Integrand size = 21, antiderivative size = 257 \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=\frac {3}{16} e^{-d+\frac {(e-b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {e+2 f x-b \log (f)}{2 \sqrt {f}}\right )-\frac {1}{16} e^{-3 d+\frac {(3 e-b \log (f))^2}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {3 e+6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )-\frac {3}{16} e^{d-\frac {(e+b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{16} e^{3 d-\frac {(3 e+b \log (f))^2}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {3 e+6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right ) \]
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Time = 0.33 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5623, 2325, 2266, 2236, 2235} \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=\frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {(e-b \log (f))^2}{4 f}-d} \text {erf}\left (\frac {-b \log (f)+e+2 f x}{2 \sqrt {f}}\right )-\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{\frac {(3 e-b \log (f))^2}{12 f}-3 d} \text {erf}\left (\frac {-b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )-\frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {(b \log (f)+e)^2}{4 f}} \text {erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{3 d-\frac {(b \log (f)+3 e)^2}{12 f}} \text {erfi}\left (\frac {b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right ) \]
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Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 5623
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{8} e^{-3 \left (d+e x+f x^2\right )} f^{a+b x}+\frac {3}{8} \exp \left (2 d+2 e x+2 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x}-\frac {3}{8} \exp \left (4 d+4 e x+4 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x}+\frac {1}{8} \exp \left (6 d+6 e x+6 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x}\right ) \, dx \\ & = -\left (\frac {1}{8} \int e^{-3 \left (d+e x+f x^2\right )} f^{a+b x} \, dx\right )+\frac {1}{8} \int \exp \left (6 d+6 e x+6 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx+\frac {3}{8} \int \exp \left (2 d+2 e x+2 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx-\frac {3}{8} \int \exp \left (4 d+4 e x+4 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx \\ & = -\left (\frac {1}{8} \int \exp \left (-3 d-3 f x^2+a \log (f)-x (3 e-b \log (f))\right ) \, dx\right )+\frac {1}{8} \int \exp \left (3 d+3 f x^2+a \log (f)+x (3 e+b \log (f))\right ) \, dx+\frac {3}{8} \int e^{-d-f x^2+a \log (f)-x (e-b \log (f))} \, dx-\frac {3}{8} \int e^{d+f x^2+a \log (f)+x (e+b \log (f))} \, dx \\ & = \frac {1}{8} \left (3 e^{-d+\frac {(e-b \log (f))^2}{4 f}} f^a\right ) \int e^{-\frac {(-e-2 f x+b \log (f))^2}{4 f}} \, dx-\frac {1}{8} \left (e^{-3 d+\frac {(3 e-b \log (f))^2}{12 f}} f^a\right ) \int e^{-\frac {(-3 e-6 f x+b \log (f))^2}{12 f}} \, dx-\frac {1}{8} \left (3 e^{d-\frac {(e+b \log (f))^2}{4 f}} f^a\right ) \int e^{\frac {(e+2 f x+b \log (f))^2}{4 f}} \, dx+\frac {1}{8} \left (e^{3 d-\frac {(3 e+b \log (f))^2}{12 f}} f^a\right ) \int e^{\frac {(3 e+6 f x+b \log (f))^2}{12 f}} \, dx \\ & = \frac {3}{16} e^{-d+\frac {(e-b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {e+2 f x-b \log (f)}{2 \sqrt {f}}\right )-\frac {1}{16} e^{-3 d+\frac {(3 e-b \log (f))^2}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {3 e+6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )-\frac {3}{16} e^{d-\frac {(e+b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{16} e^{3 d-\frac {(3 e+b \log (f))^2}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {3 e+6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right ) \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.38 \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=\frac {1}{16} e^{-\frac {3 e^2+b^2 \log ^2(f)}{4 f}} f^{a-\frac {b e+f}{2 f}} \sqrt {\frac {\pi }{3}} \left (-3 \sqrt {3} e^{\frac {e^2}{2 f}} \cosh (d) \text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right )+e^{\frac {b^2 \log ^2(f)}{6 f}} \cosh (3 d) \text {erfi}\left (\frac {3 e+6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )+3 \sqrt {3} e^{\frac {2 e^2+b^2 \log ^2(f)}{2 f}} \text {erf}\left (\frac {e+2 f x-b \log (f)}{2 \sqrt {f}}\right ) (\cosh (d)-\sinh (d))-3 \sqrt {3} e^{\frac {e^2}{2 f}} \text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right ) \sinh (d)-e^{\frac {9 e^2+2 b^2 \log ^2(f)}{6 f}} \text {erf}\left (\frac {3 e+6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right ) (\cosh (3 d)-\sinh (3 d))+e^{\frac {b^2 \log ^2(f)}{6 f}} \text {erfi}\left (\frac {3 e+6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right ) \sinh (3 d)\right ) \]
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Time = 1.38 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.03
method | result | size |
risch | \(-\frac {\operatorname {erf}\left (-\sqrt {-3 f}\, x +\frac {3 e +b \ln \left (f \right )}{2 \sqrt {-3 f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+6 \ln \left (f \right ) b e -36 d f +9 e^{2}}{12 f}}}{16 \sqrt {-3 f}}+\frac {\operatorname {erf}\left (-\sqrt {3}\, \sqrt {f}\, x +\frac {\left (b \ln \left (f \right )-3 e \right ) \sqrt {3}}{6 \sqrt {f}}\right ) \sqrt {3}\, \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {b^{2} \ln \left (f \right )^{2}-6 \ln \left (f \right ) b e -36 d f +9 e^{2}}{12 f}}}{48 \sqrt {f}}-\frac {3 \,\operatorname {erf}\left (-\sqrt {f}\, x +\frac {b \ln \left (f \right )-e}{2 \sqrt {f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {b^{2} \ln \left (f \right )^{2}-2 \ln \left (f \right ) b e -4 d f +e^{2}}{4 f}}}{16 \sqrt {f}}+\frac {3 \,\operatorname {erf}\left (-\sqrt {-f}\, x +\frac {e +b \ln \left (f \right )}{2 \sqrt {-f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+2 \ln \left (f \right ) b e -4 d f +e^{2}}{4 f}}}{16 \sqrt {-f}}\) | \(265\) |
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Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (199) = 398\).
