Integrand size = 12, antiderivative size = 156 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b \left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{2 \left (1+a^2\right )^2 x^2}-\frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 \left (1+a^2\right ) x^3}+\frac {a b^3 \text {arctanh}\left (\frac {1+a^2+a b x}{\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}}\right )}{2 \left (1+a^2\right )^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5878, 14, 744, 734, 738, 212} \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^4} \, dx=\frac {a b^3 \text {arctanh}\left (\frac {a^2+a b x+1}{\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}}\right )}{2 \left (a^2+1\right )^{5/2}}+\frac {a b \left (a^2+a b x+1\right ) \sqrt {a^2+2 a b x+b^2 x^2+1}}{2 \left (a^2+1\right )^2 x^2}-\frac {\left (a^2+2 a b x+b^2 x^2+1\right )^{3/2}}{3 \left (a^2+1\right ) x^3}-\frac {a}{3 x^3}-\frac {b}{2 x^2} \]
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Rule 14
Rule 212
Rule 734
Rule 738
Rule 744
Rule 5878
Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b x+\sqrt {1+(a+b x)^2}}{x^4} \, dx \\ & = \int \left (\frac {a}{x^4}+\frac {b}{x^3}+\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x^4}\right ) \, dx \\ & = -\frac {a}{3 x^3}-\frac {b}{2 x^2}+\int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x^4} \, dx \\ & = -\frac {a}{3 x^3}-\frac {b}{2 x^2}-\frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 \left (1+a^2\right ) x^3}-\frac {(a b) \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x^3} \, dx}{1+a^2} \\ & = -\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b \left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{2 \left (1+a^2\right )^2 x^2}-\frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 \left (1+a^2\right ) x^3}-\frac {\left (a b^3\right ) \int \frac {1}{x \sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx}{2 \left (1+a^2\right )^2} \\ & = -\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b \left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{2 \left (1+a^2\right )^2 x^2}-\frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 \left (1+a^2\right ) x^3}+\frac {\left (a b^3\right ) \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 )}{\left (1+a^2\right )^2} \\ & = -\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b \left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{2 \left (1+a^2\right )^2 x^2}-\frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 \left (1+a^2\right ) x^3}+\frac {a b^3 \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 )}{2 \left (1+a^2\right )^{5/2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^4} \, dx=\frac {1}{6} \left (-\frac {2 a}{x^3}-\frac {3 b}{x^2}-\frac {\sqrt {1+a^2+2 a b x+b^2 x^2} \left (2+2 a^4+a b x+a^3 b x+2 b^2 x^2+a^2 \left (4-b^2 x^2\right )\right )}{\left (1+a^2\right )^2 x^3}-\frac {3 a b^3 \log (x)}{\left (1+a^2\right )^{5/2}}+\frac {3 a b^3 \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{\left (1+a^2\right )^{5/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(501\) vs. \(2(136)=272\).
Time = 0.95 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.22
method | result | size |
default | \(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 \left (a^{2}+1\right ) x^{3}}-\frac {a b \left (-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {a b \left (-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{\left (a^{2}+1\right ) x}+\frac {a b \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{a^{2}+1}+\frac {2 b^{2} \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{a^{2}+1}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{2 a^{2}+2}\right )}{a^{2}+1}-\frac {a}{3 x^{3}}-\frac {b}{2 x^{2}}\) | \(502\) |
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Time = 0.25 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.47 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^4} \, dx=\frac {3 \, \sqrt {a^{2} + 1} a b^{3} x^{3} \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + \sqrt {a^{2} + 1} a + 1\right )} + {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) - 2 \, a^{7} + {\left (a^{4} - a^{2} - 2\right )} b^{3} x^{3} - 6 \, a^{5} - 6 \, a^{3} - 3 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} b x - {\left (2 \, a^{6} - {\left (a^{4} - a^{2} - 2\right )} b^{2} x^{2} + 6 \, a^{4} + {\left (a^{5} + 2 \, a^{3} + a\right )} b x + 6 \, a^{2} + 2\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, a}{6 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3}} \]
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\[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^4} \, dx=\int \frac {a + b x + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{x^{4}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (136) = 272\).
Time = 0.36 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.26 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^4} \, dx=-\frac {a^{3} b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {5}{2}}} + \frac {a b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a b^{3}}{2 \, {\left (a^{2} + 1\right )}^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} b^{2}}{2 \, {\left (a^{2} + 1\right )}^{2} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a b}{2 \, {\left (a^{2} + 1\right )}^{2} x^{2}} - \frac {b}{2 \, x^{2}} - \frac {a}{3 \, x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (a^{2} + 1\right )} x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (136) = 272\).
Time = 0.33 (sec) , antiderivative size = 715, normalized size of antiderivative = 4.58 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^4} \, dx=-\frac {a b^{3} \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 )}{2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )} \sqrt {a^{2} + 1}} - \frac {3 \, b x + 2 \, a}{6 \, x^{3}} + \frac {20 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{5} b^{3} + 12 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{7} b^{3} + 6 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{4} a^{4} b^{2} {\left | b \right |} + 24 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{6} b^{2} {\left | b \right |} + 2 \, a^{8} b^{2} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{5} a b^{3} + 32 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{3} b^{3} + 33 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{5} b^{3} + 12 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{4} a^{2} b^{2} {\left | b \right |} + 48 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{4} b^{2} {\left | b \right |} + 8 \, a^{6} b^{2} {\left | b \right |} + 12 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a b^{3} + 30 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{3} b^{3} + 6 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{4} b^{2} {\left | b \right |} + 24 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{2} b^{2} {\left | b \right |} + 12 \, a^{4} b^{2} {\left | b \right |} + 9 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a b^{3} + 8 \, a^{2} b^{2} {\left | b \right |} + 2 \, b^{2} {\left | b \right |}}{3 \, {\left (a^{4} + 2 \, a^{2} + 1\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 )}^{3}} \]
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Timed out. \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^4} \, dx=\int \frac {a+\sqrt {{\left (a+b\,x\right )}^2+1}+b\,x}{x^4} \,d x \]
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