Optimal. Leaf size=326 \[ \frac {16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {34}{3} b c^3 d^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 c^2 d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {8}{3} c^4 d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} b c^3 d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-5 b c^3 d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{27} b^2 c^6 d^3 x^3+\frac {50}{9} b^2 c^4 d^3 x-\frac {17}{3} b^2 c^3 d^3 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\frac {17}{3} b^2 c^3 d^3 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )-\frac {b^2 c^2 d^3}{3 x} \]
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Rubi [A] time = 1.02, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 12, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5739, 5684, 5653, 5717, 8, 5744, 5742, 5760, 4182, 2279, 2391, 270} \[ -\frac {17}{3} b^2 c^3 d^3 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac {17}{3} b^2 c^3 d^3 \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac {8}{3} c^4 d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} b c^3 d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-5 b c^3 d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 c^2 d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {34}{3} b c^3 d^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{27} b^2 c^6 d^3 x^3+\frac {50}{9} b^2 c^4 d^3 x-\frac {b^2 c^2 d^3}{3 x} \]
Antiderivative was successfully verified.
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Rule 8
Rule 270
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 )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (2 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx\\ &=-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (8 c^4 d^2\right ) \int \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\left (4 b c^3 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx\\ &=\frac {17}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\left (4 b c^3 d^3\right ) \int \frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac {1}{3} \left (16 c^4 d^3\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (5 b c^3 d^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx+\left (4 b c^3 d^3\right ) \int \frac {a+b \sinh ^{-1}(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 \left (a+b \sinh ^{-1}(c x)\right )}{\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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (5 b c^3 d^3\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\left (4 b c^3 d^3\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac {34}{3} b c^3 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )-\left (4 b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\left (4 b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac {34}{3} b c^3 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )-\left (4 b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\left (4 b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac {34}{3} b c^3 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac {17}{3} b^2 c^3 d^3 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\frac {17}{3} b^2 c^3 d^3 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 1.10, size = 461, normalized size = 1.41 \[ \frac {d^3 \left (9 a^2 c^6 x^6+81 a^2 c^4 x^4-81 a^2 c^2 x^2-9 a^2+18 a b c^6 x^6 \sinh ^{-1}(c x)+162 a b c^4 x^4 \sinh ^{-1}(c x)-9 a b c x \sqrt {c^2 x^2+1}-162 a b c^2 x^2 \sinh ^{-1}(c x)-6 a b c^5 x^5 \sqrt {c^2 x^2+1}-150 a b c^3 x^3 \sqrt {c^2 x^2+1}-153 a b c^3 x^3 \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )-18 a b \sinh ^{-1}(c x)+2 b^2 c^6 x^6+9 b^2 c^6 x^6 \sinh ^{-1}(c x)^2+150 b^2 c^4 x^4+81 b^2 c^4 x^4 \sinh ^{-1}(c x)^2+153 b^2 c^3 x^3 \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-153 b^2 c^3 x^3 \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+153 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-153 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-9 b^2 c^2 x^2-81 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-9 b^2 c x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)-6 b^2 c^5 x^5 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)-150 b^2 c^3 x^3 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)-9 b^2 \sinh ^{-1}(c x)^2\right )}{27 x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c^{6} d^{3} x^{6} + 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} + a^{2} d^{3} + {\left (b^{2} c^{6} d^{3} x^{6} + 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} + b^{2} d^{3}\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{6} d^{3} x^{6} + 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} + a b d^{3}\right )} \operatorname {arsinh}\left (c x\right )}{x^{4}}, x\right ) \]
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 = 0.67, size = 528, normalized size = 1.62 \[ -\frac {b^{2} c^{2} d^{3}}{3 x}+\frac {50 b^{2} c^{4} d^{3} x}{9}+\frac {2 b^{2} c^{6} d^{3} x^{3}}{27}+\frac {2 c^{6} d^{3} a b \arcsinh \left (c x \right ) x^{3}}{3}+6 c^{4} d^{3} a b \arcsinh \left (c x \right ) x -\frac {6 c^{2} d^{3} a b \arcsinh \left (c x \right )}{x}-\frac {c \,d^{3} b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 x^{2}}-\frac {2 c^{5} d^{3} b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2}}{9}-\frac {d^{3} a^{2}}{3 x^{3}}-\frac {2 c^{5} d^{3} a b \,x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {c \,d^{3} a b \sqrt {c^{2} x^{2}+1}}{3 x^{2}}+\frac {c^{6} d^{3} a^{2} x^{3}}{3}+3 c^{4} d^{3} a^{2} x -\frac {3 c^{2} d^{3} a^{2}}{x}-\frac {17 b^{2} c^{3} d^{3} \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {17 b^{2} c^{3} d^{3} \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {d^{3} b^{2} \arcsinh \left (c x \right )^{2}}{3 x^{3}}-\frac {2 d^{3} a b \arcsinh \left (c x \right )}{3 x^{3}}+3 c^{4} d^{3} b^{2} \arcsinh \left (c x \right )^{2} x -\frac {3 c^{2} d^{3} b^{2} \arcsinh \left (c x \right )^{2}}{x}-\frac {17 c^{3} d^{3} a b \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{3}-\frac {50 c^{3} d^{3} b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{9}+\frac {17 c^{3} d^{3} b^{2} \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 c^{3} d^{3} b^{2} \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {c^{6} d^{3} b^{2} \arcsinh \left (c x \right )^{2} x^{3}}{3}-\frac {50 c^{3} d^{3} a b \sqrt {c^{2} x^{2}+1}}{9} \]
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 \[ \frac {1}{3} \, a^{2} c^{6} d^{3} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{6} d^{3} + 3 \, b^{2} c^{4} d^{3} x \operatorname {arsinh}\left (c x\right )^{2} + 6 \, b^{2} c^{4} d^{3} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + 3 \, a^{2} c^{4} d^{3} x + 6 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b c^{3} d^{3} - 6 \, {\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} a b c^{2} d^{3} + \frac {1}{3} \, {\left ({\left (c^{2} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{x^{2}}\right )} c - \frac {2 \, \operatorname {arsinh}\left (c x\right )}{x^{3}}\right )} a b d^{3} - \frac {3 \, a^{2} c^{2} d^{3}}{x} - \frac {a^{2} d^{3}}{3 \, x^{3}} + \frac {{\left (b^{2} c^{6} d^{3} x^{6} - 9 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{3 \, x^{3}} - \int \frac {2 \, {\left (b^{2} c^{9} d^{3} x^{8} + b^{2} c^{7} d^{3} x^{6} - 9 \, b^{2} c^{5} d^{3} x^{4} - 10 \, b^{2} c^{3} d^{3} x^{2} - b^{2} c d^{3} + {\left (b^{2} c^{8} d^{3} x^{7} - 9 \, b^{2} c^{4} d^{3} x^{3} - b^{2} c^{2} d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{3 \, {\left (c^{3} x^{6} + c x^{4} + {\left (c^{2} x^{5} + x^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x^4} \,d x \]
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 \[ d^{3} \left (\int 3 a^{2} c^{4}\, dx + \int \frac {a^{2}}{x^{4}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{2}}\, dx + \int a^{2} c^{6} x^{2}\, dx + \int 3 b^{2} c^{4} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int 6 a b c^{4} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int b^{2} c^{6} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {6 a b c^{2} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int 2 a b c^{6} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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