Time = 0.30 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.11 \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {3} \sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + 9 \, e^{2} - 36 \, d f + 6 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{12 \, f}\right ) \operatorname {erf}\left (\frac {\sqrt {3} {\left (6 \, f x + b \log \left (f\right ) + 3 \, e\right )} \sqrt {-f}}{6 \, f}\right ) - \sqrt {3} \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + 9 \, e^{2} - 36 \, d f - 6 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{12 \, f}\right ) \operatorname {erf}\left (-\frac {\sqrt {3} {\left (6 \, f x - b \log \left (f\right ) + 3 \, e\right )}}{6 \, \sqrt {f}}\right ) - \sqrt {3} \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {\sqrt {3} {\left (6 \, f x + b \log \left (f\right ) + 3 \, e\right )} \sqrt {-f}}{6 \, f}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + 9 \, e^{2} - 36 \, d f + 6 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{12 \, f}\right ) - \sqrt {3} \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {\sqrt {3} {\left (6 \, f x - b \log \left (f\right ) + 3 \, e\right )}}{6 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + 9 \, e^{2} - 36 \, d f - 6 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{12 \, f}\right ) - 9 \, \sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f + 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \left (f\right ) + e\right )} \sqrt {-f}}{2 \, f}\right ) + 9 \, \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) \operatorname {erf}\left (-\frac {2 \, f x - b \log \left (f\right ) + e}{2 \, \sqrt {f}}\right ) + 9 \, \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \left (f\right ) + e\right )} \sqrt {-f}}{2 \, f}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f + 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) + 9 \, \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {2 \, f x - b \log \left (f\right ) + e}{2 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right )}{48 \, f} \]
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\[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=\int f^{a + b x} \sinh ^{3}{\left (d + e x + f x^{2} \right )}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.89 \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {-f} x - \frac {\sqrt {3} {\left (b \log \left (f\right ) + 3 \, e\right )}}{6 \, \sqrt {-f}}\right ) e^{\left (3 \, d - \frac {{\left (b \log \left (f\right ) + 3 \, e\right )}^{2}}{12 \, f}\right )}}{48 \, \sqrt {-f}} + \frac {3}{16} \, \sqrt {\pi } f^{a - \frac {1}{2}} \operatorname {erf}\left (\sqrt {f} x - \frac {b \log \left (f\right ) - e}{2 \, \sqrt {f}}\right ) e^{\left (-d + \frac {{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, f}\right )} - \frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {f} x - \frac {\sqrt {3} {\left (b \log \left (f\right ) - 3 \, e\right )}}{6 \, \sqrt {f}}\right ) e^{\left (-3 \, d + \frac {{\left (b \log \left (f\right ) - 3 \, e\right )}^{2}}{12 \, f}\right )}}{48 \, \sqrt {f}} - \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-f} x - \frac {b \log \left (f\right ) + e}{2 \, \sqrt {-f}}\right ) e^{\left (d - \frac {{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \, f}\right )}}{16 \, \sqrt {-f}} \]
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Time = 0.30 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.09 \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {-f} {\left (6 \, x + \frac {b \log \left (f\right ) + 3 \, e}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} + 6 \, b e \log \left (f\right ) - 12 \, a f \log \left (f\right ) + 9 \, e^{2} - 36 \, d f}{12 \, f}\right )}}{48 \, \sqrt {-f}} + \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {f} {\left (6 \, x - \frac {b \log \left (f\right ) - 3 \, e}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2} - 6 \, b e \log \left (f\right ) + 12 \, a f \log \left (f\right ) + 9 \, e^{2} - 36 \, d f}{12 \, f}\right )}}{48 \, \sqrt {f}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-f} {\left (2 \, x + \frac {b \log \left (f\right ) + e}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} + 2 \, b e \log \left (f\right ) - 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, f}\right )}}{16 \, \sqrt {-f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {f} {\left (2 \, x - \frac {b \log \left (f\right ) - e}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2} - 2 \, b e \log \left (f\right ) + 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, f}\right )}}{16 \, \sqrt {f}} \]
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Timed out. \[ \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx=\int f^{a+b\,x}\,{\mathrm {sinh}\left (f\,x^2+e\,x+d\right )}^3 \,d x \]
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