Integrand size = 15, antiderivative size = 56 \[ \int \text {erf}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=-\frac {e^{-c} \sqrt {\pi } \text {erf}(b x)^2}{8 b}+\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]
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Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6545, 6511, 6508, 30} \[ \int \text {erf}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}-\frac {\sqrt {\pi } e^{-c} \text {erf}(b x)^2}{8 b} \]
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Rule 30
Rule 6508
Rule 6511
Rule 6545
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int e^{-c-b^2 x^2} \text {erf}(b x) \, dx\right )+\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*}
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02 \[ \int \text {erf}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\frac {\pi \text {erf}(b x)^2 (-\cosh (c)+\sinh (c))+4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right ) (\cosh (c)+\sinh (c))}{8 b \sqrt {\pi }} \]
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\[\int \operatorname {erf}\left (b x \right ) \sinh \left (b^{2} x^{2}+c \right )d x\]
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\[ \int \text {erf}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\int { \operatorname {erf}\left (b x\right ) \sinh \left (b^{2} x^{2} + c\right ) \,d x } \]
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\[ \int \text {erf}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\int \sinh {\left (b^{2} x^{2} + c \right )} \operatorname {erf}{\left (b x \right )}\, dx \]
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\[ \int \text {erf}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\int { \operatorname {erf}\left (b x\right ) \sinh \left (b^{2} x^{2} + c\right ) \,d x } \]
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\[ \int \text {erf}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\int { \operatorname {erf}\left (b x\right ) \sinh \left (b^{2} x^{2} + c\right ) \,d x } \]
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Timed out. \[ \int \text {erf}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\int \mathrm {sinh}\left (b^2\,x^2+c\right )\,\mathrm {erf}\left (b\,x\right ) \,d x \]
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