Optimal. Leaf size=55 \[ \frac {b x}{2 c d^2 \sqrt {c^2 x^2+1}}-\frac {a+b \sinh ^{-1}(c x)}{2 c^2 d^2 \left (c^2 x^2+1\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5717, 191} \[ \frac {b x}{2 c d^2 \sqrt {c^2 x^2+1}}-\frac {a+b \sinh ^{-1}(c x)}{2 c^2 d^2 \left (c^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 191
Rule 5717
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^2} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{2 c^2 d^2 \left (1+c^2 x^2\right )}+\frac {b \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 c d^2}\\ &=\frac {b x}{2 c d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \sinh ^{-1}(c x)}{2 c^2 d^2 \left (1+c^2 x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 74, normalized size = 1.35 \[ -\frac {a}{2 c^2 d^2 \left (c^2 x^2+1\right )}+\frac {b x}{2 c d^2 \sqrt {c^2 x^2+1}}-\frac {b \sinh ^{-1}(c x)}{2 c^2 d^2 \left (c^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 65, normalized size = 1.18 \[ \frac {a c^{2} x^{2} + \sqrt {c^{2} x^{2} + 1} b c x - b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{2 \, {\left (c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 61, normalized size = 1.11 \[ \frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \left (-\frac {\arcsinh \left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {c x}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2}}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{4} \, b {\left (\frac {2 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 1}{c^{4} d^{2} x^{2} + c^{2} d^{2}} - 4 \, \int \frac {1}{2 \, {\left (c^{6} d^{2} x^{5} + 2 \, c^{4} d^{2} x^{3} + c^{2} d^{2} x + {\left (c^{5} d^{2} x^{4} + 2 \, c^{3} d^{2} x^{2} + c d^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac {a}{2 \, {\left (c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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