Integrand size = 23, antiderivative size = 323 \[ \int f^{a+b x+c x^2} \sinh ^3\left (d+f x^2\right ) \, dx=-\frac {3 e^{-d+\frac {b^2 \log ^2(f)}{4 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{16 \sqrt {f-c \log (f)}}+\frac {e^{-3 d+\frac {b^2 \log ^2(f)}{12 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (3 f-c \log (f))}{2 \sqrt {3 f-c \log (f)}}\right )}{16 \sqrt {3 f-c \log (f)}}-\frac {3 e^{d-\frac {b^2 \log ^2(f)}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{16 \sqrt {f+c \log (f)}}+\frac {e^{3 d-\frac {b^2 \log ^2(f)}{4 (3 f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (3 f+c \log (f))}{2 \sqrt {3 f+c \log (f)}}\right )}{16 \sqrt {3 f+c \log (f)}} \]
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Time = 0.40 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5623, 2325, 2266, 2236, 2235} \[ \int f^{a+b x+c x^2} \sinh ^3\left (d+f x^2\right ) \, dx=-\frac {3 \sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{4 f-4 c \log (f)}-d} \text {erf}\left (\frac {b \log (f)-2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{16 \sqrt {f-c \log (f)}}+\frac {\sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{12 f-4 c \log (f)}-3 d} \text {erf}\left (\frac {b \log (f)-2 x (3 f-c \log (f))}{2 \sqrt {3 f-c \log (f)}}\right )}{16 \sqrt {3 f-c \log (f)}}-\frac {3 \sqrt {\pi } f^a e^{d-\frac {b^2 \log ^2(f)}{4 (c \log (f)+f)}} \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+f)}{2 \sqrt {c \log (f)+f}}\right )}{16 \sqrt {c \log (f)+f}}+\frac {\sqrt {\pi } f^a e^{3 d-\frac {b^2 \log ^2(f)}{4 (c \log (f)+3 f)}} \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+3 f)}{2 \sqrt {c \log (f)+3 f}}\right )}{16 \sqrt {c \log (f)+3 f}} \]
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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+c x^2}+\frac {3}{8} e^{-d-f x^2} f^{a+b x+c x^2}-\frac {3}{8} e^{d+f x^2} f^{a+b x+c x^2}+\frac {1}{8} e^{3 d+3 f x^2} f^{a+b x+c x^2}\right ) \, dx \\ & = -\left (\frac {1}{8} \int e^{-3 d-3 f x^2} f^{a+b x+c x^2} \, dx\right )+\frac {1}{8} \int e^{3 d+3 f x^2} f^{a+b x+c x^2} \, dx+\frac {3}{8} \int e^{-d-f x^2} f^{a+b x+c x^2} \, dx-\frac {3}{8} \int e^{d+f x^2} f^{a+b x+c x^2} \, dx \\ & = -\left (\frac {1}{8} \int \exp \left (-3 d+a \log (f)+b x \log (f)-x^2 (3 f-c \log (f))\right ) \, dx\right )+\frac {1}{8} \int \exp \left (3 d+a \log (f)+b x \log (f)+x^2 (3 f+c \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (-d+a \log (f)+b x \log (f)-x^2 (f-c \log (f))\right ) \, dx-\frac {3}{8} \int \exp \left (d+a \log (f)+b x \log (f)+x^2 (f+c \log (f))\right ) \, dx \\ & = \frac {1}{8} \left (3 e^{-d+\frac {b^2 \log ^2(f)}{4 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (-f+c \log (f)))^2}{4 (-f+c \log (f))}\right ) \, dx-\frac {1}{8} \left (e^{-3 d+\frac {b^2 \log ^2(f)}{12 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (-3 f+c \log (f)))^2}{4 (-3 f+c \log (f))}\right ) \, dx-\frac {1}{8} \left (3 e^{d-\frac {b^2 \log ^2(f)}{4 (f+c \log (f))}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (f+c \log (f)))^2}{4 (f+c \log (f))}\right ) \, dx+\frac {1}{8} \left (e^{3 d-\frac {b^2 \log ^2(f)}{4 (3 f+c \log (f))}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (3 f+c \log (f)))^2}{4 (3 f+c \log (f))}\right ) \, dx \\ & = -\frac {3 e^{-d+\frac {b^2 \log ^2(f)}{4 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{16 \sqrt {f-c \log (f)}}+\frac {e^{-3 d+\frac {b^2 \log ^2(f)}{12 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (3 f-c \log (f))}{2 \sqrt {3 f-c \log (f)}}\right )}{16 \sqrt {3 f-c \log (f)}}-\frac {3 e^{d-\frac {b^2 \log ^2(f)}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{16 \sqrt {f+c \log (f)}}+\frac {e^{3 d-\frac {b^2 \log ^2(f)}{4 (3 f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (3 f+c \log (f))}{2 \sqrt {3 f+c \log (f)}}\right )}{16 \sqrt {3 f+c \log (f)}} \\ \end{align*}
Time = 4.73 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.56 \[ \int f^{a+b x+c x^2} \sinh ^3\left (d+f x^2\right ) \, dx=\frac {e^{-\frac {b^2 \log ^2(f) (2 f+c \log (f))}{2 (f+c \log (f)) (3 f+c \log (f))}} f^a \sqrt {\pi } \left (3 e^{\frac {1}{4} b^2 \log ^2(f) \left (\frac {1}{f-c \log (f)}+\frac {1}{f+c \log (f)}+\frac {1}{3 f+c \log (f)}\right )} \text {erf}\left (\frac {2 f x-(b+2 c x) \log (f)}{2 \sqrt {f-c \log (f)}}\right ) \sqrt {f-c \log (f)} \left (9 f^3+9 c f^2 \log (f)-c^2 f \log ^2(f)-c^3 \log ^3(f)\right ) (\cosh (d)-\sinh (d))-(f-c \log (f)) \left (e^{\frac {1}{4} b^2 \log ^2(f) \left (\frac {1}{3 f-c \log (f)}+\frac {1}{f+c \log (f)}+\frac {1}{3 f+c \log (f)}\right )} \text {erf}\left (\frac {6 f x-(b+2 c x) \log (f)}{2 \sqrt {3 f-c \log (f)}}\right ) \sqrt {3 f-c \log (f)} \left (3 f^2+4 c f \log (f)+c^2 \log ^2(f)\right ) (\cosh (3 d)-\sinh (3 d))+(3 f-c \log (f)) \left (3 e^{\frac {b^2 \log ^2(f)}{12 f+4 c \log (f)}} \text {erfi}\left (\frac {2 f x+(b+2 c x) \log (f)}{2 \sqrt {f+c \log (f)}}\right ) \sqrt {f+c \log (f)} (3 f+c \log (f)) (\cosh (d)+\sinh (d))-e^{\frac {b^2 \log ^2(f)}{4 (f+c \log (f))}} \text {erfi}\left (\frac {6 f x+(b+2 c x) \log (f)}{2 \sqrt {3 f+c \log (f)}}\right ) (f+c \log (f)) \sqrt {3 f+c \log (f)} (\cosh (3 d)+\sinh (3 d))\right )\right )\right )}{16 \left (9 f^4-10 c^2 f^2 \log ^2(f)+c^4 \log ^4(f)\right )} \]
