Integrand size = 26, antiderivative size = 326 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^3}{3 x}+\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {34}{3} b c^3 d^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \]
[Out]
Time = 0.66 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5807, 5786, 5772, 5798, 8, 5808, 5806, 5816, 4267, 2317, 2438, 276} \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {34}{3} b c^3 d^3 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {b c d^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {8}{3} c^4 d^3 x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{9} b c^3 d^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-5 b c^3 d^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )+\frac {2}{27} b^2 c^6 d^3 x^3+\frac {50}{9} b^2 c^4 d^3 x-\frac {b^2 c^2 d^3}{3 x} \]
[In]
[Out]
Rule 8
Rule 276
Rule 2317
Rule 2438
Rule 4267
Rule 5772
Rule 5786
Rule 5798
Rule 5806
Rule 5807
Rule 5808
Rule 5816
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\left (2 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx+\frac {1}{3} \left (2 b c d^3\right ) \int \frac {\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx \\ & = -\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\left (8 c^4 d^2\right ) \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx+\frac {1}{3} \left (b^2 c^2 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^2}{x^2} \, dx+\frac {1}{3} \left (5 b c^3 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx+\left (4 b c^3 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx \\ & = \frac {17}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {1}{3} \left (b^2 c^2 d^3\right ) \int \left (2 c^2+\frac {1}{x^2}+c^4 x^2\right ) \, dx+\frac {1}{3} \left (5 b c^3 d^3\right ) \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx+\left (4 b c^3 d^3\right ) \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx+\frac {1}{3} \left (16 c^4 d^3\right ) \int (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{9} \left (5 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac {1}{3} \left (4 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac {1}{3} \left (16 b c^5 d^3\right ) \int x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx \\ & = -\frac {b^2 c^2 d^3}{3 x}-\frac {11}{9} b^2 c^4 d^3 x-\frac {14}{27} b^2 c^6 d^3 x^3+\frac {17}{3} b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {1}{3} \left (5 b c^3 d^3\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx+\left (4 b c^3 d^3\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx-\frac {1}{3} \left (5 b^2 c^4 d^3\right ) \int 1 \, dx+\frac {1}{9} \left (16 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\left (4 b^2 c^4 d^3\right ) \int 1 \, dx-\frac {1}{3} \left (32 b c^5 d^3\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {b^2 c^2 d^3}{3 x}-\frac {46}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {1}{3} \left (5 b c^3 d^3\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))+\left (4 b c^3 d^3\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))+\frac {1}{3} \left (32 b^2 c^4 d^3\right ) \int 1 \, dx \\ & = -\frac {b^2 c^2 d^3}{3 x}+\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {34}{3} b c^3 d^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )+\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )-\left (4 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )+\left (4 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right ) \\ & = -\frac {b^2 c^2 d^3}{3 x}+\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {34}{3} b c^3 d^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )+\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )-\left (4 b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )+\left (4 b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right ) \\ & = -\frac {b^2 c^2 d^3}{3 x}+\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {34}{3} b c^3 d^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.41 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {d^3 \left (-9 a^2-81 a^2 c^2 x^2-9 b^2 c^2 x^2+81 a^2 c^4 x^4+150 b^2 c^4 x^4+9 a^2 c^6 x^6+2 b^2 c^6 x^6-9 a b c x \sqrt {1+c^2 x^2}-150 a b c^3 x^3 \sqrt {1+c^2 x^2}-6 a b c^5 x^5 \sqrt {1+c^2 x^2}-18 a b \text {arcsinh}(c x)-162 a b c^2 x^2 \text {arcsinh}(c x)+162 a b c^4 x^4 \text {arcsinh}(c x)+18 a b c^6 x^6 \text {arcsinh}(c x)-9 b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-150 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-6 b^2 c^5 x^5 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-9 b^2 \text {arcsinh}(c x)^2-81 b^2 c^2 x^2 \text {arcsinh}(c x)^2+81 b^2 c^4 x^4 \text {arcsinh}(c x)^2+9 b^2 c^6 x^6 \text {arcsinh}(c x)^2-153 a b c^3 x^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+153 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-153 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+153 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-153 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{27 x^3} \]
[In]
[Out]
Time = 0.26 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.44
method | result | size |
derivativedivides | \(c^{3} \left (d^{3} a^{2} \left (\frac {c^{3} x^{3}}{3}+3 c x -\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}}{9}+\frac {2 d^{3} b^{2} c^{3} x^{3}}{27}-\frac {d^{3} b^{2}}{3 c x}+\frac {50 d^{3} b^{2} c x}{9}-\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{c x}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{3}}{3}+3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c x -\frac {50 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{9}+\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(468\) |
default | \(c^{3} \left (d^{3} a^{2} \left (\frac {c^{3} x^{3}}{3}+3 c x -\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}}{9}+\frac {2 d^{3} b^{2} c^{3} x^{3}}{27}-\frac {d^{3} b^{2}}{3 c x}+\frac {50 d^{3} b^{2} c x}{9}-\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{c x}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{3}}{3}+3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c x -\frac {50 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{9}+\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(468\) |
parts | \(d^{3} a^{2} \left (\frac {c^{6} x^{3}}{3}+3 c^{4} x -\frac {3 c^{2}}{x}-\frac {1}{3 x^{3}}\right )-\frac {d^{3} b^{2} c \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 x^{2}}-\frac {2 d^{3} b^{2} c^{5} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2}}{9}-\frac {17 b^{2} c^{3} d^{3} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {17 b^{2} c^{3} d^{3} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {b^{2} c^{2} d^{3}}{3 x}-\frac {17 d^{3} b^{2} c^{3} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {2 b^{2} c^{6} d^{3} x^{3}}{27}+\frac {17 d^{3} b^{2} c^{3} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {50 d^{3} b^{2} c^{3} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{9}+\frac {50 b^{2} c^{4} d^{3} x}{9}-\frac {3 d^{3} b^{2} c^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{x}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{3 x^{3}}+\frac {d^{3} b^{2} c^{6} \operatorname {arcsinh}\left (c x \right )^{2} x^{3}}{3}+3 d^{3} b^{2} c^{4} \operatorname {arcsinh}\left (c x \right )^{2} x +2 d^{3} a b \,c^{3} \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\) | \(480\) |
[In]
[Out]
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=d^{3} \left (\int 3 a^{2} c^{4}\, dx + \int \frac {a^{2}}{x^{4}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{2}}\, dx + \int a^{2} c^{6} x^{2}\, dx + \int 3 b^{2} c^{4} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int 6 a b c^{4} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int b^{2} c^{6} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {6 a b c^{2} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int 2 a b c^{6} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
[In]
[Out]
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x^4} \,d x \]
[In]
[Out]