Optimal. Leaf size=13 \[ -\frac{1}{b \sinh ^{-1}(a+b x)} \]
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Rubi [A] time = 0.0747208, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {5867, 5675} \[ -\frac{1}{b \sinh ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 5867
Rule 5675
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2} \sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac{1}{b \sinh ^{-1}(a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0140858, size = 13, normalized size = 1. \[ -\frac{1}{b \sinh ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 14, normalized size = 1.1 \begin{align*} -{\frac{1}{b{\it Arcsinh} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.32074, size = 203, normalized size = 15.62 \begin{align*} -\frac{b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b + b\right )} x +{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}} + a}{{\left ({\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{\left (b^{2} x + a b\right )} +{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b + b\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.67799, size = 77, normalized size = 5.92 \begin{align*} -\frac{1}{b \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.73364, size = 26, normalized size = 2. \begin{align*} \begin{cases} - \frac{1}{b \operatorname{asinh}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt{a^{2} + 1} \operatorname{asinh}^{2}{\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 = 1.37482, size = 43, normalized size = 3.31 \begin{align*} -\frac{1}{b \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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