Optimal. Leaf size=292 \[ -\frac {d^2 e^{-(a+b x)^2} (a+b x)^2 (b c-a d)}{\sqrt {\pi } b^4}-\frac {d^2 e^{-(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^4}+\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}+\frac {3 d (b c-a d)^2 \text {erf}(a+b x)}{4 b^4}-\frac {e^{-(a+b x)^2} (b c-a d)^3}{\sqrt {\pi } b^4}-\frac {3 d e^{-(a+b x)^2} (a+b x) (b c-a d)^2}{2 \sqrt {\pi } b^4}+\frac {3 d^3 \text {erf}(a+b x)}{16 b^4}-\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 \sqrt {\pi } b^4}-\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 \sqrt {\pi } b^4}+\frac {(c+d x)^4 \text {erfc}(a+b x)}{4 d} \]
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Rubi [A] time = 0.27, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6362, 2226, 2205, 2209, 2212} \[ -\frac {d^2 e^{-(a+b x)^2} (a+b x)^2 (b c-a d)}{\sqrt {\pi } b^4}-\frac {d^2 e^{-(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^4}+\frac {(b c-a d)^4 \text {Erf}(a+b x)}{4 b^4 d}+\frac {3 d (b c-a d)^2 \text {Erf}(a+b x)}{4 b^4}-\frac {e^{-(a+b x)^2} (b c-a d)^3}{\sqrt {\pi } b^4}-\frac {3 d e^{-(a+b x)^2} (a+b x) (b c-a d)^2}{2 \sqrt {\pi } b^4}+\frac {3 d^3 \text {Erf}(a+b x)}{16 b^4}-\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 \sqrt {\pi } b^4}-\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 \sqrt {\pi } b^4}+\frac {(c+d x)^4 \text {Erfc}(a+b x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2209
Rule 2212
Rule 2226
Rule 6362
Rubi steps
\begin {align*} \int (c+d x)^3 \text {erfc}(a+b x) \, dx &=\frac {(c+d x)^4 \text {erfc}(a+b x)}{4 d}+\frac {b \int e^{-(a+b x)^2} (c+d x)^4 \, dx}{2 d \sqrt {\pi }}\\ &=\frac {(c+d x)^4 \text {erfc}(a+b x)}{4 d}+\frac {b \int \left (\frac {(b c-a d)^4 e^{-(a+b x)^2}}{b^4}+\frac {4 d (b c-a d)^3 e^{-(a+b x)^2} (a+b x)}{b^4}+\frac {6 d^2 (b c-a d)^2 e^{-(a+b x)^2} (a+b x)^2}{b^4}+\frac {4 d^3 (b c-a d) e^{-(a+b x)^2} (a+b x)^3}{b^4}+\frac {d^4 e^{-(a+b x)^2} (a+b x)^4}{b^4}\right ) \, dx}{2 d \sqrt {\pi }}\\ &=\frac {(c+d x)^4 \text {erfc}(a+b x)}{4 d}+\frac {d^3 \int e^{-(a+b x)^2} (a+b x)^4 \, dx}{2 b^3 \sqrt {\pi }}+\frac {\left (2 d^2 (b c-a d)\right ) \int e^{-(a+b x)^2} (a+b x)^3 \, dx}{b^3 \sqrt {\pi }}+\frac {\left (3 d (b c-a d)^2\right ) \int e^{-(a+b x)^2} (a+b x)^2 \, dx}{b^3 \sqrt {\pi }}+\frac {\left (2 (b c-a d)^3\right ) \int e^{-(a+b x)^2} (a+b x) \, dx}{b^3 \sqrt {\pi }}+\frac {(b c-a d)^4 \int e^{-(a+b x)^2} \, dx}{2 b^3 d \sqrt {\pi }}\\ &=-\frac {(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}-\frac {d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}-\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}+\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erfc}(a+b x)}{4 d}+\frac {\left (3 d^3\right ) \int e^{-(a+b x)^2} (a+b x)^2 \, dx}{4 b^3 \sqrt {\pi }}+\frac {\left (2 d^2 (b c-a d)\right ) \int e^{-(a+b x)^2} (a+b x) \, dx}{b^3 \sqrt {\pi }}+\frac {\left (3 d (b c-a d)^2\right ) \int e^{-(a+b x)^2} \, dx}{2 b^3 \sqrt {\pi }}\\ &=-\frac {d^2 (b c-a d) e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 b^4 \sqrt {\pi }}-\frac {3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}-\frac {d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}-\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}+\frac {3 d (b c-a d)^2 \text {erf}(a+b x)}{4 b^4}+\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erfc}(a+b x)}{4 d}+\frac {\left (3 d^3\right ) \int e^{-(a+b x)^2} \, dx}{8 b^3 \sqrt {\pi }}\\ &=-\frac {d^2 (b c-a d) e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 b^4 \sqrt {\pi }}-\frac {3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}-\frac {d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}-\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}+\frac {3 d^3 \text {erf}(a+b x)}{16 b^4}+\frac {3 d (b c-a d)^2 \text {erf}(a+b x)}{4 b^4}+\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erfc}(a+b x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 268, normalized size = 0.92 \[ \frac {e^{-(a+b x)^2} \left (-2 b d^2 \left (8 \left (a^2+1\right ) c+\left (2 a^2+3\right ) d x\right )+2 a \left (2 a^2+5\right ) d^3+\sqrt {\pi } e^{(a+b x)^2} \left (4 a^4 d^3-16 a^3 b c d^2+12 a^2 \left (2 b^2 c^2 d+d^3\right )-8 a \left (2 b^3 c^3+3 b c d^2\right )+3 \left (4 b^2 c^2 d+d^3\right )\right ) \text {erf}(a+b x)+4 \sqrt {\pi } b^4 x e^{(a+b x)^2} \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \text {erfc}(a+b x)+4 a b^2 d \left (6 c^2+4 c d x+d^2 x^2\right )-4 b^3 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right )}{16 \sqrt {\pi } b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 314, normalized size = 1.08 \[ \frac {4 \, \pi b^{4} d^{3} x^{4} + 16 \, \pi b^{4} c d^{2} x^{3} + 24 \, \pi b^{4} c^{2} d x^{2} + 16 \, \pi b^{4} c^{3} x - 2 \, \sqrt {\pi } {\left (2 \, b^{3} d^{3} x^{3} + 8 \, b^{3} c^{3} - 12 \, a b^{2} c^{2} d + 8 \, {\left (a^{2} + 1\right )} b c d^{2} - {\left (2 \, a^{3} + 5 \, a\right )} d^{3} + 2 \, {\left (4 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + {\left (12 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + {\left (2 \, a^{2} + 3\right )} b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} - {\left (4 \, \pi b^{4} d^{3} x^{4} + 16 \, \pi b^{4} c d^{2} x^{3} + 24 \, \pi b^{4} c^{2} d x^{2} + 16 \, \pi b^{4} c^{3} x + \pi {\left (16 \, a b^{3} c^{3} - 12 \, {\left (2 \, a^{2} + 1\right )} b^{2} c^{2} d + 8 \, {\left (2 \, a^{3} + 3 \, a\right )} b c d^{2} - {\left (4 \, a^{4} + 12 \, a^{2} + 3\right )} d^{3}\right )}\right )} \operatorname {erf}\left (b x + a\right )}{16 \, \pi b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 435, normalized size = 1.49 \[ \frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} - {\left (x \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }}\right )} c^{3} - \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {erf}\left (b x + a\right ) + \frac {\frac {\sqrt {\pi } {\left (2 \, a^{2} + 1\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi } b}\right )} c^{2} d - \frac {1}{2} \, {\left (2 \, x^{3} \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } {\left (2 \, a^{3} + 3 \, a\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {2 \, {\left (b^{2} {\left (x + \frac {a}{b}\right )}^{2} - 3 \, a b {\left (x + \frac {a}{b}\right )} + 3 \, a^{2} + 1\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi } b^{2}}\right )} c d^{2} - \frac {1}{16} \, {\left (4 \, x^{4} \operatorname {erf}\left (b x + a\right ) + \frac {\frac {\sqrt {\pi } {\left (4 \, a^{4} + 12 \, a^{2} + 3\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{3} - 8 \, a b^{2} {\left (x + \frac {a}{b}\right )}^{2} + 12 \, a^{2} b {\left (x + \frac {a}{b}\right )} - 8 \, a^{3} + 3 \, b {\left (x + \frac {a}{b}\right )} - 8 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi } b^{3}}\right )} d^{3} + c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 729, normalized size = 2.50 \[ \frac {\frac {d^{3} \mathrm {erfc}\left (b x +a \right ) \left (b x +a \right )^{4}}{4 b^{3}}-\frac {d^{3} \mathrm {erfc}\left (b x +a \right ) \left (b x +a \right )^{3} a}{b^{3}}+\frac {d^{2} \mathrm {erfc}\left (b x +a \right ) \left (b x +a \right )^{3} c}{b^{2}}+\frac {3 d^{3} \mathrm {erfc}\left (b x +a \right ) \left (b x +a \right )^{2} a^{2}}{2 b^{3}}-\frac {3 d^{2} \mathrm {erfc}\left (b x +a \right ) \left (b x +a \right )^{2} a c}{b^{2}}+\frac {3 d \,\mathrm {erfc}\left (b x +a \right ) \left (b x +a \right )^{2} c^{2}}{2 b}-\frac {d^{3} \mathrm {erfc}\left (b x +a \right ) \left (b x +a \right ) a^{3}}{b^{3}}+\frac {3 d^{2} \mathrm {erfc}\left (b x +a \right ) \left (b x +a \right ) a^{2} c}{b^{2}}-\frac {3 d \,\mathrm {erfc}\left (b x +a \right ) \left (b x +a \right ) a \,c^{2}}{b}+\mathrm {erfc}\left (b x +a \right ) \left (b x +a \right ) c^{3}+\frac {d^{3} \mathrm {erfc}\left (b x +a \right ) a^{4}}{4 b^{3}}-\frac {d^{2} \mathrm {erfc}\left (b x +a \right ) a^{3} c}{b^{2}}+\frac {3 d \,\mathrm {erfc}\left (b x +a \right ) a^{2} c^{2}}{2 b}-\mathrm {erfc}\left (b x +a \right ) a \,c^{3}+\frac {b \,\mathrm {erfc}\left (b x +a \right ) c^{4}}{4 d}+\frac {d^{4} \left (-\frac {\left (b x +a \right )^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}-\frac {3 \left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{4}+\frac {3 \sqrt {\pi }\, \erf \left (b x +a \right )}{8}\right )+\frac {a^{4} d^{4} \sqrt {\pi }\, \erf \left (b x +a \right )}{2}+\frac {b^{4} c^{4} \sqrt {\pi }\, \erf \left (b x +a \right )}{2}+2 a^{3} d^{4} {\mathrm e}^{-\left (b x +a \right )^{2}}+6 a^{2} d^{4} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )-4 a \,d^{4} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )-2 b^{3} c^{3} d \,{\mathrm e}^{-\left (b x +a \right )^{2}}+6 b^{2} c^{2} d^{2} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )+4 b c \,d^{3} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )-2 a \,b^{3} c^{3} d \sqrt {\pi }\, \erf \left (b x +a \right )+3 a^{2} b^{2} c^{2} d^{2} \sqrt {\pi }\, \erf \left (b x +a \right )-2 a^{3} b c \,d^{3} \sqrt {\pi }\, \erf \left (b x +a \right )+6 a \,b^{2} c^{2} d^{2} {\mathrm e}^{-\left (b x +a \right )^{2}}-6 a^{2} b c \,d^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}-12 a b c \,d^{3} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )}{2 \sqrt {\pi }\, b^{3} d}}{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 \[ \int {\left (d x + c\right )}^{3} \operatorname {erfc}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 352, normalized size = 1.21 \[ \frac {d^3\,x^4\,\mathrm {erfc}\left (a+b\,x\right )}{4}-\frac {\mathrm {erfc}\left (a+b\,x\right )\,\left (b^2\,\left (\frac {3\,d\,a^2\,c^2}{2}+\frac {3\,d\,c^2}{4}\right )-b\,\left (c\,a^3\,d^2+\frac {3\,c\,a\,d^2}{2}\right )+\frac {3\,d^3}{16}+\frac {3\,a^2\,d^3}{4}+\frac {a^4\,d^3}{4}-a\,b^3\,c^3\right )}{b^4}+c^3\,x\,\mathrm {erfc}\left (a+b\,x\right )+\frac {{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (2\,a^3\,d^3-8\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+5\,a\,d^3-8\,b^3\,c^3-8\,b\,c\,d^2\right )}{8\,b^4\,\sqrt {\pi }}+\frac {3\,c^2\,d\,x^2\,\mathrm {erfc}\left (a+b\,x\right )}{2}+c\,d^2\,x^3\,\mathrm {erfc}\left (a+b\,x\right )-\frac {x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (2\,a^2\,d^3-8\,a\,b\,c\,d^2+12\,b^2\,c^2\,d+3\,d^3\right )}{8\,b^3\,\sqrt {\pi }}-\frac {d^3\,x^3\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{4\,b\,\sqrt {\pi }}+\frac {x^2\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (a\,d^3-4\,b\,c\,d^2\right )}{4\,b^2\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.14, size = 746, normalized size = 2.55 \[ \begin {cases} - \frac {a^{4} d^{3} \operatorname {erfc}{\left (a + b x \right )}}{4 b^{4}} + \frac {a^{3} c d^{2} \operatorname {erfc}{\left (a + b x \right )}}{b^{3}} + \frac {a^{3} d^{3} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{4 \sqrt {\pi } b^{4}} - \frac {3 a^{2} c^{2} d \operatorname {erfc}{\left (a + b x \right )}}{2 b^{2}} - \frac {a^{2} c d^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b^{3}} - \frac {a^{2} d^{3} x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{4 \sqrt {\pi } b^{3}} - \frac {3 a^{2} d^{3} \operatorname {erfc}{\left (a + b x \right )}}{4 b^{4}} + \frac {a c^{3} \operatorname {erfc}{\left (a + b x \right )}}{b} + \frac {3 a c^{2} d e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt {\pi } b^{2}} + \frac {a c d^{2} x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b^{2}} + \frac {a d^{3} x^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{4 \sqrt {\pi } b^{2}} + \frac {3 a c d^{2} \operatorname {erfc}{\left (a + b x \right )}}{2 b^{3}} + \frac {5 a d^{3} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{8 \sqrt {\pi } b^{4}} + c^{3} x \operatorname {erfc}{\left (a + b x \right )} + \frac {3 c^{2} d x^{2} \operatorname {erfc}{\left (a + b x \right )}}{2} + c d^{2} x^{3} \operatorname {erfc}{\left (a + b x \right )} + \frac {d^{3} x^{4} \operatorname {erfc}{\left (a + b x \right )}}{4} - \frac {c^{3} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} - \frac {3 c^{2} d x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt {\pi } b} - \frac {c d^{2} x^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} - \frac {d^{3} x^{3} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{4 \sqrt {\pi } b} - \frac {3 c^{2} d \operatorname {erfc}{\left (a + b x \right )}}{4 b^{2}} - \frac {c d^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b^{3}} - \frac {3 d^{3} x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{8 \sqrt {\pi } b^{3}} - \frac {3 d^{3} \operatorname {erfc}{\left (a + b x \right )}}{16 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \operatorname {erfc}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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