Optimal. Leaf size=18 \[ -\frac{1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.0447206, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {5675} \[ -\frac{1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5675
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac{1}{b c \left (a+b \sinh ^{-1}(c x)\right )}\\ \end{align*}
Mathematica [A] time = 0.012315, size = 18, normalized size = 1. \[ -\frac{1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 19, normalized size = 1.1 \begin{align*} -{\frac{1}{bc \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16412, size = 24, normalized size = 1.33 \begin{align*} -\frac{1}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24254, size = 66, normalized size = 3.67 \begin{align*} -\frac{1}{b^{2} c \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + a b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35751, size = 41, normalized size = 2.28 \begin{align*} -\frac{1}{{\left (b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + a\right )} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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