right )}{\left (\sqrt{\pi } \cosh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )}{4 \, c}\right ) + \sqrt{\pi } \sinh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )}{4 \, c}\right )\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )}}{2 \, c}\right ) - \sqrt{-c \log \left (f\right )}{\left (\sqrt{\pi } \cosh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 4 \, e^{2} - 4 \,{\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right ) + \sqrt{\pi } \sinh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 4 \, e^{2} - 4 \,{\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + 2 \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) - \sqrt{-c \log \left (f\right )}{\left (\sqrt{\pi } \cosh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 4 \, e^{2} + 4 \,{\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right ) + \sqrt{\pi } \sinh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 4 \, e^{2} + 4 \,{\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - 2 \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right )}{8 \, c \log \left (f\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 ^{2}{\left (d + e x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28009, size = 304, normalized size = 1.39 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{4 \, \sqrt{-c \log \left (f\right )}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b \log \left (f\right ) - 2 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 8 \, c d \log \left (f\right ) - 4 \, b e \log \left (f\right ) + 4 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{8 \, \sqrt{-c \log \left (f\right )}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b \log \left (f\right ) + 2 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 8 \, c d \log \left (f\right ) + 4 \, b e \log \left (f\right ) + 4 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{8 \, \sqrt{-c \log \left (f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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