Optimal. Leaf size=51 \[ \frac {(a+b x) e^{\sin ^{-1}(a+b x)}}{2 b}+\frac {\sqrt {1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4836, 4433} \[ \frac {(a+b x) e^{\sin ^{-1}(a+b x)}}{2 b}+\frac {\sqrt {1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{2 b} \]
Antiderivative was successfully verified.
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Rule 4433
Rule 4836
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int e^x \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {e^{\sin ^{-1}(a+b x)} (a+b x)}{2 b}+\frac {e^{\sin ^{-1}(a+b x)} \sqrt {1-(a+b x)^2}}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.69 \[ \frac {\left (\sqrt {1-(a+b x)^2}+a+b x\right ) e^{\sin ^{-1}(a+b x)}}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 39, normalized size = 0.76 \[ \frac {{\left (b x + a + \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.99, size = 43, normalized size = 0.84 \[ \frac {{\left (b x + a\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{2 \, b} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} e^{\left (\arcsin \left (b x + a\right )\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\arcsin \left (b x +a \right )}\, dx \]
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 \[ \int e^{\left (\arcsin \left (b x + a\right )\right )}\,{d x} \]
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 \[ \int {\mathrm {e}}^{\mathrm {asin}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 65, normalized size = 1.27 \[ \begin {cases} \frac {a e^{\operatorname {asin}{\left (a + b x \right )}}}{2 b} + \frac {x e^{\operatorname {asin}{\left (a + b x \right )}}}{2} + \frac {\sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{2 b} & \text {for}\: b \neq 0 \\x e^{\operatorname {asin}{\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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