Integrand size = 19, antiderivative size = 115 \[ \int f^{a+b x} \sinh \left (d+e x+f x^2\right ) \, dx=-\frac {1}{4} e^{-d+\frac {(e-b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {e+2 f x-b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{4} e^{d-\frac {(e+b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right ) \]
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Time = 0.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5623, 2325, 2266, 2236, 2235} \[ \int f^{a+b x} \sinh \left (d+e x+f x^2\right ) \, dx=\frac {1}{4} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {(b \log (f)+e)^2}{4 f}} \text {erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )-\frac {1}{4} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {(e-b \log (f))^2}{4 f}-d} \text {erf}\left (\frac {-b \log (f)+e+2 f x}{2 \sqrt {f}}\right ) \]
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Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 5623
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2} e^{-d-e x-f x^2} f^{a+b x}+\frac {1}{2} e^{d+e x+f x^2} f^{a+b x}\right ) \, dx \\ & = -\left (\frac {1}{2} \int e^{-d-e x-f x^2} f^{a+b x} \, dx\right )+\frac {1}{2} \int e^{d+e x+f x^2} f^{a+b x} \, dx \\ & = -\left (\frac {1}{2} \int e^{-d-f x^2+a \log (f)-x (e-b \log (f))} \, dx\right )+\frac {1}{2} \int e^{d+f x^2+a \log (f)+x (e+b \log (f))} \, dx \\ & = -\left (\frac {1}{2} \left (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\right )+\frac {1}{2} \left (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}{4} 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}{4} 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 ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.08 \[ \int f^{a+b x} \sinh \left (d+e x+f x^2\right ) \, dx=\frac {1}{4} e^{-\frac {e^2+b^2 \log ^2(f)}{4 f}} f^{a-\frac {b e+f}{2 f}} \sqrt {\pi } \left (-e^{\frac {e^2+b^2 \log ^2(f)}{2 f}} \text {erf}\left (\frac {e+2 f x-b \log (f)}{2 \sqrt {f}}\right ) (\cosh (d)-\sinh (d))+\text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right ) (\cosh (d)+\sinh (d))\right ) \]
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Time = 0.49 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.10
method | result | size |
risch | \(-\frac {\operatorname {erf}\left (-\sqrt {-f}\, x +\frac {e +b \ln \left (f \right )}{2 \sqrt {-f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+2 \ln \left (f \right ) b e -4 d f +e^{2}}{4 f}}}{4 \sqrt {-f}}+\frac {\operatorname {erf}\left (-\sqrt {f}\, x +\frac {b \ln \left (f \right )-e}{2 \sqrt {f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {b^{2} \ln \left (f \right )^{2}-2 \ln \left (f \right ) b e -4 d f +e^{2}}{4 f}}}{4 \sqrt {f}}\) | \(126\) |
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Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (90) = 180\).
Time = 0.29 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.20 \[ \int f^{a+b x} \sinh \left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f + 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \left (f\right ) + e\right )} \sqrt {-f}}{2 \, f}\right ) - \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) \operatorname {erf}\left (-\frac {2 \, f x - b \log \left (f\right ) + e}{2 \, \sqrt {f}}\right ) - \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \left (f\right ) + e\right )} \sqrt {-f}}{2 \, f}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f + 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) - \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {2 \, f x - b \log \left (f\right ) + e}{2 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right )}{4 \, f} \]
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\[ \int f^{a+b x} \sinh \left (d+e x+f x^2\right ) \, dx=\int f^{a + b x} \sinh {\left (d + e x + f x^{2} \right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89 \[ \int f^{a+b x} \sinh \left (d+e x+f x^2\right ) \, dx=-\frac {1}{4} \, \sqrt {\pi } f^{a - \frac {1}{2}} \operatorname {erf}\left (\sqrt {f} x - \frac {b \log \left (f\right ) - e}{2 \, \sqrt {f}}\right ) e^{\left (-d + \frac {{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, f}\right )} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-f} x - \frac {b \log \left (f\right ) + e}{2 \, \sqrt {-f}}\right ) e^{\left (d - \frac {{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \, f}\right )}}{4 \, \sqrt {-f}} \]
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Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.15 \[ \int f^{a+b x} \sinh \left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-f} {\left (2 \, x + \frac {b \log \left (f\right ) + e}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} + 2 \, b e \log \left (f\right ) - 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, f}\right )}}{4 \, \sqrt {-f}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {f} {\left (2 \, x - \frac {b \log \left (f\right ) - e}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2} - 2 \, b e \log \left (f\right ) + 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, f}\right )}}{4 \, \sqrt {f}} \]
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Timed out. \[ \int f^{a+b x} \sinh \left (d+e x+f x^2\right ) \, dx=\int f^{a+b\,x}\,\mathrm {sinh}\left (f\,x^2+e\,x+d\right ) \,d x \]
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