Optimal. Leaf size=561 \[ -\frac {b c d^2 \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {5 c^2 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac {5 b c^5 d^2 x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {c^2 x^2+1}}+\frac {5}{2} c^4 d^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 c^3 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {c^2 x^2+1}}+\frac {7 c^3 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {c^2 x^2+1}}+\frac {7}{3} b c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {14 b c^3 d^2 \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {c^2 x^2+1}}-\frac {b^2 c^2 d^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}}{3 x}+\frac {7}{12} b^2 c^4 d^2 x \sqrt {c^2 d x^2+d}-\frac {7 b^2 c^3 d^2 \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {c^2 x^2+1}}-\frac {23 b^2 c^3 d^2 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{12 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.85, antiderivative size = 561, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 15, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {5739, 5682, 5675, 5661, 321, 215, 5726, 5659, 3716, 2190, 2279, 2391, 195, 5728, 277} \[ \frac {7 b^2 c^3 d^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {c^2 x^2+1}}-\frac {5 b c^5 d^2 x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {c^2 x^2+1}}+\frac {5}{2} c^4 d^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 c^3 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {c^2 x^2+1}}-\frac {7 c^3 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {c^2 x^2+1}}+\frac {7}{3} b c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {14 b c^3 d^2 \sqrt {c^2 d x^2+d} \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {c^2 x^2+1}}-\frac {5 c^2 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {7}{12} b^2 c^4 d^2 x \sqrt {c^2 d x^2+d}-\frac {b^2 c^2 d^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}}{3 x}-\frac {23 b^2 c^3 d^2 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{12 \sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Rule 195
Rule 215
Rule 277
Rule 321
Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5659
Rule 5661
Rule 5675
Rule 5682
Rule 5726
Rule 5728
Rule 5739
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (5 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx+\frac {\left (2 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (5 c^4 d^2\right ) \int \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (4 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (10 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {\left (4 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (10 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (5 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{3 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^5 d^2 \sqrt {d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {2}{3} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {1+c^2 x^2}}+\frac {\left (4 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (10 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{6 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^6 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}-\frac {2 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {7 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {1+c^2 x^2}}-\frac {\left (8 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (20 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}-\frac {23 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{12 \sqrt {1+c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {7 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {1+c^2 x^2}}+\frac {14 b c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (4 b^2 c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (10 b^2 c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}-\frac {23 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{12 \sqrt {1+c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {7 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {1+c^2 x^2}}+\frac {14 b c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}-\frac {23 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{12 \sqrt {1+c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {7 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {1+c^2 x^2}}+\frac {14 b c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}+\frac {7 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 2.36, size = 616, normalized size = 1.10 \[ \frac {d^2 \left (-56 a^2 c^2 x^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}-8 a^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+12 a^2 c^4 x^4 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+60 a^2 c^3 \sqrt {d} x^3 \sqrt {c^2 x^2+1} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )-8 a b c x \sqrt {c^2 d x^2+d}+112 a b c^3 x^3 \sqrt {c^2 d x^2+d} \log (c x)+2 b \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left (30 a c^3 x^3+3 b c^3 x^3 \sinh \left (2 \sinh ^{-1}(c x)\right )-4 b \left (-7 c^3 x^3+7 c^2 x^2 \sqrt {c^2 x^2+1}+\sqrt {c^2 x^2+1}\right )\right )-6 a b c^3 x^3 \sqrt {c^2 d x^2+d} \cosh \left (2 \sinh ^{-1}(c x)\right )-2 b \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \left (-6 a c^3 x^3 \sinh \left (2 \sinh ^{-1}(c x)\right )+56 a c^2 x^2 \sqrt {c^2 x^2+1}+8 a \sqrt {c^2 x^2+1}-56 b c^3 x^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )+3 b c^3 x^3 \cosh \left (2 \sinh ^{-1}(c x)\right )+4 b c x\right )-8 b^2 c^2 x^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}-56 b^2 c^3 x^3 \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )+20 b^2 c^3 x^3 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)^3+3 b^2 c^3 x^3 \sqrt {c^2 d x^2+d} \sinh \left (2 \sinh ^{-1}(c x)\right )\right )}{24 x^3 \sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{4} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} + 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}}{x^{4}}, x\right ) \]
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 [B] time = 0.55, size = 3311, normalized size = 5.90 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^4} \,d x \]
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 \[ \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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