Optimal. Leaf size=268 \[ \frac{\sqrt{\frac{\pi }{2}} e^{\frac{b^2}{2 c}-2 a} \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} b^2 e^{\frac{b^2}{2 c}-2 a} \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}-\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} b^2 e^{2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}-\frac{b \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{x \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac{x^3}{6} \]
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Rubi [A] time = 0.244567, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {5394, 5387, 5374, 2234, 2204, 2205, 5383, 5375} \[ \frac{\sqrt{\frac{\pi }{2}} e^{\frac{b^2}{2 c}-2 a} \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} b^2 e^{\frac{b^2}{2 c}-2 a} \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}-\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} b^2 e^{2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}-\frac{b \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{x \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac{x^3}{6} \]
Antiderivative was successfully verified.
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Rule 5394
Rule 5387
Rule 5374
Rule 2234
Rule 2204
Rule 2205
Rule 5383
Rule 5375
Rubi steps
\begin{align*} \int x^2 \sinh ^2\left (a+b x+c x^2\right ) \, dx &=\int \left (-\frac{x^2}{2}+\frac{1}{2} x^2 \cosh \left (2 a+2 b x+2 c x^2\right )\right ) \, dx\\ &=-\frac{x^3}{6}+\frac{1}{2} \int x^2 \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx\\ &=-\frac{x^3}{6}+\frac{x \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac{\int \sinh \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c}-\frac{b \int x \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{4 c}\\ &=-\frac{x^3}{6}-\frac{b \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{x \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{b^2 \int \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c^2}+\frac{\int e^{-2 a-2 b x-2 c x^2} \, dx}{16 c}-\frac{\int e^{2 a+2 b x+2 c x^2} \, dx}{16 c}\\ &=-\frac{x^3}{6}-\frac{b \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{x \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{b^2 \int e^{-2 a-2 b x-2 c x^2} \, dx}{16 c^2}+\frac{b^2 \int e^{2 a+2 b x+2 c x^2} \, dx}{16 c^2}-\frac{e^{2 a-\frac{b^2}{2 c}} \int e^{\frac{(2 b+4 c x)^2}{8 c}} \, dx}{16 c}+\frac{e^{-2 a+\frac{b^2}{2 c}} \int e^{-\frac{(-2 b-4 c x)^2}{8 c}} \, dx}{16 c}\\ &=-\frac{x^3}{6}+\frac{e^{-2 a+\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{e^{2 a-\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{b \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{x \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{\left (b^2 e^{2 a-\frac{b^2}{2 c}}\right ) \int e^{\frac{(2 b+4 c x)^2}{8 c}} \, dx}{16 c^2}+\frac{\left (b^2 e^{-2 a+\frac{b^2}{2 c}}\right ) \int e^{-\frac{(-2 b-4 c x)^2}{8 c}} \, dx}{16 c^2}\\ &=-\frac{x^3}{6}+\frac{b^2 e^{-2 a+\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{e^{-2 a+\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}+\frac{b^2 e^{2 a-\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}-\frac{e^{2 a-\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{b \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{x \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}\\ \end{align*}
Mathematica [A] time = 0.743297, size = 176, normalized size = 0.66 \[ \frac{3 \sqrt{2 \pi } \left (b^2+c\right ) \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\cosh \left (2 a-\frac{b^2}{2 c}\right )-\sinh \left (2 a-\frac{b^2}{2 c}\right )\right )+3 \sqrt{2 \pi } \left (b^2-c\right ) \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\sinh \left (2 a-\frac{b^2}{2 c}\right )+\cosh \left (2 a-\frac{b^2}{2 c}\right )\right )-4 \sqrt{c} \left (3 (b-2 c x) \sinh (2 (a+x (b+c x)))+8 c^2 x^3\right )}{192 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.11, size = 281, normalized size = 1.1 \begin{align*} -{\frac{{x}^{3}}{6}}-{\frac{x{{\rm e}^{-2\,c{x}^{2}-2\,bx-2\,a}}}{16\,c}}+{\frac{b{{\rm e}^{-2\,c{x}^{2}-2\,bx-2\,a}}}{32\,{c}^{2}}}+{\frac{{b}^{2}\sqrt{\pi }\sqrt{2}}{64}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( \sqrt{2}\sqrt{c}x+{\frac{b\sqrt{2}}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{5}{2}}}}+{\frac{\sqrt{\pi }\sqrt{2}}{64}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( \sqrt{2}\sqrt{c}x+{\frac{b\sqrt{2}}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}+{\frac{x{{\rm e}^{2\,c{x}^{2}+2\,bx+2\,a}}}{16\,c}}-{\frac{b{{\rm e}^{2\,c{x}^{2}+2\,bx+2\,a}}}{32\,{c}^{2}}}-{\frac{{b}^{2}\sqrt{\pi }}{32\,{c}^{2}}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( -\sqrt{-2\,c}x+{b{\frac{1}{\sqrt{-2\,c}}}} \right ){\frac{1}{\sqrt{-2\,c}}}}+{\frac{\sqrt{\pi }}{32\,c}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( -\sqrt{-2\,c}x+{b{\frac{1}{\sqrt{-2\,c}}}} \right ){\frac{1}{\sqrt{-2\,c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.33611, size = 397, normalized size = 1.48 \begin{align*} -\frac{1}{6} \, x^{3} + \frac{\sqrt{2}{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b^{2}{\left (\operatorname{erf}\left (\sqrt{\frac{1}{2}} \sqrt{-\frac{{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt{-\frac{{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac{5}{2}}} - \frac{2 \, \sqrt{2} b e^{\left (\frac{{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{c^{\frac{3}{2}}} - \frac{2 \,{\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac{3}{2}, -\frac{{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}{\left (-\frac{{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac{3}{2}} c^{\frac{5}{2}}}\right )} e^{\left (2 \, a - \frac{b^{2}}{2 \, c}\right )}}{64 \, \sqrt{c}} - \frac{\sqrt{2}{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b^{2}{\left (\operatorname{erf}\left (\sqrt{\frac{1}{2}} \sqrt{\frac{{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt{\frac{{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac{5}{2}}} + \frac{2 \, \sqrt{2} b c e^{\left (-\frac{{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac{5}{2}}} - \frac{2 \,{\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac{3}{2}, \frac{{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}{\left (\frac{{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac{3}{2}} \left (-c\right )^{\frac{5}{2}}}\right )} e^{\left (-2 \, a + \frac{b^{2}}{2 \, c}\right )}}{64 \, \sqrt{-c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.17034, size = 1905, normalized size = 7.11 \begin{align*} \text{result too large to display} \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 x^{2} \sinh ^{2}{\left (a + b x + c x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.4333, size = 246, normalized size = 0.92 \begin{align*} -\frac{1}{6} \, x^{3} - \frac{\frac{\sqrt{2} \sqrt{\pi }{\left (b^{2} + c\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (\frac{b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt{c}} + 2 \,{\left (c{\left (2 \, x + \frac{b}{c}\right )} - 2 \, b\right )} e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{64 \, c^{2}} - \frac{\frac{\sqrt{2} \sqrt{\pi }{\left (b^{2} - c\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{-c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt{-c}} - 2 \,{\left (c{\left (2 \, x + \frac{b}{c}\right )} - 2 \, b\right )} e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{64 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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