Integrand size = 26, antiderivative size = 337 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {19}{48} d^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {d^3 (a+b \text {arcsinh}(c x))^3}{3 b}+d^3 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right ) \]
[Out]
Time = 0.52 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5808, 5775, 3797, 2221, 2611, 2320, 6724, 5785, 5783, 30, 5786, 14, 267} \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=-\frac {1}{18} b c d^3 x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {7}{36} b c d^3 x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {19}{24} b c d^3 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-b d^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {d^3 (a+b \text {arcsinh}(c x))^3}{3 b}-\frac {19}{48} d^3 (a+b \text {arcsinh}(c x))^2+d^3 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right )+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {71}{144} b^2 c^2 d^3 x^2+\frac {1}{108} b^2 d^3 \left (c^2 x^2+1\right )^3 \]
[In]
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Rule 14
Rule 30
Rule 267
Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5775
Rule 5783
Rule 5785
Rule 5786
Rule 5808
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+d \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x} \, dx-\frac {1}{3} \left (b c d^3\right ) \int \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx \\ & = -\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+d^2 \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx-\frac {1}{18} \left (5 b c d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx-\frac {1}{2} \left (b c d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx+\frac {1}{18} \left (b^2 c^2 d^3\right ) \int x \left (1+c^2 x^2\right )^2 \, dx \\ & = \frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+d^3 \int \frac {(a+b \text {arcsinh}(c x))^2}{x} \, dx-\frac {1}{24} \left (5 b c d^3\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx-\frac {1}{8} \left (3 b c d^3\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx-\left (b c d^3\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx+\frac {1}{72} \left (5 b^2 c^2 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx+\frac {1}{8} \left (b^2 c^2 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx \\ & = \frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {d^3 \text {Subst}\left (\int x^2 \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}-\frac {1}{48} \left (5 b c d^3\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{16} \left (3 b c d^3\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{2} \left (b c d^3\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{72} \left (5 b^2 c^2 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac {1}{48} \left (5 b^2 c^2 d^3\right ) \int x \, dx+\frac {1}{8} \left (b^2 c^2 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac {1}{16} \left (3 b^2 c^2 d^3\right ) \int x \, dx+\frac {1}{2} \left (b^2 c^2 d^3\right ) \int x \, dx \\ & = \frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {19}{48} d^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {d^3 (a+b \text {arcsinh}(c x))^3}{3 b}+\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x^2}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b} \\ & = \frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {19}{48} d^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {d^3 (a+b \text {arcsinh}(c x))^3}{3 b}+d^3 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\left (2 d^3\right ) \text {Subst}\left (\int x \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right ) \\ & = \frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {19}{48} d^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {d^3 (a+b \text {arcsinh}(c x))^3}{3 b}+d^3 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )+\left (b d^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right ) \\ & = \frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {19}{48} d^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {d^3 (a+b \text {arcsinh}(c x))^3}{3 b}+d^3 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \left (b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ & = \frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {19}{48} d^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {d^3 (a+b \text {arcsinh}(c x))^3}{3 b}+d^3 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {d^3 \left (5184 a^2 c^2 x^2+2592 a^2 c^4 x^4+576 a^2 c^6 x^6-3600 a b c x \sqrt {1+c^2 x^2}-1056 a b c^3 x^3 \sqrt {1+c^2 x^2}-192 a b c^5 x^5 \sqrt {1+c^2 x^2}+10368 a b c^2 x^2 \text {arcsinh}(c x)+5184 a b c^4 x^4 \text {arcsinh}(c x)+1152 a b c^6 x^6 \text {arcsinh}(c x)-3456 a b \text {arcsinh}(c x)^2-1152 b^2 \text {arcsinh}(c x)^3+783 b^2 \cosh (2 \text {arcsinh}(c x))+1566 b^2 \text {arcsinh}(c x)^2 \cosh (2 \text {arcsinh}(c x))+27 b^2 \cosh (4 \text {arcsinh}(c x))+216 b^2 \text {arcsinh}(c x)^2 \cosh (4 \text {arcsinh}(c x))+b^2 \cosh (6 \text {arcsinh}(c x))+18 b^2 \text {arcsinh}(c x)^2 \cosh (6 \text {arcsinh}(c x))+6912 a b \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+3456 b^2 \text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+3456 a^2 \log (c x)-3600 a b \log \left (-c x+\sqrt {1+c^2 x^2}\right )+3456 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-1728 b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )-1566 b^2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))-108 b^2 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))-6 b^2 \text {arcsinh}(c x) \sinh (6 \text {arcsinh}(c x))\right )}{3456} \]
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[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(686\) vs. \(2(334)=668\).
