Integrand size = 18, antiderivative size = 239 \[ \int f^{a+b x} \sinh ^3\left (d+f x^2\right ) \, dx=\frac {3}{16} e^{-d+\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right )-\frac {1}{16} e^{-3 d+\frac {b^2 \log ^2(f)}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )-\frac {3}{16} e^{d-\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {2 f x+b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{16} e^{3 d-\frac {b^2 \log ^2(f)}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5623, 2325, 2266, 2236, 2235} \[ \int f^{a+b x} \sinh ^3\left (d+f x^2\right ) \, dx=\frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{4 f}-d} \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right )-\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{12 f}-3 d} \text {erf}\left (\frac {6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )-\frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {b^2 \log ^2(f)}{4 f}} \text {erfi}\left (\frac {b \log (f)+2 f x}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{3 d-\frac {b^2 \log ^2(f)}{12 f}} \text {erfi}\left (\frac {b \log (f)+6 f x}{2 \sqrt {3} \sqrt {f}}\right ) \]
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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 d-3 f x^2} f^{a+b x}+\frac {3}{8} e^{-d-f x^2} f^{a+b x}-\frac {3}{8} e^{d+f x^2} f^{a+b x}+\frac {1}{8} e^{3 d+3 f x^2} f^{a+b x}\right ) \, dx \\ & = -\left (\frac {1}{8} \int e^{-3 d-3 f x^2} f^{a+b x} \, dx\right )+\frac {1}{8} \int e^{3 d+3 f x^2} f^{a+b x} \, dx+\frac {3}{8} \int e^{-d-f x^2} f^{a+b x} \, dx-\frac {3}{8} \int e^{d+f x^2} f^{a+b x} \, dx \\ & = -\left (\frac {1}{8} \int e^{-3 d-3 f x^2+a \log (f)+b x \log (f)} \, dx\right )+\frac {1}{8} \int e^{3 d+3 f x^2+a \log (f)+b x \log (f)} \, dx+\frac {3}{8} \int e^{-d-f x^2+a \log (f)+b x \log (f)} \, dx-\frac {3}{8} \int e^{d+f x^2+a \log (f)+b x \log (f)} \, dx \\ & = -\left (\frac {1}{8} \left (3 e^{d-\frac {b^2 \log ^2(f)}{4 f}} f^a\right ) \int e^{\frac {(2 f x+b \log (f))^2}{4 f}} \, dx\right )+\frac {1}{8} \left (e^{3 d-\frac {b^2 \log ^2(f)}{12 f}} f^a\right ) \int e^{\frac {(6 f x+b \log (f))^2}{12 f}} \, dx-\frac {1}{8} \left (e^{-3 d+\frac {b^2 \log ^2(f)}{12 f}} f^a\right ) \int e^{-\frac {(-6 f x+b \log (f))^2}{12 f}} \, dx+\frac {1}{8} \left (3 e^{-d+\frac {b^2 \log ^2(f)}{4 f}} f^a\right ) \int e^{-\frac {(-2 f x+b \log (f))^2}{4 f}} \, dx \\ & = \frac {3}{16} e^{-d+\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right )-\frac {1}{16} e^{-3 d+\frac {b^2 \log ^2(f)}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )-\frac {3}{16} e^{d-\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {2 f x+b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{16} e^{3 d-\frac {b^2 \log ^2(f)}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right ) \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.20 \[ \int f^{a+b x} \sinh ^3\left (d+f x^2\right ) \, dx=\frac {1}{16} e^{-\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \left (-3 \sqrt {3} \cosh (d) \text {erfi}\left (\frac {2 f x+b \log (f)}{2 \sqrt {f}}\right )+e^{\frac {b^2 \log ^2(f)}{6 f}} \cosh (3 d) \text {erfi}\left (\frac {6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )+3 \sqrt {3} e^{\frac {b^2 \log ^2(f)}{2 f}} \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right ) (\cosh (d)-\sinh (d))-3 \sqrt {3} \text {erfi}\left (\frac {2 f x+b \log (f)}{2 \sqrt {f}}\right ) \sinh (d)-e^{\frac {b^2 \log ^2(f)}{3 f}} \text {erf}\left (\frac {6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right ) (\cosh (3 d)-\sinh (3 d))+e^{\frac {b^2 \log ^2(f)}{6 f}} \text {erfi}\left (\frac {6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right ) \sinh (3 d)\right ) \]
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Time = 1.54 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {\operatorname {erf}\left (-\sqrt {-3 f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-3 f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-36 d f}{12 f}}}{16 \sqrt {-3 f}}+\frac {\operatorname {erf}\left (-\sqrt {3}\, \sqrt {f}\, x +\frac {\ln \left (f \right ) b \sqrt {3}}{6 \sqrt {f}}\right ) \sqrt {3}\, \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {b^{2} \ln \left (f \right )^{2}-36 d f}{12 f}}}{48 \sqrt {f}}-\frac {3 \,\operatorname {erf}\left (-\sqrt {f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {b^{2} \ln \left (f \right )^{2}-4 d f}{4 f}}}{16 \sqrt {f}}+\frac {3 \,\operatorname {erf}\left (-\sqrt {-f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-4 d f}{4 f}}}{16 \sqrt {-f}}\) | \(207\) |
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Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (181) = 362\).
