Optimal. Leaf size=54 \[ -\frac{\sqrt{\pi } a \text{Erf}(a+b x)}{4 b^2}-\frac{\sqrt{\pi } a \text{Erfi}(a+b x)}{4 b^2}+\frac{\sinh \left ((a+b x)^2\right )}{2 b^2} \]
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Rubi [A] time = 0.0530845, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5365, 6742, 5299, 2204, 2205, 5321, 2637} \[ -\frac{\sqrt{\pi } a \text{Erf}(a+b x)}{4 b^2}-\frac{\sqrt{\pi } a \text{Erfi}(a+b x)}{4 b^2}+\frac{\sinh \left ((a+b x)^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 5365
Rule 6742
Rule 5299
Rule 2204
Rule 2205
Rule 5321
Rule 2637
Rubi steps
\begin{align*} \int x \cosh \left ((a+b x)^2\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (-a+x) \cosh \left (x^2\right ) \, dx,x,a+b x\right )}{b^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a \cosh \left (x^2\right )+x \cosh \left (x^2\right )\right ) \, dx,x,a+b x\right )}{b^2}\\ &=\frac{\operatorname{Subst}\left (\int x \cosh \left (x^2\right ) \, dx,x,a+b x\right )}{b^2}-\frac{a \operatorname{Subst}\left (\int \cosh \left (x^2\right ) \, dx,x,a+b x\right )}{b^2}\\ &=\frac{\operatorname{Subst}\left (\int \cosh (x) \, dx,x,(a+b x)^2\right )}{2 b^2}-\frac{a \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,a+b x\right )}{2 b^2}-\frac{a \operatorname{Subst}\left (\int e^{x^2} \, dx,x,a+b x\right )}{2 b^2}\\ &=-\frac{a \sqrt{\pi } \text{erf}(a+b x)}{4 b^2}-\frac{a \sqrt{\pi } \text{erfi}(a+b x)}{4 b^2}+\frac{\sinh \left ((a+b x)^2\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0292377, size = 39, normalized size = 0.72 \[ \frac{2 \sinh \left ((a+b x)^2\right )-\sqrt{\pi } a (\text{Erf}(a+b x)+\text{Erfi}(a+b x))}{4 b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.033, size = 66, normalized size = 1.2 \begin{align*} -{\frac{{{\rm e}^{- \left ( bx+a \right ) ^{2}}}}{4\,{b}^{2}}}-{\frac{a{\it Erf} \left ( bx+a \right ) \sqrt{\pi }}{4\,{b}^{2}}}+{\frac{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}{4\,{b}^{2}}}+{\frac{{\frac{i}{4}}a\sqrt{\pi }{\it Erf} \left ( ibx+ia \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.5552, size = 945, normalized size = 17.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97037, size = 319, normalized size = 5.91 \begin{align*} -\frac{{\left (\sqrt{\pi } a \sqrt{b^{2}} \operatorname{erf}\left (\frac{\sqrt{b^{2}}{\left (b x + a\right )}}{b}\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} + \sqrt{\pi } a \sqrt{b^{2}} \operatorname{erfi}\left (\frac{\sqrt{b^{2}}{\left (b x + a\right )}}{b}\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - b e^{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2}\right )} + b\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh{\left (a^{2} + 2 a b x + b^{2} x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.30638, size = 134, normalized size = 2.48 \begin{align*} -\frac{-\frac{i \, \sqrt{\pi } a \operatorname{erf}\left (i \, b{\left (x + \frac{a}{b}\right )}\right )}{b} - \frac{e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}}{b}}{4 \, b} + \frac{\frac{\sqrt{\pi } a \operatorname{erf}\left (-b{\left (x + \frac{a}{b}\right )}\right )}{b} - \frac{e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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