Integrand size = 24, antiderivative size = 174 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {8}{3} b c^3 d^3 \sqrt {1+c^2 x^2}-\frac {b c d^3 \sqrt {1+c^2 x^2}}{6 x^2}-\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))-\frac {17}{6} b c^3 d^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 5803, 12, 1813, 1635, 911, 1167, 214} \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+3 c^4 d^3 x (a+b \text {arcsinh}(c x))-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {17}{6} b c^3 d^3 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {b c d^3 \sqrt {c^2 x^2+1}}{6 x^2}-\frac {1}{9} b c^3 d^3 \left (c^2 x^2+1\right )^{3/2}-\frac {8}{3} b c^3 d^3 \sqrt {c^2 x^2+1} \]
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Rule 12
Rule 214
Rule 276
Rule 911
Rule 1167
Rule 1635
Rule 1813
Rule 5803
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))-(b c) \int \frac {d^3 \left (-1-9 c^2 x^2+9 c^4 x^4+c^6 x^6\right )}{3 x^3 \sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))-\frac {1}{3} \left (b c d^3\right ) \int \frac {-1-9 c^2 x^2+9 c^4 x^4+c^6 x^6}{x^3 \sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} \left (b c d^3\right ) \text {Subst}\left (\int \frac {-1-9 c^2 x+9 c^4 x^2+c^6 x^3}{x^2 \sqrt {1+c^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {b c d^3 \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {1}{6} \left (b c d^3\right ) \text {Subst}\left (\int \frac {\frac {17 c^2}{2}-9 c^4 x-c^6 x^2}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {b c d^3 \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {\left (b d^3\right ) \text {Subst}\left (\int \frac {\frac {33 c^2}{2}-7 c^2 x^2-c^2 x^4}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 c} \\ & = -\frac {b c d^3 \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {\left (b d^3\right ) \text {Subst}\left (\int \left (-8 c^4-c^4 x^2+\frac {17 c^2}{2 \left (-\frac {1}{c^2}+\frac {x^2}{c^2}\right )}\right ) \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 c} \\ & = -\frac {8}{3} b c^3 d^3 \sqrt {1+c^2 x^2}-\frac {b c d^3 \sqrt {1+c^2 x^2}}{6 x^2}-\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {1}{6} \left (17 b c d^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right ) \\ & = -\frac {8}{3} b c^3 d^3 \sqrt {1+c^2 x^2}-\frac {b c d^3 \sqrt {1+c^2 x^2}}{6 x^2}-\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2}-\frac {d^3 (a+b \text {arcsinh}(c x))}{3 x^3}-\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arcsinh}(c x))+\frac {1}{3} c^6 d^3 x^3 (a+b \text {arcsinh}(c x))-\frac {17}{6} b c^3 d^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {d^3 \left (-6 a-54 a c^2 x^2+54 a c^4 x^4+6 a c^6 x^6-3 b c x \sqrt {1+c^2 x^2}-50 b c^3 x^3 \sqrt {1+c^2 x^2}-2 b c^5 x^5 \sqrt {1+c^2 x^2}+6 b \left (-1-9 c^2 x^2+9 c^4 x^4+c^6 x^6\right ) \text {arcsinh}(c x)+51 b c^3 x^3 \log (x)-51 b c^3 x^3 \log \left (1+\sqrt {1+c^2 x^2}\right )\right )}{18 x^3} \]
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Time = 0.16 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.88
method | result | size |
parts | \(d^{3} a \left (\frac {c^{6} x^{3}}{3}+3 c^{4} x -\frac {3 c^{2}}{x}-\frac {1}{3 x^{3}}\right )+d^{3} b \,c^{3} \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\) | \(153\) |
derivativedivides | \(c^{3} \left (d^{3} a \left (\frac {c^{3} x^{3}}{3}+3 c x -\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(155\) |
default | \(c^{3} \left (d^{3} a \left (\frac {c^{3} x^{3}}{3}+3 c x -\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(155\) |
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Time = 0.29 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.66 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {6 \, a c^{6} d^{3} x^{6} + 54 \, a c^{4} d^{3} x^{4} - 51 \, b c^{3} d^{3} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1} + 1\right ) + 51 \, b c^{3} d^{3} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1} - 1\right ) - 54 \, a c^{2} d^{3} x^{2} - 6 \, {\left (b c^{6} + 9 \, b c^{4} - 9 \, b c^{2} - b\right )} d^{3} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, a d^{3} + 6 \, {\left (b c^{6} d^{3} x^{6} + 9 \, b c^{4} d^{3} x^{4} - 9 \, b c^{2} d^{3} x^{2} - {\left (b c^{6} + 9 \, b c^{4} - 9 \, b c^{2} - b\right )} d^{3} x^{3} - b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (2 \, b c^{5} d^{3} x^{5} + 50 \, b c^{3} d^{3} x^{3} + 3 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}}{18 \, x^{3}} \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=d^{3} \left (\int 3 a c^{4}\, dx + \int \frac {a}{x^{4}}\, dx + \int \frac {3 a c^{2}}{x^{2}}\, dx + \int a c^{6} x^{2}\, dx + \int 3 b c^{4} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {3 b c^{2} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int b c^{6} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
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Time = 0.19 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.20 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {1}{3} \, a c^{6} d^{3} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{6} d^{3} + 3 \, a c^{4} d^{3} x + 3 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b c^{3} d^{3} - 3 \, {\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} b c^{2} d^{3} + \frac {1}{6} \, {\left ({\left (c^{2} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{x^{2}}\right )} c - \frac {2 \, \operatorname {arsinh}\left (c x\right )}{x^{3}}\right )} b d^{3} - \frac {3 \, a c^{2} d^{3}}{x} - \frac {a d^{3}}{3 \, x^{3}} \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3}{x^4} \,d x \]
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