Optimal. Leaf size=63 \[ \frac{b c \sqrt{c^2 x^2+1} \log (x)}{\sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{d x} \]
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Rubi [A] time = 0.0891756, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {5723, 29} \[ \frac{b c \sqrt{c^2 x^2+1} \log (x)}{\sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{d x} \]
Antiderivative was successfully verified.
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Rule 5723
Rule 29
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^2 \sqrt{d+c^2 d x^2}} \, dx &=-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x} \, dx}{\sqrt{d+c^2 d x^2}}\\ &=-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac{b c \sqrt{1+c^2 x^2} \log (x)}{\sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.158615, size = 67, normalized size = 1.06 \[ \frac{b c \log (x) \sqrt{d \left (c^2 x^2+1\right )}}{d \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{d x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.118, size = 183, normalized size = 2.9 \begin{align*} -{\frac{a}{dx}\sqrt{{c}^{2}d{x}^{2}+d}}-{\frac{b{\it Arcsinh} \left ( cx \right ) c}{d}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{b{\it Arcsinh} \left ( cx \right ) x{c}^{2}}{d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{ \left ({c}^{2}{x}^{2}+1 \right ) dx}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{bc}{d}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2}-1 \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \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 = 3.12669, size = 301, normalized size = 4.78 \begin{align*} \frac{b c \sqrt{d} x \log \left (\frac{c^{2} d x^{6} + c^{2} d x^{2} + d x^{4} + \sqrt{c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} + 1}{\left (x^{4} - 1\right )} \sqrt{d} + d}{c^{2} x^{4} + x^{2}}\right ) - 2 \, \sqrt{c^{2} d x^{2} + d} b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 2 \, \sqrt{c^{2} d x^{2} + d} a}{2 \, d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asinh}{\left (c x \right )}}{x^{2} \sqrt{d \left (c^{2} x^{2} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\left (c x\right ) + a}{\sqrt{c^{2} d x^{2} + d} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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