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Time = 1.10 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.01
method | result | size |
risch | \(-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-3 f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )-3 f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-12 d \ln \left (f \right ) c -36 d f}{4 \left (3 f +c \ln \left (f \right )\right )}}}{16 \sqrt {-c \ln \left (f \right )-3 f}}+\frac {\operatorname {erf}\left (-x \sqrt {3 f -c \ln \left (f \right )}+\frac {\ln \left (f \right ) b}{2 \sqrt {3 f -c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+12 d \ln \left (f \right ) c -36 d f}{4 \left (c \ln \left (f \right )-3 f \right )}}}{16 \sqrt {3 f -c \ln \left (f \right )}}-\frac {3 \,\operatorname {erf}\left (-x \sqrt {f -c \ln \left (f \right )}+\frac {\ln \left (f \right ) b}{2 \sqrt {f -c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+4 d \ln \left (f \right ) c -4 d f}{4 \left (c \ln \left (f \right )-f \right )}}}{16 \sqrt {f -c \ln \left (f \right )}}+\frac {3 \,\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )-f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-4 d \ln \left (f \right ) c -4 d f}{4 \left (f +c \ln \left (f \right )\right )}}}{16 \sqrt {-c \ln \left (f \right )-f}}\) | \(326\) |
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Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (275) = 550\).
Time = 0.30 (sec) , antiderivative size = 852, normalized size of antiderivative = 2.64 \[ \int f^{a+b x+c x^2} \sinh ^3\left (d+f x^2\right ) \, dx=\text {Too large to display} \]
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\[ \int f^{a+b x+c x^2} \sinh ^3\left (d+f x^2\right ) \, dx=\int f^{a + b x + c x^{2}} \sinh ^{3}{\left (d + f x^{2} \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.89 \[ \int f^{a+b x+c x^2} \sinh ^3\left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 3 \, f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) - 3 \, f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) + 3 \, f\right )}} + 3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) - 3 \, f}} - \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) - f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) + f\right )}} + d\right )}}{16 \, \sqrt {-c \log \left (f\right ) - f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) + f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) - f\right )}} - d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + f}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 3 \, f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) + 3 \, f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) - 3 \, f\right )}} - 3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + 3 \, f}} \]
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Time = 0.29 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.14 \[ \int f^{a+b x+c x^2} \sinh ^3\left (d+f x^2\right ) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - 3 \, f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) + 3 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 12 \, a f \log \left (f\right ) - 36 \, d f}{4 \, {\left (c \log \left (f\right ) + 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) - 3 \, f}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) + f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) - f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) - f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) + 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) + f}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + 3 \, f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) - 3 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) + 12 \, a f \log \left (f\right ) - 36 \, d f}{4 \, {\left (c \log \left (f\right ) - 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) + 3 \, f}} \]
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Timed out. \[ \int f^{a+b x+c x^2} \sinh ^3\left (d+f x^2\right ) \, dx=\int f^{c\,x^2+b\,x+a}\,{\mathrm {sinh}\left (f\,x^2+d\right )}^3 \,d x \]
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