Integrand size = 24, antiderivative size = 249 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^3} \, dx=-\frac {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{32} b c^2 d^3 \text {arcsinh}(c x)+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))^2}{2 b}+3 c^2 d^3 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\frac {3}{2} b c^2 d^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5802, 283, 201, 221, 5801, 5775, 3797, 2221, 2317, 2438} \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^3} \, dx=-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {3}{4} c^2 d^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))+\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))^2}{2 b}+3 c^2 d^3 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-\frac {3}{2} b c^2 d^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {3}{32} b c^2 d^3 \text {arcsinh}(c x)-\frac {b c d^3 \left (c^2 x^2+1\right )^{5/2}}{2 x}+\frac {7}{16} b c^3 d^3 x \left (c^2 x^2+1\right )^{3/2}-\frac {3}{32} b c^3 d^3 x \sqrt {c^2 x^2+1} \]
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Rule 201
Rule 221
Rule 283
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5801
Rule 5802
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{2 x^2}+\left (3 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x} \, dx+\frac {1}{2} \left (b c d^3\right ) \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2} \, dx \\ & = -\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{2 x^2}+\left (3 c^2 d^2\right ) \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx-\frac {1}{4} \left (3 b c^3 d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx+\frac {1}{2} \left (5 b c^3 d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx \\ & = \frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{2 x^2}+\left (3 c^2 d^3\right ) \int \frac {a+b \text {arcsinh}(c x)}{x} \, dx-\frac {1}{16} \left (9 b c^3 d^3\right ) \int \sqrt {1+c^2 x^2} \, dx-\frac {1}{2} \left (3 b c^3 d^3\right ) \int \sqrt {1+c^2 x^2} \, dx+\frac {1}{8} \left (15 b c^3 d^3\right ) \int \sqrt {1+c^2 x^2} \, dx \\ & = -\frac {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {\left (3 c^2 d^3\right ) \text {Subst}\left (\int x \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}-\frac {1}{32} \left (9 b c^3 d^3\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{4} \left (3 b c^3 d^3\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{16} \left (15 b c^3 d^3\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{32} b c^2 d^3 \text {arcsinh}(c x)+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))^2}{2 b}+\frac {\left (6 c^2 d^3\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b} \\ & = -\frac {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{32} b c^2 d^3 \text {arcsinh}(c x)+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))^2}{2 b}+3 c^2 d^3 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\left (3 c^2 d^3\right ) \text {Subst}\left (\int \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right ) \\ & = -\frac {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{32} b c^2 d^3 \text {arcsinh}(c x)+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))^2}{2 b}+3 c^2 d^3 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+\frac {1}{2} \left (3 b c^2 d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ & = -\frac {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{32} b c^2 d^3 \text {arcsinh}(c x)+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))^2}{2 b}+3 c^2 d^3 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\frac {3}{2} b c^2 d^3 \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\frac {d^3 \left (-16 a b-48 a^2 c^2 x^2+48 a b c^4 x^4+8 a b c^6 x^6-16 b^2 c x \sqrt {1+c^2 x^2}-21 b^2 c^3 x^3 \sqrt {1+c^2 x^2}-2 b^2 c^5 x^5 \sqrt {1+c^2 x^2}-48 b^2 c^2 x^2 \text {arcsinh}(c x)^2+96 a b c^2 x^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+b \text {arcsinh}(c x) \left (-96 a c^2 x^2+b \left (-16+21 c^2 x^2+48 c^4 x^4+8 c^6 x^6\right )+96 b c^2 x^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )\right )+48 b^2 c^2 x^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )}{32 b x^2} \]
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Time = 0.29 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(c^{2} \left (d^{3} a \left (\frac {c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+3 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+\frac {d^{3} b}{2}+\frac {21 b \,d^{3} \operatorname {arcsinh}\left (c x \right )}{32}-\frac {3 d^{3} b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+3 d^{3} b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{16}-\frac {21 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{32}+\frac {d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} b \,\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {d^{3} b \sqrt {c^{2} x^{2}+1}}{2 c x}+3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )\right )\) | \(289\) |
default | \(c^{2} \left (d^{3} a \left (\frac {c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+3 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+\frac {d^{3} b}{2}+\frac {21 b \,d^{3} \operatorname {arcsinh}\left (c x \right )}{32}-\frac {3 d^{3} b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+3 d^{3} b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{16}-\frac {21 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{32}+\frac {d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} b \,\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {d^{3} b \sqrt {c^{2} x^{2}+1}}{2 c x}+3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )\right )\) | \(289\) |
parts | \(d^{3} a \left (\frac {c^{6} x^{4}}{4}+\frac {3 c^{4} x^{2}}{2}-\frac {1}{2 x^{2}}+3 c^{2} \ln \left (x \right )\right )+\frac {d^{3} b \,c^{2}}{2}+3 d^{3} b \,c^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b \,c^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b \,c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b \,c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {3 d^{3} b \,c^{4} \operatorname {arcsinh}\left (c x \right ) x^{2}}{2}-\frac {d^{3} b c \sqrt {c^{2} x^{2}+1}}{2 x}-\frac {d^{3} b \,\operatorname {arcsinh}\left (c x \right )}{2 x^{2}}+\frac {d^{3} b \,c^{6} \operatorname {arcsinh}\left (c x \right ) x^{4}}{4}-\frac {d^{3} b \,c^{5} x^{3} \sqrt {c^{2} x^{2}+1}}{16}-\frac {21 b \,c^{3} d^{3} x \sqrt {c^{2} x^{2}+1}}{32}-\frac {3 d^{3} b \,c^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{2}+\frac {21 b \,c^{2} d^{3} \operatorname {arcsinh}\left (c x \right )}{32}\) | \(301\) |
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\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^3} \, dx=d^{3} \left (\int \frac {a}{x^{3}}\, dx + \int \frac {3 a c^{2}}{x}\, dx + \int 3 a c^{4} x\, dx + \int a c^{6} x^{3}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b c^{2} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 3 b c^{4} x \operatorname {asinh}{\left (c x \right )}\, dx + \int b c^{6} x^{3} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3}{x^3} \,d x \]
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