Optimal. Leaf size=56 \[ \frac {\sqrt {\pi } e^c \text {erf}(b x)^2}{8 b}-\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]
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Rubi [A] time = 0.06, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6410, 6373, 30, 6376} \[ \frac {\sqrt {\pi } e^c \text {Erf}(b x)^2}{8 b}-\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6373
Rule 6376
Rule 6410
Rubi steps
\begin {align*} \int \text {erf}(b x) \sinh \left (c-b^2 x^2\right ) \, dx &=\frac {1}{2} \int e^{c-b^2 x^2} \text {erf}(b x) \, dx-\frac {1}{2} \int e^{-c+b^2 x^2} \text {erf}(b x) \, dx\\ &=-\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}+\frac {\left (e^c \sqrt {\pi }\right ) \operatorname {Subst}(\int x \, dx,x,\text {erf}(b x))}{4 b}\\ &=\frac {e^c \sqrt {\pi } \text {erf}(b x)^2}{8 b}-\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 61, normalized size = 1.09 \[ \frac {(\cosh (c)-\sinh (c)) \left (\pi \text {erf}(b x)^2 (\sinh (2 c)+\cosh (2 c))-4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )\right )}{8 \sqrt {\pi } b} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\operatorname {erf}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right ), x\right ) \]
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 \[ \int -\operatorname {erf}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int -\erf \left (b x \right ) \sinh \left (b^{2} x^{2}-c \right )\, dx \]
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 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (b x\right )^{2} e^{c}}{8 \, b} - \frac {1}{2} \, \int \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} - c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {sinh}\left (c-b^2\,x^2\right )\,\mathrm {erf}\left (b\,x\right ) \,d x \]
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 \[ - \int \sinh {\left (b^{2} x^{2} - c \right )} \operatorname {erf}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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