Optimal. Leaf size=57 \[ \frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi }}+\frac {\sqrt {\pi } e^{-c} \text {erfi}(b x)^2}{8 b} \]
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Rubi [A] time = 0.05, antiderivative size = 57, 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 = {6415, 6378, 6375, 30} \[ \frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi }}+\frac {\sqrt {\pi } e^{-c} \text {Erfi}(b x)^2}{8 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6375
Rule 6378
Rule 6415
Rubi steps
\begin {align*} \int \cosh \left (c-b^2 x^2\right ) \text {erfi}(b x) \, dx &=\frac {1}{2} \int e^{c-b^2 x^2} \text {erfi}(b x) \, dx+\frac {1}{2} \int e^{-c+b^2 x^2} \text {erfi}(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 {erfi}(b x))}{4 b}\\ &=\frac {e^{-c} \sqrt {\pi } \text {erfi}(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 = 1.96, size = 72, normalized size = 1.26 \[ \frac {\pi \text {erfi}(b x) (2 \text {erf}(b x) (\sinh (c)+\cosh (c))+\text {erfi}(b x) (\cosh (c)-\sinh (c)))-4 b^2 x^2 (\sinh (c)+\cosh (c)) \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{8 \sqrt {\pi } b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b^{2} x^{2} - c\right ) \operatorname {erfi}\left (b x\right ), x\right ) \]
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 \[ \int \cosh \left (b^{2} x^{2} - c\right ) \operatorname {erfi}\left (b x\right )\,{d x} \]
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 \[ \int \cosh \left (b^{2} x^{2}-c \right ) \erfi \left (b x \right )\, dx \]
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 \[ \int \cosh \left (b^{2} x^{2} - c\right ) \operatorname {erfi}\left (b x\right )\,{d x} \]
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 \[ \int \mathrm {cosh}\left (c-b^2\,x^2\right )\,\mathrm {erfi}\left (b\,x\right ) \,d x \]
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 \[ \int \cosh {\left (b^{2} x^{2} - c \right )} \operatorname {erfi}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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