Integrand size = 10, antiderivative size = 129 \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=-\frac {b \sqrt {1+(a+b x)^2}}{6 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2}}{2 \left (1+a^2\right )^2 x}-\frac {\text {arcsinh}(a+b x)}{3 x^3}+\frac {\left (1-2 a^2\right ) b^3 \text {arctanh}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{6 \left (1+a^2\right )^{5/2}} \]
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Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5859, 5828, 759, 821, 739, 212} \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=\frac {\left (1-2 a^2\right ) b^3 \text {arctanh}\left (\frac {a (a+b x)+1}{\sqrt {a^2+1} \sqrt {(a+b x)^2+1}}\right )}{6 \left (a^2+1\right )^{5/2}}+\frac {a b^2 \sqrt {(a+b x)^2+1}}{2 \left (a^2+1\right )^2 x}-\frac {b \sqrt {(a+b x)^2+1}}{6 \left (a^2+1\right ) x^2}-\frac {\text {arcsinh}(a+b x)}{3 x^3} \]
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Rule 212
Rule 739
Rule 759
Rule 821
Rule 5828
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\text {arcsinh}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^4} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\text {arcsinh}(a+b x)}{3 x^3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {b \sqrt {1+(a+b x)^2}}{6 \left (1+a^2\right ) x^2}-\frac {\text {arcsinh}(a+b x)}{3 x^3}-\frac {b^2 \text {Subst}\left (\int \frac {\frac {2 a}{b}+\frac {x}{b}}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sqrt {1+x^2}} \, dx,x,a+b x\right )}{6 \left (1+a^2\right )} \\ & = -\frac {b \sqrt {1+(a+b x)^2}}{6 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2}}{2 \left (1+a^2\right )^2 x}-\frac {\text {arcsinh}(a+b x)}{3 x^3}-\frac {\left (\left (1-2 a^2\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1+x^2}} \, dx,x,a+b x\right )}{6 \left (1+a^2\right )^2} \\ & = -\frac {b \sqrt {1+(a+b x)^2}}{6 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2}}{2 \left (1+a^2\right )^2 x}-\frac {\text {arcsinh}(a+b x)}{3 x^3}+\frac {\left (\left (1-2 a^2\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{b^2}+\frac {a^2}{b^2}-x^2} \, dx,x,\frac {\frac {1}{b}+\frac {a (a+b x)}{b}}{\sqrt {1+(a+b x)^2}}\right )}{6 \left (1+a^2\right )^2} \\ & = -\frac {b \sqrt {1+(a+b x)^2}}{6 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2}}{2 \left (1+a^2\right )^2 x}-\frac {\text {arcsinh}(a+b x)}{3 x^3}+\frac {\left (1-2 a^2\right ) b^3 \text {arctanh}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{6 \left (1+a^2\right )^{5/2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.16 \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=\frac {-\sqrt {1+a^2} b x \left (1+a^2-3 a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}-2 \left (1+a^2\right )^{5/2} \text {arcsinh}(a+b x)+\left (-1+2 a^2\right ) b^3 x^3 \log (x)+\left (1-2 a^2\right ) b^3 x^3 \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{6 \left (1+a^2\right )^{5/2} x^3} \]
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Time = 0.02 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.57
method | result | size |
parts | \(-\frac {\operatorname {arcsinh}\left (b x +a \right )}{3 x^{3}}+\frac {b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {3 a b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{3}\) | \(203\) |
derivativedivides | \(b^{3} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{3 b^{3} x^{3}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{6 \left (a^{2}+1\right ) b^{2} x^{2}}-\frac {a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{6 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(213\) |
default | \(b^{3} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{3 b^{3} x^{3}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{6 \left (a^{2}+1\right ) b^{2} x^{2}}-\frac {a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{6 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(213\) |
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Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (111) = 222\).
Time = 0.29 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.21 \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=\frac {{\left (2 \, a^{2} - 1\right )} \sqrt {a^{2} + 1} b^{3} x^{3} \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} - \sqrt {a^{2} + 1} a + 1\right )} - {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) + 3 \, {\left (a^{3} + a\right )} b^{3} x^{3} + 2 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 2 \, {\left (a^{6} + 3 \, a^{4} - {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3} + 3 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (3 \, {\left (a^{3} + a\right )} b^{2} x^{2} - {\left (a^{4} + 2 \, a^{2} + 1\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{6 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3}} \]
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\[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=\int \frac {\operatorname {asinh}{\left (a + b x \right )}}{x^{4}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (111) = 222\).
Time = 0.18 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.20 \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=-\frac {1}{6} \, {\left (\frac {3 \, a^{2} b^{2} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {5}{2}}} - \frac {b^{2} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a b}{{\left (a^{2} + 1\right )}^{2} x} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (a^{2} + 1\right )} x^{2}}\right )} b - \frac {\operatorname {arsinh}\left (b x + a\right )}{3 \, x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (111) = 222\).
Time = 0.38 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.95 \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=\frac {1}{6} \, b {\left (\frac {{\left (2 \, a^{2} b^{2} - b^{2}\right )} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} + 1\right )} \sqrt {a^{2} + 1}} - \frac {2 \, {\left (2 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{2} b^{2} - 6 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{4} b^{2} - 4 \, a^{5} b {\left | b \right |} - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} b^{2} - 7 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{2} b^{2} - 8 \, a^{3} b {\left | b \right |} - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} b^{2} - 4 \, a b {\left | b \right |}\right )}}{{\left (a^{4} + 2 \, a^{2} + 1\right )} {\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} - a^{2} - 1\right )}^{2}}\right )} - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}{3 \, x^{3}} \]
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Timed out. \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=\int \frac {\mathrm {asinh}\left (a+b\,x\right )}{x^4} \,d x \]
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