Time = 0.33 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.04
method | result | size |
parts | \(-\frac {d^{3} a b \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{18}-\frac {11 d^{3} a b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{36}-\frac {25 d^{3} a b c x \sqrt {c^{2} x^{2}+1}}{24}-2 d^{3} b^{2} \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{48}-2 d^{3} b^{2} \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{6} x^{6}}{6}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{5} x^{5}}{18}-\frac {11 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}-\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{24}+\frac {811 d^{3} b^{2}}{3456}+d^{3} a^{2} \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (x \right )\right )+\frac {25 b^{2} c^{2} d^{3} x^{2}}{48}+\frac {11 b^{2} c^{4} d^{3} x^{4}}{144}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {25 d^{3} a b \,\operatorname {arcsinh}\left (c x \right )}{24}-d^{3} a b \operatorname {arcsinh}\left (c x \right )^{2}+2 d^{3} a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} c^{6} x^{6}}{108}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(687\) |
derivativedivides | \(d^{3} a^{2} \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (c x \right )\right )-\frac {d^{3} a b \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{18}-\frac {11 d^{3} a b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{36}-\frac {25 d^{3} a b c x \sqrt {c^{2} x^{2}+1}}{24}-2 d^{3} b^{2} \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{48}-2 d^{3} b^{2} \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{6} x^{6}}{6}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{5} x^{5}}{18}-\frac {11 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}-\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{24}+\frac {811 d^{3} b^{2}}{3456}+\frac {25 b^{2} c^{2} d^{3} x^{2}}{48}+\frac {11 b^{2} c^{4} d^{3} x^{4}}{144}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {25 d^{3} a b \,\operatorname {arcsinh}\left (c x \right )}{24}-d^{3} a b \operatorname {arcsinh}\left (c x \right )^{2}+2 d^{3} a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} c^{6} x^{6}}{108}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(689\) |
default | \(d^{3} a^{2} \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (c x \right )\right )-\frac {d^{3} a b \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{18}-\frac {11 d^{3} a b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{36}-\frac {25 d^{3} a b c x \sqrt {c^{2} x^{2}+1}}{24}-2 d^{3} b^{2} \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{48}-2 d^{3} b^{2} \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{6} x^{6}}{6}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{5} x^{5}}{18}-\frac {11 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}-\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{24}+\frac {811 d^{3} b^{2}}{3456}+\frac {25 b^{2} c^{2} d^{3} x^{2}}{48}+\frac {11 b^{2} c^{4} d^{3} x^{4}}{144}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {25 d^{3} a b \,\operatorname {arcsinh}\left (c x \right )}{24}-d^{3} a b \operatorname {arcsinh}\left (c x \right )^{2}+2 d^{3} a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} c^{6} x^{6}}{108}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(689\) |
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\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=d^{3} \left (\int \frac {a^{2}}{x}\, dx + \int 3 a^{2} c^{2} x\, dx + \int 3 a^{2} c^{4} x^{3}\, dx + \int a^{2} c^{6} x^{5}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{2} x \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{4} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx + \int 6 a b c^{4} x^{3} \operatorname {asinh}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{5} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x} \,d x \]
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