Optimal. Leaf size=75 \[ -\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}+\frac {\sqrt {\pi } e^{-c} \text {erfc}(b x)^2}{8 b}+\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{4 b} \]
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Rubi [A] time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6411, 6377, 2204, 6376, 6374, 30} \[ -\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}+\frac {\sqrt {\pi } e^{-c} \text {Erfc}(b x)^2}{8 b}+\frac {\sqrt {\pi } e^c \text {Erfi}(b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2204
Rule 6374
Rule 6376
Rule 6377
Rule 6411
Rubi steps
\begin {align*} \int \text {erfc}(b x) \sinh \left (c+b^2 x^2\right ) \, dx &=-\left (\frac {1}{2} \int e^{-c-b^2 x^2} \text {erfc}(b x) \, dx\right )+\frac {1}{2} \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx\\ &=\frac {1}{2} \int e^{c+b^2 x^2} \, dx-\frac {1}{2} \int e^{c+b^2 x^2} \text {erf}(b x) \, dx+\frac {\left (e^{-c} \sqrt {\pi }\right ) \operatorname {Subst}(\int x \, dx,x,\text {erfc}(b x))}{4 b}\\ &=\frac {e^{-c} \sqrt {\pi } \text {erfc}(b x)^2}{8 b}+\frac {e^c \sqrt {\pi } \text {erfi}(b x)}{4 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 = 0.20, size = 83, normalized size = 1.11 \[ \frac {\pi \left (\text {erf}(b x)^2 (\cosh (c)-\sinh (c))-2 \text {erf}(b x) (\cosh (c)-\sinh (c))+2 \text {erfi}(b x) (\sinh (c)+\cosh (c))\right )-4 b^2 x^2 (\sinh (c)+\cosh (c)) \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{8 \sqrt {\pi } b} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\operatorname {erf}\left (b x\right ) - 1\right )} \sinh \left (b^{2} x^{2} + c\right ), x\right ) \]
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 \[ \int \operatorname {erfc}\left (b x\right ) \sinh \left (b^{2} x^{2} + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \mathrm {erfc}\left (b x \right ) \sinh \left (b^{2} x^{2}+c \right )\, dx \]
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 \[ \int \operatorname {erfc}\left (b x\right ) \sinh \left (b^{2} x^{2} + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {sinh}\left (b^2\,x^2+c\right )\,\mathrm {erfc}\left (b\,x\right ) \,d x \]
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 \[ \int \sinh {\left (b^{2} x^{2} + c \right )} \operatorname {erfc}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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