Optimal. Leaf size=34 \[ \frac{(a+b x) \sinh ^{-1}(a+b x)}{b}-\frac{\sqrt{(a+b x)^2+1}}{b} \]
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Rubi [A] time = 0.0139822, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5863, 5653, 261} \[ \frac{(a+b x) \sinh ^{-1}(a+b x)}{b}-\frac{\sqrt{(a+b x)^2+1}}{b} \]
Antiderivative was successfully verified.
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Rule 5863
Rule 5653
Rule 261
Rubi steps
\begin{align*} \int \sinh ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \sinh ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \sinh ^{-1}(a+b x)}{b}-\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\sqrt{1+(a+b x)^2}}{b}+\frac{(a+b x) \sinh ^{-1}(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.026631, size = 40, normalized size = 1.18 \[ \frac{(a+b x) \sinh ^{-1}(a+b x)-\sqrt{a^2+2 a b x+b^2 x^2+1}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 31, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ( bx+a \right ){\it Arcsinh} \left ( bx+a \right ) -\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08046, size = 41, normalized size = 1.21 \begin{align*} \frac{{\left (b x + a\right )} \operatorname{arsinh}\left (b x + a\right ) - \sqrt{{\left (b x + a\right )}^{2} + 1}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.4697, size = 135, normalized size = 3.97 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.183704, size = 46, normalized size = 1.35 \begin{align*} \begin{cases} \frac{a \operatorname{asinh}{\left (a + b x \right )}}{b} + x \operatorname{asinh}{\left (a + b x \right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{b} & \text{for}\: b \neq 0 \\x \operatorname{asinh}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28512, size = 124, normalized size = 3.65 \begin{align*} -b{\left (\frac{a \log \left (-a b -{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{\left | b \right |}\right )}{b{\left | b \right |}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{2}}\right )} + x \log \left (b x + a + \sqrt{{\left (b x + a\right )}^{2} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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