Optimal. Leaf size=354 \[ -3 b c^2 d^3 \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{4} c^2 d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac {c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}-\frac {3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+3 c^2 d^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {7}{8} b c^3 d^3 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {3}{16} b c^3 d^3 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {21}{32} b^2 c^4 d^3 x^2-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (e^{-2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x) \]
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Rubi [A] time = 0.75, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 15, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {5739, 5744, 5659, 3716, 2190, 2531, 2282, 6589, 5682, 5675, 30, 5684, 14, 266, 43} \[ 3 b c^2 d^3 \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {3}{2} b^2 c^2 d^3 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )+\frac {7}{8} b c^3 d^3 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {3}{16} b c^3 d^3 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{4} c^2 d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}-\frac {3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+3 c^2 d^3 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {21}{32} b^2 c^4 d^3 x^2+b^2 c^2 d^3 \log (x) \]
Warning: Unable to verify antiderivative.
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Rule 14
Rule 30
Rule 43
Rule 266
Rule 2190
Rule 2282
Rule 2531
Rule 3716
Rule 5659
Rule 5675
Rule 5682
Rule 5684
Rule 5739
Rule 5744
Rule 6589
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^3} \, dx &=-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\left (3 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} \, dx+\left (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^2} \, dx\\ &=-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {3}{4} c^2 d^3 \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}{2 x^2}+\left (3 c^2 d^2\right ) \int \frac {\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx+\left (b^2 c^2 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^2}{x} \, dx-\frac {1}{2} \left (3 b c^3 d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\left (5 b c^3 d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=\frac {7}{8} b c^3 d^3 x \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 )}{x}+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{4} c^2 d^3 \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}{2 x^2}+\left (3 c^2 d^3\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx+\frac {1}{2} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\left (1+c^2 x\right )^2}{x} \, dx,x,x^2\right )-\frac {1}{8} \left (9 b c^3 d^3\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\left (3 b c^3 d^3\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {1}{4} \left (15 b c^3 d^3\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {1}{8} \left (3 b^2 c^4 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx-\frac {1}{4} \left (5 b^2 c^4 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx\\ &=-\frac {3}{16} b c^3 d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {7}{8} b c^3 d^3 x \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 )}{x}+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{4} c^2 d^3 \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}{2 x^2}+\left (3 c^2 d^3\right ) \operatorname {Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{2} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \left (2 c^2+\frac {1}{x}+c^4 x\right ) \, dx,x,x^2\right )-\frac {1}{16} \left (9 b c^3 d^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{2} \left (3 b c^3 d^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{8} \left (15 b c^3 d^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{8} \left (3 b^2 c^4 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac {1}{16} \left (9 b^2 c^4 d^3\right ) \int x \, dx-\frac {1}{4} \left (5 b^2 c^4 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac {1}{2} \left (3 b^2 c^4 d^3\right ) \int x \, dx-\frac {1}{8} \left (15 b^2 c^4 d^3\right ) \int x \, dx\\ &=\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4-\frac {3}{16} b c^3 d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {7}{8} b c^3 d^3 x \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 )}{x}-\frac {3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{4} c^2 d^3 \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}{2 x^2}-\frac {c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}+b^2 c^2 d^3 \log (x)-\left (6 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)^2}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4-\frac {3}{16} b c^3 d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {7}{8} b c^3 d^3 x \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 )}{x}-\frac {3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{4} c^2 d^3 \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}{2 x^2}-\frac {c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)-\left (6 b c^2 d^3\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4-\frac {3}{16} b c^3 d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {7}{8} b c^3 d^3 x \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 )}{x}-\frac {3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{4} c^2 d^3 \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}{2 x^2}-\frac {c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 b c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\left (3 b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4-\frac {3}{16} b c^3 d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {7}{8} b c^3 d^3 x \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 )}{x}-\frac {3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{4} c^2 d^3 \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}{2 x^2}-\frac {c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 b c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} \left (3 b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )\\ &=\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4-\frac {3}{16} b c^3 d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {7}{8} b c^3 d^3 x \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 )}{x}-\frac {3}{32} c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{4} c^2 d^3 \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}{2 x^2}-\frac {c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 b c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 1.20, size = 459, normalized size = 1.30 \[ \frac {1}{256} d^3 \left (64 a^2 c^6 x^4+384 a^2 c^4 x^2+768 a^2 c^2 \log (x)-\frac {128 a^2}{x^2}+128 a b c^6 x^4 \sinh ^{-1}(c x)+768 a b c^4 x^2 \sinh ^{-1}(c x)-768 a b c^2 \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )-\frac {256 a b c \sqrt {c^2 x^2+1}}{x}+768 a b c^2 \sinh ^{-1}(c x)^2+336 a b c^2 \sinh ^{-1}(c x)+1536 a b c^2 \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-32 a b c^5 x^3 \sqrt {c^2 x^2+1}-336 a b c^3 x \sqrt {c^2 x^2+1}-\frac {256 a b \sinh ^{-1}(c x)}{x^2}+768 b^2 c^2 \sinh ^{-1}(c x) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-384 b^2 c^2 \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac {256 b^2 c \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{x}+256 b^2 c^2 \log (c x)-256 b^2 c^2 \sinh ^{-1}(c x)^3-160 b^2 c^2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )-4 b^2 c^2 \sinh ^{-1}(c x) \sinh \left (4 \sinh ^{-1}(c x)\right )+768 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+80 b^2 c^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+160 b^2 c^2 \sinh ^{-1}(c x)^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+b^2 c^2 \cosh \left (4 \sinh ^{-1}(c x)\right )+8 b^2 c^2 \sinh ^{-1}(c x)^2 \cosh \left (4 \sinh ^{-1}(c x)\right )-\frac {128 b^2 \sinh ^{-1}(c x)^2}{x^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c^{6} d^{3} x^{6} + 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} + a^{2} d^{3} + {\left (b^{2} c^{6} d^{3} x^{6} + 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} + b^{2} d^{3}\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{6} d^{3} x^{6} + 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} + a b d^{3}\right )} \operatorname {arsinh}\left (c x\right )}{x^{3}}, x\right ) \]
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 [B] time = 0.80, size = 838, normalized size = 2.37 \[ \frac {21 b^{2} c^{4} d^{3} x^{2}}{32}+\frac {b^{2} c^{6} d^{3} x^{4}}{32}+6 c^{2} d^{3} b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {c^{6} d^{3} b^{2} \arcsinh \left (c x \right )^{2} x^{4}}{4}+\frac {3 c^{4} d^{3} b^{2} \arcsinh \left (c x \right )^{2} x^{2}}{2}+3 c^{2} d^{3} b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+3 c^{2} d^{3} b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+6 c^{2} d^{3} b^{2} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-3 c^{2} d^{3} a b \arcsinh \left (c x \right )^{2}+\frac {21 c^{2} d^{3} a b \arcsinh \left (c x \right )}{16}+6 c^{2} d^{3} a b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} a b \arcsinh \left (c x \right )}{x^{2}}+6 c^{2} d^{3} a b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {81 d^{3} b^{2} c^{2}}{256}-\frac {d^{3} a^{2}}{2 x^{2}}-\frac {c^{5} d^{3} a b \,x^{3} \sqrt {c^{2} x^{2}+1}}{8}-\frac {21 c^{3} d^{3} a b x \sqrt {c^{2} x^{2}+1}}{16}-\frac {c \,d^{3} a b \sqrt {c^{2} x^{2}+1}}{x}+3 c^{4} d^{3} a b \arcsinh \left (c x \right ) x^{2}-\frac {c^{5} d^{3} b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3}}{8}-\frac {21 c^{3} d^{3} b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{16}+6 c^{2} d^{3} a b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+6 c^{2} d^{3} a b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {c \,d^{3} b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{x}+\frac {c^{6} d^{3} a b \arcsinh \left (c x \right ) x^{4}}{2}+\frac {c^{6} d^{3} a^{2} x^{4}}{4}+\frac {3 c^{4} d^{3} a^{2} x^{2}}{2}+3 c^{2} d^{3} a^{2} \ln \left (c x \right )+c^{2} d^{3} b^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+c^{2} d^{3} b^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 c^{2} d^{3} b^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \arcsinh \left (c x \right )^{2}}{2 x^{2}}+\frac {21 c^{2} d^{3} b^{2} \arcsinh \left (c x \right )^{2}}{32}-c^{2} d^{3} b^{2} \arcsinh \left (c x \right )^{3}-6 c^{2} d^{3} b^{2} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )-6 c^{2} d^{3} b^{2} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+c^{2} d^{3} b^{2} \arcsinh \left (c x \right )+d^{3} a b \,c^{2} \]
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 \[ \frac {1}{4} \, a^{2} c^{6} d^{3} x^{4} + \frac {3}{2} \, a^{2} c^{4} d^{3} x^{2} + 3 \, a^{2} c^{2} d^{3} \log \relax (x) - a b d^{3} {\left (\frac {\sqrt {c^{2} x^{2} + 1} c}{x} + \frac {\operatorname {arsinh}\left (c x\right )}{x^{2}}\right )} - \frac {a^{2} d^{3}}{2 \, x^{2}} + \int b^{2} c^{6} d^{3} x^{3} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, a b c^{6} d^{3} x^{3} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 3 \, b^{2} c^{4} d^{3} x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, a b c^{4} d^{3} x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \frac {3 \, b^{2} c^{2} d^{3} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{x} + \frac {6 \, a b c^{2} d^{3} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x} + \frac {b^{2} d^{3} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x^3} \,d x \]
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 \[ d^{3} \left (\int \frac {a^{2}}{x^{3}}\, dx + \int \frac {3 a^{2} c^{2}}{x}\, dx + \int 3 a^{2} c^{4} x\, dx + \int a^{2} c^{6} x^{3}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{4} x \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {6 a b c^{2} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 6 a b c^{4} x \operatorname {asinh}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{3} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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