Integrand size = 26, antiderivative size = 105 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {c \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b \sqrt {1+c^2 x^2}}+\frac {b c \sqrt {d+c^2 d x^2} \log (x)}{\sqrt {1+c^2 x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5805, 29, 5783} \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {c \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{2 b \sqrt {c^2 x^2+1}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{x}+\frac {b c \log (x) \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}} \]
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Rule 29
Rule 5783
Rule 5805
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {\left (b c \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{x} \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (c^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {c \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b \sqrt {1+c^2 x^2}}+\frac {b c \sqrt {d+c^2 d x^2} \log (x)}{\sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=-\frac {a \sqrt {d \left (1+c^2 x^2\right )}}{x}+\frac {b c \sqrt {d \left (1+c^2 x^2\right )} \left (-\frac {2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{c x}+\text {arcsinh}(c x)^2+2 \log (c x)\right )}{2 \sqrt {1+c^2 x^2}}+a c \sqrt {d} \log \left (c d x+\sqrt {d} \sqrt {d \left (1+c^2 x^2\right )}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(250\) vs. \(2(93)=186\).
Time = 0.21 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.39
method | result | size |
default | \(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+a \,c^{2} x \sqrt {c^{2} d \,x^{2}+d}+\frac {a \,c^{2} d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} c}{2 \sqrt {c^{2} x^{2}+1}}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) c}{\sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{x \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}}\right )\) | \(251\) |
parts | \(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+a \,c^{2} x \sqrt {c^{2} d \,x^{2}+d}+\frac {a \,c^{2} d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} c}{2 \sqrt {c^{2} x^{2}+1}}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) c}{\sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{x \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}}\right )\) | \(251\) |
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\[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int { \frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int { \frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d}}{x^2} \,d x \]
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