Integrand size = 21, antiderivative size = 140 \[ \int f^{a+c x^2} \sinh \left (d+e x+f x^2\right ) \, dx=-\frac {e^{-d+\frac {e^2}{4 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {e+2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}}+\frac {e^{d-\frac {e^2}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{4 \sqrt {f+c \log (f)}} \]
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Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5623, 2325, 2266, 2236, 2235} \[ \int f^{a+c x^2} \sinh \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^a e^{d-\frac {e^2}{4 (c \log (f)+f)}} \text {erfi}\left (\frac {2 x (c \log (f)+f)+e}{2 \sqrt {c \log (f)+f}}\right )}{4 \sqrt {c \log (f)+f}}-\frac {\sqrt {\pi } f^a e^{\frac {e^2}{4 f-4 c \log (f)}-d} \text {erf}\left (\frac {2 x (f-c \log (f))+e}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 5623
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2} e^{-d-e x-f x^2} f^{a+c x^2}+\frac {1}{2} e^{d+e x+f x^2} f^{a+c x^2}\right ) \, dx \\ & = -\left (\frac {1}{2} \int e^{-d-e x-f x^2} f^{a+c x^2} \, dx\right )+\frac {1}{2} \int e^{d+e x+f x^2} f^{a+c x^2} \, dx \\ & = -\left (\frac {1}{2} \int e^{-d-e x+a \log (f)-x^2 (f-c \log (f))} \, dx\right )+\frac {1}{2} \int e^{d+e x+a \log (f)+x^2 (f+c \log (f))} \, dx \\ & = -\left (\frac {1}{2} \left (e^{-d+\frac {e^2}{4 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-e+2 x (-f+c \log (f)))^2}{4 (-f+c \log (f))}\right ) \, dx\right )+\frac {1}{2} \left (e^{d-\frac {e^2}{4 (f+c \log (f))}} f^a\right ) \int \exp \left (\frac {(e+2 x (f+c \log (f)))^2}{4 (f+c \log (f))}\right ) \, dx \\ & = -\frac {e^{-d+\frac {e^2}{4 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {e+2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}}+\frac {e^{d-\frac {e^2}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{4 \sqrt {f+c \log (f)}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.19 \[ \int f^{a+c x^2} \sinh \left (d+e x+f x^2\right ) \, dx=\frac {e^{-\frac {e^2}{4 (f+c \log (f))}} f^a \sqrt {\pi } \left (-e^{\frac {e^2 f}{2 f^2-2 c^2 \log ^2(f)}} \text {erf}\left (\frac {e+2 f x-2 c x \log (f)}{2 \sqrt {f-c \log (f)}}\right ) \sqrt {f+c \log (f)} (\cosh (d)-\sinh (d))+\text {erfi}\left (\frac {e+2 f x+2 c x \log (f)}{2 \sqrt {f+c \log (f)}}\right ) \sqrt {f-c \log (f)} (\cosh (d)+\sinh (d))\right )}{4 \sqrt {f-c \log (f)} \sqrt {f+c \log (f)}} \]
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Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.05
method | result | size |
risch | \(-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-f}\, x +\frac {e}{2 \sqrt {-c \ln \left (f \right )-f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 d \ln \left (f \right ) c +4 d f -e^{2}}{4 f +4 c \ln \left (f \right )}}}{4 \sqrt {-c \ln \left (f \right )-f}}-\frac {\operatorname {erf}\left (x \sqrt {f -c \ln \left (f \right )}+\frac {e}{2 \sqrt {f -c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d \ln \left (f \right ) c -4 d f +e^{2}}{4 \left (c \ln \left (f \right )-f \right )}}}{4 \sqrt {f -c \ln \left (f \right )}}\) | \(147\) |
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Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (119) = 238\).
Time = 0.31 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.30 \[ \int f^{a+c x^2} \sinh \left (d+e x+f x^2\right ) \, dx=\frac {{\left (\sqrt {\pi } {\left (c \log \left (f\right ) + f\right )} \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} - e^{2} + 4 \, d f - 4 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right ) + \sqrt {\pi } {\left (c \log \left (f\right ) + f\right )} \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} - e^{2} + 4 \, d f - 4 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) + f} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - 2 \, f x - e\right )} \sqrt {-c \log \left (f\right ) + f}}{2 \, {\left (c \log \left (f\right ) - f\right )}}\right ) - {\left (\sqrt {\pi } {\left (c \log \left (f\right ) - f\right )} \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} - e^{2} + 4 \, d f + 4 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right ) + \sqrt {\pi } {\left (c \log \left (f\right ) - f\right )} \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} - e^{2} + 4 \, d f + 4 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) - f} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + 2 \, f x + e\right )} \sqrt {-c \log \left (f\right ) - f}}{2 \, {\left (c \log \left (f\right ) + f\right )}}\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} - f^{2}\right )}} \]
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\[ \int f^{a+c x^2} \sinh \left (d+e x+f x^2\right ) \, dx=\int f^{a + c x^{2}} \sinh {\left (d + e x + f x^{2} \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int f^{a+c x^2} \sinh \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - f} x - \frac {e}{2 \, \sqrt {-c \log \left (f\right ) - f}}\right ) e^{\left (d - \frac {e^{2}}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )}}{4 \, \sqrt {-c \log \left (f\right ) - f}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + f} x + \frac {e}{2 \, \sqrt {-c \log \left (f\right ) + f}}\right ) e^{\left (-d - \frac {e^{2}}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )}}{4 \, \sqrt {-c \log \left (f\right ) + f}} \]
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Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.23 \[ \int f^{a+c x^2} \sinh \left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - f} {\left (2 \, x + \frac {e}{c \log \left (f\right ) + f}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) + 4 \, a f \log \left (f\right ) - e^{2} + 4 \, d f}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )}}{4 \, \sqrt {-c \log \left (f\right ) - f}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + f} {\left (2 \, x - \frac {e}{c \log \left (f\right ) - f}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - 4 \, a f \log \left (f\right ) - e^{2} + 4 \, d f}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )}}{4 \, \sqrt {-c \log \left (f\right ) + f}} \]
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Timed out. \[ \int f^{a+c x^2} \sinh \left (d+e x+f x^2\right ) \, dx=\int f^{c\,x^2+a}\,\mathrm {sinh}\left (f\,x^2+e\,x+d\right ) \,d x \]
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