Integrand size = 12, antiderivative size = 99 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=-\frac {a}{x}-\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x}+b \text {arcsinh}(a+b x)-\frac {a b \text {arctanh}\left (\frac {1+a^2+a b x}{\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}}\right )}{\sqrt {1+a^2}}+b \log (x) \]
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Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5878, 14, 746, 857, 633, 221, 738, 212} \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=-\frac {a b \text {arctanh}\left (\frac {a^2+a b x+1}{\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}}\right )}{\sqrt {a^2+1}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2+1}}{x}+b \text {arcsinh}(a+b x)-\frac {a}{x}+b \log (x) \]
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Rule 14
Rule 212
Rule 221
Rule 633
Rule 738
Rule 746
Rule 857
Rule 5878
Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b x+\sqrt {1+(a+b x)^2}}{x^2} \, dx \\ & = \int \left (\frac {a}{x^2}+\frac {b}{x}+\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x^2}\right ) \, dx \\ & = -\frac {a}{x}+b \log (x)+\int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x^2} \, dx \\ & = -\frac {a}{x}-\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x}+b \log (x)+\frac {1}{2} \int \frac {2 a b+2 b^2 x}{x \sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx \\ & = -\frac {a}{x}-\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x}+b \log (x)+(a b) \int \frac {1}{x \sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx+b^2 \int \frac {1}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx \\ & = -\frac {a}{x}-\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x}+b \log (x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )-(2 a b) \text {Subst}\left (\int \frac {1}{4 \left (1+a^2\right )-x^2} \, dx,x,\frac {2 \left (1+a^2\right )+2 a b x}{\sqrt {1+a^2+2 a b x+b^2 x^2}}\right ) \\ & = -\frac {a}{x}-\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x}+b \text {arcsinh}(a+b x)-\frac {a b \text {arctanh}\left (\frac {1+a^2+a b x}{\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}}\right )}{\sqrt {1+a^2}}+b \log (x) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=b \text {arcsinh}(a+b x)-\frac {a+\sqrt {1+a^2+2 a b x+b^2 x^2}+\left (-1-\frac {a}{\sqrt {1+a^2}}\right ) b x \log (x)+\frac {a b x \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{\sqrt {1+a^2}}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(283\) vs. \(2(91)=182\).
Time = 0.76 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.87
method | result | size |
default | \(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{\left (a^{2}+1\right ) x}+\frac {a b \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{a^{2}+1}+\frac {2 b^{2} \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{a^{2}+1}+b \ln \left (x \right )-\frac {a}{x}\) | \(284\) |
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Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (91) = 182\).
Time = 0.26 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.85 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=\frac {\sqrt {a^{2} + 1} a b x \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} - \sqrt {a^{2} + 1} a + 1\right )} - {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) - {\left (a^{2} + 1\right )} b x \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (a^{2} + 1\right )} b x \log \left (x\right ) - a^{3} - {\left (a^{2} + 1\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} - a}{{\left (a^{2} + 1\right )} x} \]
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\[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=\int \frac {a + b x + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{x^{2}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.72 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=-\frac {a b \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{\sqrt {a^{2} + 1}} + b \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right ) + b \log \left (x\right ) - \frac {a}{x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (91) = 182\).
Time = 0.36 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.36 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=\frac {a b \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{\sqrt {a^{2} + 1}} - \frac {b^{2} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |}\right )}{{\left | b \right |}} + b \log \left ({\left | x \right |}\right ) - \frac {a}{x} + \frac {2 \, {\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a b^{5} + a^{2} b^{4} {\left | b \right |} + b^{4} {\left | b \right |}\right )}}{{\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} - a^{2} - 1\right )} b^{4}} \]
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Time = 3.66 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.72 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=b\,\ln \left (x\right )-\frac {a}{x}+\ln \left (\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}+\frac {x\,b^2+a\,b}{\sqrt {b^2}}\right )\,\sqrt {b^2}-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{x\,\left (a^2+1\right )}+\frac {a^3\,b\,\mathrm {atanh}\left (\frac {a^2+b\,x\,a+1}{\sqrt {a^2+1}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}\right )}{{\left (a^2+1\right )}^{3/2}}-\frac {a^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{x\,\left (a^2+1\right )}+\frac {a\,b\,\mathrm {atanh}\left (\frac {a^2+b\,x\,a+1}{\sqrt {a^2+1}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}\right )}{{\left (a^2+1\right )}^{3/2}}-\frac {2\,a\,b\,\ln \left (a\,b+\frac {a^2+1}{x}+\frac {\sqrt {a^2+1}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{x}\right )}{\sqrt {a^2+1}} \]
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