Optimal. Leaf size=355 \[ -\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}+\frac {2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {Erf}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {Erf}(a+b x)}{3 x}+\frac {2}{3} a b^2 \sqrt {b^2-d} e^{c+\frac {a^2 d}{b^2-d}} \text {Erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )+\frac {4 a^2 b^3 \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}-\frac {2 b \left (b^2-d\right ) \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {4 b d \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {4}{3} d^2 \text {Int}\left (e^{c+d x^2} \text {Erf}(a+b x),x\right ) \]
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Rubi [A]
time = 0.61, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {e^{c+d x^2} \text {Erf}(a+b x)}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^4} \, dx &=-\frac {e^{c+d x^2} \text {erf}(a+b x)}{3 x^3}+\frac {1}{3} (2 d) \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^2} \, dx+\frac {(2 b) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x^3} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+d x^2} \text {erf}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erf}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erf}(a+b x) \, dx-\frac {\left (2 a b^2\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x^2} \, dx}{3 \sqrt {\pi }}-\frac {\left (2 b \left (b^2-d\right )\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {(4 b d) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}+\frac {2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erf}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erf}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erf}(a+b x) \, dx+\frac {\left (4 a^2 b^3\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}-\frac {\left (2 b \left (b^2-d\right )\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (4 a b^2 \left (b^2-d\right )\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{3 \sqrt {\pi }}+\frac {(4 b d) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}+\frac {2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erf}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erf}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erf}(a+b x) \, dx+\frac {\left (4 a^2 b^3\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}-\frac {\left (2 b \left (b^2-d\right )\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {(4 b d) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (4 a b^2 \left (b^2-d\right ) e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}+\frac {2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erf}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erf}(a+b x)}{3 x}+\frac {2}{3} a b^2 \sqrt {b^2-d} e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erf}(a+b x) \, dx+\frac {\left (4 a^2 b^3\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}-\frac {\left (2 b \left (b^2-d\right )\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {(4 b d) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{c+d x^2} \text {Erf}(a+b x)}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{d \,x^{2}+c} \erf \left (b x +a \right )}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{c} \int \frac {e^{d x^{2}} \operatorname {erf}{\left (a + b x \right )}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {erf}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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