Optimal. Leaf size=71 \[ -\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b}+\frac {(a+b x) \text {erfc}(a+b x)^2}{b}-\frac {2 e^{-(a+b x)^2} \text {erfc}(a+b x)}{\sqrt {\pi } b} \]
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Rubi [A] time = 0.16, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6353, 6383, 2205} \[ -\frac {\sqrt {\frac {2}{\pi }} \text {Erf}\left (\sqrt {2} (a+b x)\right )}{b}+\frac {(a+b x) \text {Erfc}(a+b x)^2}{b}-\frac {2 e^{-(a+b x)^2} \text {Erfc}(a+b x)}{\sqrt {\pi } b} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 6353
Rule 6383
Rubi steps
\begin {align*} \int \text {erfc}(a+b x)^2 \, dx &=\frac {(a+b x) \text {erfc}(a+b x)^2}{b}+\frac {4 \int e^{-(a+b x)^2} (a+b x) \text {erfc}(a+b x) \, dx}{\sqrt {\pi }}\\ &=\frac {(a+b x) \text {erfc}(a+b x)^2}{b}+\frac {4 \operatorname {Subst}\left (\int e^{-x^2} x \text {erfc}(x) \, dx,x,a+b x\right )}{b \sqrt {\pi }}\\ &=-\frac {2 e^{-(a+b x)^2} \text {erfc}(a+b x)}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfc}(a+b x)^2}{b}-\frac {4 \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,a+b x\right )}{b \pi }\\ &=-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b}-\frac {2 e^{-(a+b x)^2} \text {erfc}(a+b x)}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfc}(a+b x)^2}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 66, normalized size = 0.93 \[ \frac {\text {erfc}(a+b x) \left ((a+b x) \text {erfc}(a+b x)-\frac {2 e^{-(a+b x)^2}}{\sqrt {\pi }}\right )-\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 141, normalized size = 1.99 \[ -\frac {2 \, \pi b^{2} x \operatorname {erf}\left (b x + a\right ) - \pi b^{2} x + 2 \, \pi a \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - {\left (\pi b^{2} x + \pi a b\right )} \operatorname {erf}\left (b x + a\right )^{2} + \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, \sqrt {\pi } {\left (b \operatorname {erf}\left (b x + a\right ) - b\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{\pi b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfc}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 59, normalized size = 0.83 \[ \frac {\left (b x +a \right ) \erf \left (b x +a \right )^{2}+\frac {2 \erf \left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \erf \left (\left (b x +a \right ) \sqrt {2}\right )}{\sqrt {\pi }}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfc}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {erfc}\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int \operatorname {erfc}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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