Optimal. Leaf size=80 \[ -\frac{a+b \sinh ^{-1}(c x)}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac{b x}{6 c d^3 \sqrt{c^2 x^2+1}}+\frac{b x}{12 c d^3 \left (c^2 x^2+1\right )^{3/2}} \]
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Rubi [A] time = 0.053746, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {5717, 192, 191} \[ -\frac{a+b \sinh ^{-1}(c x)}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac{b x}{6 c d^3 \sqrt{c^2 x^2+1}}+\frac{b x}{12 c d^3 \left (c^2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5717
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^3} \, dx &=-\frac{a+b \sinh ^{-1}(c x)}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac{b \int \frac{1}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 c d^3}\\ &=\frac{b x}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{a+b \sinh ^{-1}(c x)}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac{b \int \frac{1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{6 c d^3}\\ &=\frac{b x}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{b x}{6 c d^3 \sqrt{1+c^2 x^2}}-\frac{a+b \sinh ^{-1}(c x)}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0699482, size = 56, normalized size = 0.7 \[ \frac{-3 a+b c x \sqrt{c^2 x^2+1} \left (2 c^2 x^2+3\right )-3 b \sinh ^{-1}(c x)}{12 d^3 \left (c^3 x^2+c\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 76, normalized size = 1. \begin{align*}{\frac{1}{{c}^{2}} \left ( -{\frac{a}{4\,{d}^{3} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{b}{{d}^{3}} \left ( -{\frac{{\it Arcsinh} \left ( cx \right ) }{4\, \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{cx}{12} \left ({c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}+{\frac{cx}{6}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{16} \, b{\left (\frac{4 \, \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 1}{c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}} - 16 \, \int \frac{1}{4 \,{\left (c^{8} d^{3} x^{7} + 3 \, c^{6} d^{3} x^{5} + 3 \, c^{4} d^{3} x^{3} + c^{2} d^{3} x +{\left (c^{7} d^{3} x^{6} + 3 \, c^{5} d^{3} x^{4} + 3 \, c^{3} d^{3} x^{2} + c d^{3}\right )} \sqrt{c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac{a}{4 \,{\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.34944, size = 207, normalized size = 2.59 \begin{align*} \frac{3 \, a c^{4} x^{4} + 6 \, a c^{2} x^{2} - 3 \, b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (2 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt{c^{2} x^{2} + 1}}{12 \,{\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \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*} \frac{\int \frac{a x}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac{b x \operatorname{asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \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{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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