Optimal. Leaf size=61 \[ \frac {a+b \cosh ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5716, 39} \[ \frac {a+b \cosh ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 39
Rule 5716
Rubi steps
\begin {align*} \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {a+b \cosh ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 c d^2}\\ &=-\frac {b x}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 53, normalized size = 0.87 \[ \frac {a+b c x \sqrt {c x-1} \sqrt {c x+1}+b \cosh ^{-1}(c x)}{2 c^2 d^2-2 c^4 d^2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 65, normalized size = 1.07 \[ -\frac {a c^{2} x^{2} + \sqrt {c^{2} x^{2} - 1} b c x + b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 64, normalized size = 1.05 \[ \frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\mathrm {arccosh}\left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {c x}{2 \sqrt {c x -1}\, \sqrt {c x +1}}\right )}{d^{2}}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 134, normalized size = 2.20 \[ -\frac {1}{4} \, {\left ({\left (\frac {\sqrt {c^{2} x^{2} - 1} c^{2} d^{2}}{c^{7} d^{4} x + c^{6} d^{4}} + \frac {\sqrt {c^{2} x^{2} - 1} c^{2} d^{2}}{c^{7} d^{4} x - c^{6} d^{4}}\right )} c^{2} + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{c^{4} d^{2} x^{2} - c^{2} d^{2}}\right )} b - \frac {a}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a x}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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