Optimal. Leaf size=37 \[ \frac {\sqrt {\pi } \text {erf}(a+b x)}{4 b}+\frac {\sqrt {\pi } \text {erfi}(a+b x)}{4 b} \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5311, 5299, 2204, 2205} \[ \frac {\sqrt {\pi } \text {Erf}(a+b x)}{4 b}+\frac {\sqrt {\pi } \text {Erfi}(a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5299
Rule 5311
Rubi steps
\begin {align*} \int \cosh \left ((a+b x)^2\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \cosh \left (x^2\right ) \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,a+b x\right )}{2 b}+\frac {\operatorname {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*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 0.68 \[ \frac {\sqrt {\pi } (\text {erf}(a+b x)+\text {erfi}(a+b x))}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 54, normalized size = 1.46 \[ \frac {\sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + \sqrt {\pi } \sqrt {b^{2}} \operatorname {erfi}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right )}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.13, size = 39, normalized size = 1.05 \[ -\frac {i \, \sqrt {\pi } \operatorname {erf}\left (i \, b {\left (x + \frac {a}{b}\right )}\right )}{4 \, b} - \frac {\sqrt {\pi } \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 36, normalized size = 0.97 \[ \frac {\erf \left (b x +a \right ) \sqrt {\pi }}{4 b}-\frac {i \sqrt {\pi }\, \erf \left (i b x +i a \right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 478, normalized size = 12.92 \[ -\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 ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {cosh}\left ({\left (a+b\,x\right )}^2\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\left (\left (a + b x\right )^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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