Time = 0.32 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.86 \[ \int f^{a+b x} \sinh ^3\left (d+f x^2\right ) \, dx=-\frac {\sqrt {3} \sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} - 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right ) \operatorname {erf}\left (\frac {\sqrt {3} {\left (6 \, f x + b \log \left (f\right )\right )} \sqrt {-f}}{6 \, f}\right ) - \sqrt {3} \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right ) \operatorname {erf}\left (-\frac {\sqrt {3} {\left (6 \, f x - b \log \left (f\right )\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 )\right )}}{6 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right ) - \sqrt {3} \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {\sqrt {3} {\left (6 \, f x + b \log \left (f\right )\right )} \sqrt {-f}}{6 \, f}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} - 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right ) - 9 \, \sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} - 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right ) \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \left (f\right )\right )} \sqrt {-f}}{2 \, f}\right ) + 9 \, \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right ) \operatorname {erf}\left (-\frac {2 \, f x - b \log \left (f\right )}{2 \, \sqrt {f}}\right ) + 9 \, \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {2 \, f x - b \log \left (f\right )}{2 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right ) + 9 \, \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \left (f\right )\right )} \sqrt {-f}}{2 \, f}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} - 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right )}{48 \, f} \]
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\[ \int f^{a+b x} \sinh ^3\left (d+f x^2\right ) \, dx=\int f^{a + b x} \sinh ^{3}{\left (d + f x^{2} \right )}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.84 \[ \int f^{a+b x} \sinh ^3\left (d+f x^2\right ) \, dx=\frac {3}{16} \, \sqrt {\pi } f^{a - \frac {1}{2}} \operatorname {erf}\left (\sqrt {f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {f}}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2}}{4 \, f} - d\right )} - \frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {f} x - \frac {\sqrt {3} b \log \left (f\right )}{6 \, \sqrt {f}}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2}}{12 \, f} - 3 \, d\right )}}{48 \, \sqrt {f}} + \frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {-f} x - \frac {\sqrt {3} b \log \left (f\right )}{6 \, \sqrt {-f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{12 \, f} + 3 \, d\right )}}{48 \, \sqrt {-f}} - \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, f} + d\right )}}{16 \, \sqrt {-f}} \]
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Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.93 \[ \int f^{a+b x} \sinh ^3\left (d+f x^2\right ) \, dx=\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {f} {\left (6 \, x - \frac {b \log \left (f\right )}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2} + 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right )}}{48 \, \sqrt {f}} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {-f} {\left (6 \, x + \frac {b \log \left (f\right )}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right )}}{48 \, \sqrt {-f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {f} {\left (2 \, x - \frac {b \log \left (f\right )}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right )}}{16 \, \sqrt {f}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-f} {\left (2 \, x + \frac {b \log \left (f\right )}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right )}}{16 \, \sqrt {-f}} \]
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Timed out. \[ \int f^{a+b x} \sinh ^3\left (d+f x^2\right ) \, dx=\int f^{a+b\,x}\,{\mathrm {sinh}\left (f\,x^2+d\right )}^3 \,d x \]
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