Optimal. Leaf size=307 \[ \frac {122}{25} b^2 c^2 d^3 x+\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-2 b^2 c d^3 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^3 \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.47, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 12, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5807, 5786,
5772, 5798, 8, 200, 5808, 5806, 5816, 4267, 2317, 2438} \begin {gather*} \frac {6}{5} c^2 d^3 x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{5} c^2 d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2}{25} b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {22}{5} b c d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2-4 b c d^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{125} b^2 c^6 d^3 x^5+\frac {14}{75} b^2 c^4 d^3 x^3+\frac {122}{25} b^2 c^2 d^3 x-2 b^2 c d^3 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^3 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 200
Rule 2317
Rule 2438
Rule 4267
Rule 5772
Rule 5786
Rule 5798
Rule 5806
Rule 5807
Rule 5808
Rule 5816
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^2} \, dx &=-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (6 c^2 d\right ) \int \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\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} \, dx\\ &=\frac {2}{5} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {1}{5} \left (24 c^2 d^2\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^3\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx-\frac {1}{5} \left (2 b^2 c^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \, dx-\frac {1}{5} \left (12 b c^3 d^3\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=\frac {2}{3} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \int \frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac {1}{5} \left (16 c^2 d^3\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{5} \left (2 b^2 c^2 d^3\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx+\frac {1}{25} \left (12 b^2 c^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \, dx-\frac {1}{3} \left (2 b^2 c^2 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac {1}{5} \left (16 b c^3 d^3\right ) \int x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac {16}{15} b^2 c^2 d^3 x-\frac {22}{45} b^2 c^4 d^3 x^3-\frac {2}{25} b^2 c^6 d^3 x^5+2 b c d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx+\frac {1}{25} \left (12 b^2 c^2 d^3\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx+\frac {1}{15} \left (16 b^2 c^2 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\left (2 b^2 c^2 d^3\right ) \int 1 \, dx-\frac {1}{5} \left (32 b c^3 d^3\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {38}{25} b^2 c^2 d^3 x+\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{5} \left (32 b^2 c^2 d^3\right ) \int 1 \, dx\\ &=\frac {122}{25} b^2 c^2 d^3 x+\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (2 b^2 c d^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\left (2 b^2 c d^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {122}{25} b^2 c^2 d^3 x+\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (2 b^2 c d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\left (2 b^2 c d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )\\ &=\frac {122}{25} b^2 c^2 d^3 x+\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-2 b^2 c d^3 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^3 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.86, size = 466, normalized size = 1.52 \begin {gather*} \frac {1}{720} d^3 \left (-\frac {720 a^2}{x}+2160 a^2 c^2 x+3460 b^2 c^2 x+720 a^2 c^4 x^3+144 a^2 c^6 x^5-\frac {17568}{5} a b c \sqrt {1+c^2 x^2}-\frac {2016}{5} a b c^3 x^2 \sqrt {1+c^2 x^2}-\frac {288}{5} a b c^5 x^4 \sqrt {1+c^2 x^2}-\frac {1440 a b \sinh ^{-1}(c x)}{x}+4320 a b c^2 x \sinh ^{-1}(c x)+1440 a b c^4 x^3 \sinh ^{-1}(c x)+288 a b c^6 x^5 \sinh ^{-1}(c x)-3420 b^2 c \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)-\frac {720 b^2 \sinh ^{-1}(c x)^2}{x}+1890 b^2 c^2 x \sinh ^{-1}(c x)^2-1440 a b c \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )+80 b^2 c^2 x \cosh \left (2 \sinh ^{-1}(c x)\right )+360 b^2 c^2 x \sinh ^{-1}(c x)^2 \cosh \left (2 \sinh ^{-1}(c x)\right )-90 b^2 c \sinh ^{-1}(c x) \cosh \left (3 \sinh ^{-1}(c x)\right )-\frac {18}{5} b^2 c \sinh ^{-1}(c x) \cosh \left (5 \sinh ^{-1}(c x)\right )+1440 b^2 c \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-1440 b^2 c \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )+1440 b^2 c \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-1440 b^2 c \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )-10 b^2 c \sinh \left (3 \sinh ^{-1}(c x)\right )-45 b^2 c \sinh ^{-1}(c x)^2 \sinh \left (3 \sinh ^{-1}(c x)\right )+\frac {18}{25} b^2 c \sinh \left (5 \sinh ^{-1}(c x)\right )+9 b^2 c \sinh ^{-1}(c x)^2 \sinh \left (5 \sinh ^{-1}(c x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.66, size = 463, normalized size = 1.51
method | result | size |
derivativedivides | \(c \left (d^{3} a^{2} \left (\frac {c^{5} x^{5}}{5}+c^{3} x^{3}+3 c x -\frac {1}{c x}\right )-2 b^{2} d^{3} \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {b^{2} d^{3} \arcsinh \left (c x \right )^{2} c^{5} x^{5}}{5}+b^{2} d^{3} \arcsinh \left (c x \right )^{2} c^{3} x^{3}+3 b^{2} d^{3} \arcsinh \left (c x \right )^{2} c x -\frac {122 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{25}-2 b^{2} d^{3} \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {122 b^{2} d^{3} c x}{25}+\frac {14 b^{2} d^{3} c^{3} x^{3}}{75}+\frac {2 b^{2} d^{3} c^{5} x^{5}}{125}+2 b^{2} d^{3} \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {2 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{25}-\frac {14 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}}{25}+2 b^{2} d^{3} \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {b^{2} d^{3} \arcsinh \left (c x \right )^{2}}{c x}+2 d^{3} a b \left (\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\arcsinh \left (c x \right ) c^{3} x^{3}+3 \arcsinh \left (c x \right ) c x -\frac {\arcsinh \left (c x \right )}{c x}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{25}-\frac {61 \sqrt {c^{2} x^{2}+1}}{25}-\arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\) | \(463\) |
default | \(c \left (d^{3} a^{2} \left (\frac {c^{5} x^{5}}{5}+c^{3} x^{3}+3 c x -\frac {1}{c x}\right )-2 b^{2} d^{3} \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {b^{2} d^{3} \arcsinh \left (c x \right )^{2} c^{5} x^{5}}{5}+b^{2} d^{3} \arcsinh \left (c x \right )^{2} c^{3} x^{3}+3 b^{2} d^{3} \arcsinh \left (c x \right )^{2} c x -\frac {122 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{25}-2 b^{2} d^{3} \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {122 b^{2} d^{3} c x}{25}+\frac {14 b^{2} d^{3} c^{3} x^{3}}{75}+\frac {2 b^{2} d^{3} c^{5} x^{5}}{125}+2 b^{2} d^{3} \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {2 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{25}-\frac {14 b^{2} d^{3} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}}{25}+2 b^{2} d^{3} \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {b^{2} d^{3} \arcsinh \left (c x \right )^{2}}{c x}+2 d^{3} a b \left (\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\arcsinh \left (c x \right ) c^{3} x^{3}+3 \arcsinh \left (c x \right ) c x -\frac {\arcsinh \left (c x \right )}{c x}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{25}-\frac {61 \sqrt {c^{2} x^{2}+1}}{25}-\arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\) | \(463\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} d^{3} \left (\int 3 a^{2} c^{2}\, dx + \int \frac {a^{2}}{x^{2}}\, dx + \int 3 a^{2} c^{4} x^{2}\, dx + \int a^{2} c^{6} x^{4}\, dx + \int 3 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 6 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 3 b^{2} c^{4} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{4} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{4} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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