Optimal. Leaf size=271 \[ -\frac{3 \sqrt{\pi } f^a e^{-\frac{e^2}{4 c \log (f)}-d} \text{Erfi}\left (\frac{e-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{-\frac{9 e^2}{4 c \log (f)}-3 d} \text{Erfi}\left (\frac{3 e-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{3 \sqrt{\pi } f^a e^{d-\frac{e^2}{4 c \log (f)}} \text{Erfi}\left (\frac{2 c x \log (f)+e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{3 d-\frac{9 e^2}{4 c \log (f)}} \text{Erfi}\left (\frac{2 c x \log (f)+3 e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.345477, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5512, 2287, 2234, 2204} \[ -\frac{3 \sqrt{\pi } f^a e^{-\frac{e^2}{4 c \log (f)}-d} \text{Erfi}\left (\frac{e-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{-\frac{9 e^2}{4 c \log (f)}-3 d} \text{Erfi}\left (\frac{3 e-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{3 \sqrt{\pi } f^a e^{d-\frac{e^2}{4 c \log (f)}} \text{Erfi}\left (\frac{2 c x \log (f)+e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{3 d-\frac{9 e^2}{4 c \log (f)}} \text{Erfi}\left (\frac{2 c x \log (f)+3 e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5512
Rule 2287
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int f^{a+c x^2} \sinh ^3(d+e x) \, dx &=\int \left (-\frac{1}{8} e^{-3 d-3 e x} f^{a+c x^2}+\frac{3}{8} e^{-d-e x} f^{a+c x^2}-\frac{3}{8} e^{d+e x} f^{a+c x^2}+\frac{1}{8} e^{3 d+3 e x} f^{a+c x^2}\right ) \, dx\\ &=-\left (\frac{1}{8} \int e^{-3 d-3 e x} f^{a+c x^2} \, dx\right )+\frac{1}{8} \int e^{3 d+3 e x} f^{a+c x^2} \, dx+\frac{3}{8} \int e^{-d-e x} f^{a+c x^2} \, dx-\frac{3}{8} \int e^{d+e x} f^{a+c x^2} \, dx\\ &=-\left (\frac{1}{8} \int e^{-3 d-3 e x+a \log (f)+c x^2 \log (f)} \, dx\right )+\frac{1}{8} \int e^{3 d+3 e x+a \log (f)+c x^2 \log (f)} \, dx+\frac{3}{8} \int e^{-d-e x+a \log (f)+c x^2 \log (f)} \, dx-\frac{3}{8} \int e^{d+e x+a \log (f)+c x^2 \log (f)} \, dx\\ &=-\left (\frac{1}{8} \left (e^{-3 d-\frac{9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac{(-3 e+2 c x \log (f))^2}{4 c \log (f)}} \, dx\right )+\frac{1}{8} \left (e^{3 d-\frac{9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac{(3 e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac{1}{8} \left (3 e^{-d-\frac{e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac{(-e+2 c x \log (f))^2}{4 c \log (f)}} \, dx-\frac{1}{8} \left (3 e^{d-\frac{e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac{(e+2 c x \log (f))^2}{4 c \log (f)}} \, dx\\ &=-\frac{3 e^{-d-\frac{e^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{e-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{e^{-3 d-\frac{9 e^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{3 e-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{3 e^{d-\frac{e^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{e+2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{e^{3 d-\frac{9 e^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{3 e+2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}\\ \end{align*}
Mathematica [A] time = 0.45102, size = 214, normalized size = 0.79 \[ \frac{\sqrt{\pi } f^a e^{-\frac{9 e^2}{4 c \log (f)}} \left ((\sinh (d)+\cosh (d)) \left (3 (\cosh (2 d)-\sinh (2 d)) e^{\frac{2 e^2}{c \log (f)}} \text{Erfi}\left (\frac{2 c x \log (f)-e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+(\sinh (2 d)+\cosh (2 d)) \text{Erfi}\left (\frac{2 c x \log (f)+3 e}{2 \sqrt{c} \sqrt{\log (f)}}\right )-3 e^{\frac{2 e^2}{c \log (f)}} \text{Erfi}\left (\frac{2 c x \log (f)+e}{2 \sqrt{c} \sqrt{\log (f)}}\right )\right )+(\sinh (3 d)-\cosh (3 d)) \text{Erfi}\left (\frac{2 c x \log (f)-3 e}{2 \sqrt{c} \sqrt{\log (f)}}\right )\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.146, size = 234, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{\pi }{f}^{a}}{16}{{\rm e}^{{\frac{12\,d\ln \left ( f \right ) c-9\,{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{3\,e}{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{\sqrt{\pi }{f}^{a}}{16}{{\rm e}^{-{\frac{12\,d\ln \left ( f \right ) c+9\,{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) }x+{\frac{3\,e}{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{\frac{3\,\sqrt{\pi }{f}^{a}}{16}{{\rm e}^{-{\frac{4\,d\ln \left ( f \right ) c+{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) }x+{\frac{e}{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{\frac{3\,\sqrt{\pi }{f}^{a}}{16}{{\rm e}^{{\frac{4\,d\ln \left ( f \right ) c-{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{e}{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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.08158, size = 285, normalized size = 1.05 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x - \frac{3 \, e}{2 \, \sqrt{-c \log \left (f\right )}}\right ) e^{\left (3 \, d - \frac{9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt{-c \log \left (f\right )}} - \frac{3 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x - \frac{e}{2 \, \sqrt{-c \log \left (f\right )}}\right ) e^{\left (d - \frac{e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt{-c \log \left (f\right )}} + \frac{3 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x + \frac{e}{2 \, \sqrt{-c \log \left (f\right )}}\right ) e^{\left (-d - \frac{e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt{-c \log \left (f\right )}} - \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x + \frac{3 \, e}{2 \, \sqrt{-c \log \left (f\right )}}\right ) e^{\left (-3 \, d - \frac{9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt{-c \log \left (f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.83901, size = 1202, normalized size = 4.44 \begin{align*} -\frac{\sqrt{-c \log \left (f\right )}{\left (\sqrt{\pi } \cosh \left (\frac{4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt{\pi } \sinh \left (\frac{4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x \log \left (f\right ) + 3 \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) - 3 \, \sqrt{-c \log \left (f\right )}{\left (\sqrt{\pi } \cosh \left (\frac{4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt{\pi } \sinh \left (\frac{4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x \log \left (f\right ) + e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) + 3 \, \sqrt{-c \log \left (f\right )}{\left (\sqrt{\pi } \cosh \left (\frac{4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt{\pi } \sinh \left (\frac{4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x \log \left (f\right ) - 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{4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt{\pi } \sinh \left (\frac{4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x \log \left (f\right ) - 3 \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right )}{16 \, c \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + c x^{2}} \sinh ^{3}{\left (d + e x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21843, size = 356, normalized size = 1.31 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{3 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt{-c \log \left (f\right )}} + \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt{-c \log \left (f\right )}} - \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x - \frac{e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \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{3 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt{-c \log \left (f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]