Integrand size = 23, antiderivative size = 225 \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 d+\frac {b^2 \log ^2(f)}{8 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (2 f-c \log (f))}{2 \sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d-\frac {b^2 \log ^2(f)}{8 f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (2 f+c \log (f))}{2 \sqrt {2 f+c \log (f)}}\right )}{8 \sqrt {2 f+c \log (f)}} \]
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Time = 0.28 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5624, 2266, 2235, 2325, 2236} \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=-\frac {\sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{8 f-4 c \log (f)}-2 d} \text {erf}\left (\frac {b \log (f)-2 x (2 f-c \log (f))}{2 \sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {b^2 \log ^2(f)}{4 c \log (f)+8 f}} \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+2 f)}{2 \sqrt {c \log (f)+2 f}}\right )}{8 \sqrt {c \log (f)+2 f}}+\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 5624
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} f^{a+b x+c x^2}+\frac {1}{4} e^{-2 d-2 f x^2} f^{a+b x+c x^2}+\frac {1}{4} e^{2 d+2 f x^2} f^{a+b x+c x^2}\right ) \, dx \\ & = \frac {1}{4} \int e^{-2 d-2 f x^2} f^{a+b x+c x^2} \, dx+\frac {1}{4} \int e^{2 d+2 f x^2} f^{a+b x+c x^2} \, dx+\frac {1}{2} \int f^{a+b x+c x^2} \, dx \\ & = \frac {1}{4} \int \exp \left (-2 d+a \log (f)+b x \log (f)-x^2 (2 f-c \log (f))\right ) \, dx+\frac {1}{4} \int \exp \left (2 d+a \log (f)+b x \log (f)+x^2 (2 f+c \log (f))\right ) \, dx+\frac {1}{2} f^{a-\frac {b^2}{4 c}} \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx \\ & = \frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {1}{4} \left (e^{-2 d+\frac {b^2 \log ^2(f)}{8 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (-2 f+c \log (f)))^2}{4 (-2 f+c \log (f))}\right ) \, dx+\frac {1}{4} \left (e^{2 d-\frac {b^2 \log ^2(f)}{8 f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (2 f+c \log (f)))^2}{4 (2 f+c \log (f))}\right ) \, dx \\ & = \frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 d+\frac {b^2 \log ^2(f)}{8 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (2 f-c \log (f))}{2 \sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d-\frac {b^2 \log ^2(f)}{8 f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (2 f+c \log (f))}{2 \sqrt {2 f+c \log (f)}}\right )}{8 \sqrt {2 f+c \log (f)}} \\ \end{align*}
Time = 1.58 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.14 \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=\frac {1}{8} f^a \sqrt {\pi } \left (\frac {2 f^{-\frac {b^2}{4 c}} \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{\sqrt {c} \sqrt {\log (f)}}-\frac {e^{-\frac {b^2 \log ^2(f)}{8 f+4 c \log (f)}} \left (e^{\frac {b^2 f \log ^2(f)}{4 f^2-c^2 \log ^2(f)}} \text {erf}\left (\frac {4 f x-(b+2 c x) \log (f)}{2 \sqrt {2 f-c \log (f)}}\right ) \sqrt {2 f-c \log (f)} (2 f+c \log (f)) (\cosh (2 d)-\sinh (2 d))+\text {erfi}\left (\frac {4 f x+(b+2 c x) \log (f)}{2 \sqrt {2 f+c \log (f)}}\right ) (2 f-c \log (f)) \sqrt {2 f+c \log (f)} (\cosh (2 d)+\sinh (2 d))\right )}{-4 f^2+c^2 \log ^2(f)}\right ) \]
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Time = 0.40 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-\frac {\operatorname {erf}\left (-x \sqrt {2 f -c \ln \left (f \right )}+\frac {\ln \left (f \right ) b}{2 \sqrt {2 f -c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+8 d \ln \left (f \right ) c -16 d f}{4 \left (c \ln \left (f \right )-2 f \right )}}}{8 \sqrt {2 f -c \ln \left (f \right )}}-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-2 f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )-2 f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-8 d \ln \left (f \right ) c -16 d f}{4 \left (2 f +c \ln \left (f \right )\right )}}}{8 \sqrt {-c \ln \left (f \right )-2 f}}-\frac {f^{a} \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(217\) |
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Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (185) = 370\).
Time = 0.27 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.07 \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=-\frac {{\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f + 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f + 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f} \operatorname {erf}\left (-\frac {{\left (4 \, f x - {\left (2 \, c x + b\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f}}{2 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f - 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f - 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f} \operatorname {erf}\left (\frac {{\left (4 \, f x + {\left (2 \, c x + b\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f}}{2 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right ) + 2 \, {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )}{4 \, c}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )}{4 \, c}\right )\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{8 \, {\left (c^{3} \log \left (f\right )^{3} - 4 \, c f^{2} \log \left (f\right )\right )}} \]
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\[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=\int f^{a + b x + c x^{2}} \cosh ^{2}{\left (d + f x^{2} \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.88 \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) - 2 \, f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}} + 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) + 2 \, f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}} - 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right )}}\right )}{4 \, \sqrt {-c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} \]
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Time = 0.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.06 \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - 2 \, f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) + 2 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 8 \, c d \log \left (f\right ) - 8 \, a f \log \left (f\right ) - 16 \, d f}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + 2 \, f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) - 2 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 8 \, c d \log \left (f\right ) + 8 \, a f \log \left (f\right ) - 16 \, d f}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{4 \, \sqrt {-c \log \left (f\right )}} \]
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Timed out. \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=\int f^{c\,x^2+b\,x+a}\,{\mathrm {cosh}\left (f\,x^2+d\right )}^2 \,d x \]
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