Optimal. Leaf size=56 \[ \frac {e^c \sqrt {\pi } \text {Erf}(b x)^2}{8 b}-\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]
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Rubi [A]
time = 0.04, 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 = {6545, 6508, 30,
6511} \begin {gather*} \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 }} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 6508
Rule 6511
Rule 6545
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 ) \text {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 = 0.03, size = 61, normalized size = 1.09 \begin {gather*} \frac {(\cosh (c)-\sinh (c)) \left (-4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )+\pi \text {Erf}(b x)^2 (\cosh (2 c)+\sinh (2 c))\right )}{8 b \sqrt {\pi }} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \sinh {\left (b^{2} x^{2} - c \right )} \operatorname {erf}{\left (b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \mathrm {sinh}\left (c-b^2\,x^2\right )\,\mathrm {erf}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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