Integrand size = 9, antiderivative size = 39 \[ \int \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=-\frac {1}{4} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (-1-2 x)\right )+\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 x)\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5483, 2266, 2235, 2236} \[ \int \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 x+1)\right )-\frac {1}{4} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (-2 x-1)\right ) \]
[In]
[Out]
Rule 2235
Rule 2236
Rule 2266
Rule 5483
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int e^{-\frac {1}{4}-x-x^2} \, dx+\frac {1}{2} \int e^{\frac {1}{4}+x+x^2} \, dx \\ & = \frac {1}{2} \int e^{-\frac {1}{4} (-1-2 x)^2} \, dx+\frac {1}{2} \int e^{\frac {1}{4} (1+2 x)^2} \, dx \\ & = -\frac {1}{4} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (-1-2 x)\right )+\frac {1}{4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 x)\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.56 \[ \int \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{4} \sqrt {\pi } \left (\text {erf}\left (\frac {1}{2}+x\right )+\text {erfi}\left (\frac {1}{2}+x\right )\right ) \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.64
method | result | size |
risch | \(\frac {\operatorname {erf}\left (\frac {1}{2}+x \right ) \sqrt {\pi }}{4}-\frac {i \sqrt {\pi }\, \operatorname {erf}\left (i x +\frac {1}{2} i\right )}{4}\) | \(25\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.36 \[ \int \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{4} \, \sqrt {\pi } {\left (\operatorname {erf}\left (x + \frac {1}{2}\right ) + \operatorname {erfi}\left (x + \frac {1}{2}\right )\right )} \]
[In]
[Out]
\[ \int \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\int \cosh {\left (x^{2} + x + \frac {1}{4} \right )}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.41 \[ \int \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=-\frac {{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{2 \, {\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac {3}{2}}} + \frac {{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{2 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac {3}{2}}} + x \cosh \left (x^{2} + x + \frac {1}{4}\right ) + \frac {1}{4} \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )} + \frac {1}{4} \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.54 \[ \int \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{4} \, \sqrt {\pi } \operatorname {erf}\left (x + \frac {1}{2}\right ) + \frac {1}{4} i \, \sqrt {\pi } \operatorname {erf}\left (-i \, x - \frac {1}{2} i\right ) \]
[In]
[Out]
Timed out. \[ \int \cosh \left (\frac {1}{4}+x+x^2\right ) \, dx=\int \mathrm {cosh}\left (x^2+x+\frac {1}{4}\right ) \,d x \]
[In]
[Out]