Optimal. Leaf size=307 \[ \frac {6}{5} c^2 d^3 x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{5} c^2 d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2}{25} b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {22}{5} b c d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2-4 b c d^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{125} b^2 c^6 d^3 x^5+\frac {14}{75} b^2 c^4 d^3 x^3+\frac {122}{25} b^2 c^2 d^3 x-2 b^2 c d^3 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^3 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right ) \]
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Rubi [A] time = 0.74, 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 = {5739, 5684, 5653, 5717, 8, 194, 5744, 5742, 5760, 4182, 2279, 2391} \[ -2 b^2 c d^3 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^3 \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac {6}{5} c^2 d^3 x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{5} c^2 d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2}{25} b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {22}{5} b c d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2-4 b c d^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{125} b^2 c^6 d^3 x^5+\frac {14}{75} b^2 c^4 d^3 x^3+\frac {122}{25} b^2 c^2 d^3 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 194
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^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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\left (2 b^2 c d^3\right ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\left (2 b^2 c d^3\right ) \operatorname {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 = 1.43, size = 466, normalized size = 1.52 \[ \frac {1}{720} d^3 \left (144 a^2 c^6 x^5+720 a^2 c^4 x^3+2160 a^2 c^2 x-\frac {720 a^2}{x}+288 a b c^6 x^5 \sinh ^{-1}(c x)+1440 a b c^4 x^3 \sinh ^{-1}(c x)-\frac {17568}{5} a b c \sqrt {c^2 x^2+1}-1440 a b c \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )+4320 a b c^2 x \sinh ^{-1}(c x)-\frac {288}{5} a b c^5 x^4 \sqrt {c^2 x^2+1}-\frac {2016}{5} a b c^3 x^2 \sqrt {c^2 x^2+1}-\frac {1440 a b \sinh ^{-1}(c x)}{x}-3420 b^2 c \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+3460 b^2 c^2 x+1890 b^2 c^2 x \sinh ^{-1}(c x)^2+360 b^2 c^2 x \sinh ^{-1}(c x)^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+80 b^2 c^2 x \cosh \left (2 \sinh ^{-1}(c x)\right )+1440 b^2 c \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-1440 b^2 c \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )-10 b^2 c \sinh \left (3 \sinh ^{-1}(c x)\right )-45 b^2 c \sinh ^{-1}(c x)^2 \sinh \left (3 \sinh ^{-1}(c x)\right )+\frac {18}{25} b^2 c \sinh \left (5 \sinh ^{-1}(c x)\right )+9 b^2 c \sinh ^{-1}(c x)^2 \sinh \left (5 \sinh ^{-1}(c x)\right )-\frac {720 b^2 \sinh ^{-1}(c x)^2}{x}+1440 b^2 c \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-1440 b^2 c \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-90 b^2 c \sinh ^{-1}(c x) \cosh \left (3 \sinh ^{-1}(c x)\right )-\frac {18}{5} b^2 c \sinh ^{-1}(c x) \cosh \left (5 \sinh ^{-1}(c x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c^{6} d^{3} x^{6} + 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} + a^{2} d^{3} + {\left (b^{2} c^{6} d^{3} x^{6} + 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} + b^{2} d^{3}\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{6} d^{3} x^{6} + 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} + a b d^{3}\right )} \operatorname {arsinh}\left (c x\right )}{x^{2}}, x\right ) \]
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 = 0.30, size = 516, normalized size = 1.68 \[ \frac {2 d^{3} a b \arcsinh \left (c x \right ) c^{6} x^{5}}{5}+2 d^{3} a b \arcsinh \left (c x \right ) c^{4} x^{3}+6 d^{3} a b \arcsinh \left (c x \right ) c^{2} x -\frac {2 d^{3} b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{5} x^{4}}{25}-\frac {14 d^{3} b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{2}}{25}-\frac {2 d^{3} a b \,c^{5} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {14 d^{3} a b \,c^{3} x^{2} \sqrt {c^{2} x^{2}+1}}{25}+\frac {d^{3} a^{2} c^{6} x^{5}}{5}+d^{3} a^{2} c^{4} x^{3}+3 d^{3} a^{2} c^{2} x -2 b^{2} c \,d^{3} \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} c \,d^{3} \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \arcsinh \left (c x \right )^{2}}{x}+\frac {122 b^{2} c^{2} d^{3} x}{25}+\frac {14 b^{2} c^{4} d^{3} x^{3}}{75}+\frac {2 b^{2} c^{6} d^{3} x^{5}}{125}-\frac {d^{3} a^{2}}{x}-\frac {122 c \,d^{3} a b \sqrt {c^{2} x^{2}+1}}{25}-\frac {2 d^{3} a b \arcsinh \left (c x \right )}{x}-\frac {122 c \,d^{3} b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{25}+2 c \,d^{3} b^{2} \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-2 c \,d^{3} b^{2} \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 c \,d^{3} a b \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+\frac {d^{3} b^{2} \arcsinh \left (c x \right )^{2} c^{6} x^{5}}{5}+d^{3} b^{2} \arcsinh \left (c x \right )^{2} c^{4} x^{3}+3 d^{3} b^{2} \arcsinh \left (c x \right )^{2} c^{2} x \]
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 \[ \frac {1}{5} \, a^{2} c^{6} d^{3} x^{5} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{6} d^{3} + a^{2} c^{4} d^{3} x^{3} + \frac {2}{3} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{4} d^{3} + 3 \, b^{2} c^{2} d^{3} x \operatorname {arsinh}\left (c x\right )^{2} + 6 \, b^{2} c^{2} d^{3} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + 3 \, a^{2} c^{2} d^{3} x + 6 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b c d^{3} - 2 \, {\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} a b d^{3} - \frac {a^{2} d^{3}}{x} + \frac {{\left (b^{2} c^{6} d^{3} x^{6} + 5 \, b^{2} c^{4} d^{3} x^{4} - 5 \, b^{2} d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{5 \, x} - \int \frac {2 \, {\left (b^{2} c^{9} d^{3} x^{8} + 6 \, b^{2} c^{7} d^{3} x^{6} + 5 \, b^{2} c^{5} d^{3} x^{4} - 5 \, b^{2} c^{3} d^{3} x^{2} - 5 \, b^{2} c d^{3} + {\left (b^{2} c^{8} d^{3} x^{7} + 5 \, b^{2} c^{6} d^{3} x^{5} - 5 \, b^{2} c^{2} d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{5 \, {\left (c^{3} x^{4} + c x^{2} + {\left (c^{2} x^{3} + x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x^2} \,d x \]
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 \[ d^{3} \left (\int 3 a^{2} c^{2}\, dx + \int \frac {a^{2}}{x^{2}}\, dx + \int 3 a^{2} c^{4} x^{2}\, dx + \int a^{2} c^{6} x^{4}\, dx + \int 3 b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int 6 a b c^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int 3 b^{2} c^{4} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{4} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{4} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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