Optimal. Leaf size=229 \[ \frac {4}{3} c^2 d^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2}{9} b c d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {10}{3} b c d^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-4 b c d^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{27} b^2 c^4 d^2 x^3+\frac {32}{9} b^2 c^2 d^2 x-2 b^2 c d^2 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^2 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right ) \]
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Rubi [A] time = 0.52, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5739, 5684, 5653, 5717, 8, 5744, 5742, 5760, 4182, 2279, 2391} \[ -2 b^2 c d^2 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^2 \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac {4}{3} c^2 d^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2}{9} b c d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {10}{3} b c d^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2-4 b c d^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{27} b^2 c^4 d^2 x^3+\frac {32}{9} b^2 c^2 d^2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2279
Rule 2391
Rule 4182
Rule 5653
Rule 5684
Rule 5717
Rule 5739
Rule 5742
Rule 5744
Rule 5760
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (4 c^2 d\right ) \int \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\left (2 b c d^2\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx\\ &=\frac {2}{3} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \int \frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac {1}{3} \left (8 c^2 d^2\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{3} \left (2 b^2 c^2 d^2\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac {1}{3} \left (8 b c^3 d^2\right ) \int x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac {2}{3} b^2 c^2 d^2 x-\frac {2}{9} b^2 c^4 d^2 x^3+2 b c d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx+\frac {1}{9} \left (8 b^2 c^2 d^2\right ) \int \left (1+c^2 x^2\right ) \, dx-\left (2 b^2 c^2 d^2\right ) \int 1 \, dx-\frac {1}{3} \left (16 b c^3 d^2\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {16}{9} b^2 c^2 d^2 x+\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{3} \left (16 b^2 c^2 d^2\right ) \int 1 \, dx\\ &=\frac {32}{9} b^2 c^2 d^2 x+\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (2 b^2 c d^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\left (2 b^2 c d^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {32}{9} b^2 c^2 d^2 x+\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (2 b^2 c d^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\left (2 b^2 c d^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )\\ &=\frac {32}{9} b^2 c^2 d^2 x+\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {8}{3} c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-2 b^2 c d^2 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^2 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 1.17, size = 306, normalized size = 1.34 \[ \frac {1}{54} d^2 \left (18 a^2 c^4 x^3+108 a^2 c^2 x-\frac {54 a^2}{x}+36 a b c^4 x^3 \sinh ^{-1}(c x)-12 a b c \left (c^2 x^2-2\right ) \sqrt {c^2 x^2+1}+216 a b c \left (c x \sinh ^{-1}(c x)-\sqrt {c^2 x^2+1}\right )-\frac {108 a b \left (c x \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )+\sinh ^{-1}(c x)\right )}{x}+2 b^2 c^2 x \left (2 c^2 x^2+9 c^2 x^2 \sinh ^{-1}(c x)^2-12\right )-189 b^2 c \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+108 b^2 c^2 x \left (\sinh ^{-1}(c x)^2+2\right )+108 b^2 c \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-108 b^2 c \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )-\frac {54 b^2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+2 c x \left (\log \left (e^{-\sinh ^{-1}(c x)}+1\right )-\log \left (1-e^{-\sinh ^{-1}(c x)}\right )\right )\right )}{x}-3 b^2 c \sinh ^{-1}(c x) \cosh \left (3 \sinh ^{-1}(c x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c^{4} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} + 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname {arsinh}\left (c x\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 400, normalized size = 1.75 \[ \frac {d^{2} a^{2} c^{4} x^{3}}{3}+2 d^{2} a^{2} c^{2} x -\frac {d^{2} a^{2}}{x}+\frac {32 b^{2} c^{2} d^{2} x}{9}+\frac {2 b^{2} c^{4} d^{2} x^{3}}{27}+\frac {d^{2} b^{2} \arcsinh \left (c x \right )^{2} c^{4} x^{3}}{3}+2 d^{2} b^{2} \arcsinh \left (c x \right )^{2} c^{2} x -\frac {32 c \,d^{2} b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{9}-\frac {d^{2} b^{2} \arcsinh \left (c x \right )^{2}}{x}-2 c \,d^{2} b^{2} \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 c \,d^{2} b^{2} \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} c \,d^{2} \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {2 d^{2} b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{2}}{9}-2 b^{2} c \,d^{2} \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {2 d^{2} a b \arcsinh \left (c x \right ) c^{4} x^{3}}{3}+4 d^{2} a b \arcsinh \left (c x \right ) c^{2} x -\frac {2 d^{2} a b \arcsinh \left (c x \right )}{x}-\frac {2 d^{2} a b \,c^{3} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {32 c \,d^{2} a b \sqrt {c^{2} x^{2}+1}}{9}-2 c \,d^{2} a b \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a^{2} c^{4} d^{2} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{4} d^{2} + 2 \, b^{2} c^{2} d^{2} x \operatorname {arsinh}\left (c x\right )^{2} + 4 \, b^{2} c^{2} d^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + 2 \, a^{2} c^{2} d^{2} x + 4 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b c d^{2} - 2 \, {\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} a b d^{2} - \frac {a^{2} d^{2}}{x} + \frac {{\left (b^{2} c^{4} d^{2} x^{4} - 3 \, b^{2} d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{3 \, x} - \int \frac {2 \, {\left (b^{2} c^{7} d^{2} x^{6} + b^{2} c^{5} d^{2} x^{4} - 3 \, b^{2} c^{3} d^{2} x^{2} - 3 \, b^{2} c d^{2} + {\left (b^{2} c^{6} d^{2} x^{5} - 3 \, b^{2} c^{2} d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{3 \, {\left (c^{3} x^{4} + c x^{2} + {\left (c^{2} x^{3} + x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ d^{2} \left (\int 2 a^{2} c^{2}\, dx + \int \frac {a^{2}}{x^{2}}\, dx + \int a^{2} c^{4} x^{2}\, dx + \int 2 b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int 4 a b c^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int b^{2} c^{4} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{4} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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