Integrand size = 22, antiderivative size = 55 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {b x}{2 c d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 d^2 \left (1+c^2 x^2\right )} \]
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Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5798, 197} \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {b x}{2 c d^2 \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 d^2 \left (c^2 x^2+1\right )} \]
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Rule 197
Rule 5798
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arcsinh}(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 \text {arcsinh}(c x)}{2 c^2 d^2 \left (1+c^2 x^2\right )} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.35 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=-\frac {a}{2 c^2 d^2 \left (1+c^2 x^2\right )}+\frac {b x}{2 c d^2 \sqrt {1+c^2 x^2}}-\frac {b \text {arcsinh}(c x)}{2 c^2 d^2 \left (1+c^2 x^2\right )} \]
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Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {c x}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2}}}{c^{2}}\) | \(61\) |
default | \(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {c x}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2}}}{c^{2}}\) | \(61\) |
parts | \(-\frac {a}{2 d^{2} c^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {c x}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2} c^{2}}\) | \(63\) |
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Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {a c^{2} x^{2} + \sqrt {c^{2} x^{2} + 1} b c x - b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{2 \, {\left (c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}} \]
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\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
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\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]
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