Optimal. Leaf size=65 \[ \frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{4 \sqrt [4]{e} b}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{4 \sqrt [4]{e} b} \]
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Rubi [A] time = 0.06, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5896, 5513, 2234, 2204} \[ \frac {\sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{4 \sqrt [4]{e} b}+\frac {\sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{4 \sqrt [4]{e} b} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 5513
Rule 5896
Rubi steps
\begin {align*} \int e^{\sinh ^{-1}(a+b x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int e^{x^2} \cosh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2} e^{-x+x^2}+\frac {e^{x+x^2}}{2}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int e^{-x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b}+\frac {\operatorname {Subst}\left (\int e^{x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int e^{\frac {1}{4} (-1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b \sqrt [4]{e}}+\frac {\operatorname {Subst}\left (\int e^{\frac {1}{4} (1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b \sqrt [4]{e}}\\ &=\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (-1+2 \sinh ^{-1}(a+b x)\right )\right )}{4 b \sqrt [4]{e}}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (1+2 \sinh ^{-1}(a+b x)\right )\right )}{4 b \sqrt [4]{e}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.68 \[ \frac {\sqrt {\pi } \left (\text {erfi}\left (\sinh ^{-1}(a+b x)+\frac {1}{2}\right )+\text {erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )\right )}{4 \sqrt [4]{e} b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (\operatorname {arsinh}\left (b x + a\right )^{2}\right )}, x\right ) \]
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 \[ \int e^{\left (\operatorname {arsinh}\left (b x + a\right )^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\arcsinh \left (b x +a \right )^{2}}\, dx \]
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 \[ \int e^{\left (\operatorname {arsinh}\left (b x + a\right )^{2}\right )}\,{d x} \]
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 \[ \int {\mathrm {e}}^{{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int e^{\operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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