Optimal. Leaf size=15 \[ \frac{\sinh ^{-1}(a+b x)^3}{3 b} \]
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Rubi [A] time = 0.0680001, antiderivative size = 15, 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} \[ \frac{\sinh ^{-1}(a+b x)^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 5867
Rule 5675
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a+b x)^2}{\sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sinh ^{-1}(x)^2}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\sinh ^{-1}(a+b x)^3}{3 b}\\ \end{align*}
Mathematica [A] time = 0.017671, size = 15, normalized size = 1. \[ \frac{\sinh ^{-1}(a+b x)^3}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{3}}{3\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.58049, size = 78, normalized size = 5.2 \begin{align*} \frac{\log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.988957, size = 26, normalized size = 1.73 \begin{align*} \begin{cases} \frac{\operatorname{asinh}^{3}{\left (a + b x \right )}}{3 b} & \text{for}\: b \neq 0 \\\frac{x \operatorname{asinh}^{2}{\left (a \right )}}{\sqrt{a^{2} + 1}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.50459, size = 43, normalized size = 2.87 \begin{align*} \frac{\log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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