Integrand size = 8, antiderivative size = 37 \[ \int \cosh \left ((a+b x)^2\right ) \, dx=\frac {\sqrt {\pi } \text {erf}(a+b x)}{4 b}+\frac {\sqrt {\pi } \text {erfi}(a+b x)}{4 b} \]
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Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5419, 5407, 2235, 2236} \[ \int \cosh \left ((a+b x)^2\right ) \, dx=\frac {\sqrt {\pi } \text {erf}(a+b x)}{4 b}+\frac {\sqrt {\pi } \text {erfi}(a+b x)}{4 b} \]
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Rule 2235
Rule 2236
Rule 5407
Rule 5419
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \cosh \left (x^2\right ) \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,a+b x\right )}{2 b}+\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,a+b x\right )}{2 b} \\ & = \frac {\sqrt {\pi } \text {erf}(a+b x)}{4 b}+\frac {\sqrt {\pi } \text {erfi}(a+b x)}{4 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.68 \[ \int \cosh \left ((a+b x)^2\right ) \, dx=\frac {\sqrt {\pi } (\text {erf}(a+b x)+\text {erfi}(a+b x))}{4 b} \]
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Result contains complex when optimal does not.
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {\operatorname {erf}\left (b x +a \right ) \sqrt {\pi }}{4 b}-\frac {i \sqrt {\pi }\, \operatorname {erf}\left (i x b +i a \right )}{4 b}\) | \(36\) |
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.59 \[ \int \cosh \left ((a+b x)^2\right ) \, dx=-\frac {\sqrt {\pi } \sqrt {-b^{2}} \operatorname {erf}\left (\frac {\sqrt {-b^{2}} {\left (b x + a\right )}}{b}\right ) - \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right )}{4 \, b^{2}} \]
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\[ \int \cosh \left ((a+b x)^2\right ) \, dx=\int \cosh {\left (\left (a + b x\right )^{2} \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (29) = 58\).
Time = 0.39 (sec) , antiderivative size = 478, normalized size of antiderivative = 12.92 \[ \int \cosh \left ((a+b x)^2\right ) \, dx=-\frac {1}{2} \, {\left (\frac {{\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a b^{2} {\left (\operatorname {erf}\left (\frac {\sqrt {{\left (b^{2} x + a b\right )}^{2}}}{b}\right ) - 1\right )}}{\sqrt {{\left (b^{2} x + a b\right )}^{2}} \left (-b^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} e^{\left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac {3}{2}}}\right )} a}{\sqrt {-b^{2}}} + \frac {{\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{2} b^{3} {\left (\operatorname {erf}\left (\frac {\sqrt {{\left (b^{2} x + a b\right )}^{2}}}{b}\right ) - 1\right )}}{\sqrt {{\left (b^{2} x + a b\right )}^{2}} \left (-b^{2}\right )^{\frac {5}{2}}} - \frac {{\left (b^{2} x + a b\right )}^{3} b^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{{\left ({\left (b^{2} x + a b\right )}^{2}\right )}^{\frac {3}{2}} \left (-b^{2}\right )^{\frac {5}{2}}} + \frac {2 \, a b^{3} e^{\left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac {5}{2}}}\right )} b}{\sqrt {-b^{2}}} - \frac {a {\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}\right ) - 1\right )}}{b^{2} \sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}} - \frac {e^{\left (\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{b}\right )}}{b} + \frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{2} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}\right ) - 1\right )}}{b^{3} \sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}} - \frac {2 \, a e^{\left (\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{b^{2}} - \frac {{\left (b^{2} x + a b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{b^{5} \left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )^{\frac {3}{2}}}\right )} b + x \cosh \left ({\left (b x + a\right )}^{2}\right ) \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int \cosh \left ((a+b x)^2\right ) \, dx=-\frac {i \, \sqrt {\pi } \operatorname {erf}\left (i \, b {\left (x + \frac {a}{b}\right )}\right )}{4 \, b} - \frac {\sqrt {\pi } \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{4 \, b} \]
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Timed out. \[ \int \cosh \left ((a+b x)^2\right ) \, dx=\int \mathrm {cosh}\left ({\left (a+b\,x\right )}^2\right ) \,d x \]
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