Optimal. Leaf size=545 \[ -\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.40, antiderivative size = 545, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 12, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {4787, 4783,
4803, 4268, 2611, 2320, 6724, 4715, 267, 4739, 455, 45} \begin {gather*} \frac {2 i b d \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2+d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2-\frac {2 d \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{\sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{9 \sqrt {1-c^2 x^2}}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c d x \text {ArcSin}(c x) \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 267
Rule 455
Rule 2320
Rule 2611
Rule 4268
Rule 4715
Rule 4739
Rule 4783
Rule 4787
Rule 4803
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx &=\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+d \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c d \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {c^2 x}{3}}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-c^2 x}}+\frac {1}{3} \sqrt {1-c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.60, size = 576, normalized size = 1.06 \begin {gather*} -\frac {1}{3} a^2 d \left (-4+c^2 x^2\right ) \sqrt {d-c^2 d x^2}+a^2 d^{3/2} \log (c x)-a^2 d^{3/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {2 a b d \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {1-c^2 x^2} \text {ArcSin}(c x)+\text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )-\text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )+i \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-i \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 d \sqrt {d-c^2 d x^2} \left (2 \sqrt {1-c^2 x^2}+2 c x \text {ArcSin}(c x)-\sqrt {1-c^2 x^2} \text {ArcSin}(c x)^2-\text {ArcSin}(c x)^2 \left (\log \left (1-e^{i \text {ArcSin}(c x)}\right )-\log \left (1+e^{i \text {ArcSin}(c x)}\right )\right )-2 i \text {ArcSin}(c x) \left (\text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )+2 \left (\text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {a b d \sqrt {d-c^2 d x^2} \left (9 c x-3 \text {ArcSin}(c x) \left (3 \sqrt {1-c^2 x^2}+\cos (3 \text {ArcSin}(c x))\right )+\sin (3 \text {ArcSin}(c x))\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b^2 d \sqrt {d-c^2 d x^2} \left (27 \sqrt {1-c^2 x^2} \left (-2+\text {ArcSin}(c x)^2\right )+\left (-2+9 \text {ArcSin}(c x)^2\right ) \cos (3 \text {ArcSin}(c x))-6 \text {ArcSin}(c x) (9 c x+\sin (3 \text {ArcSin}(c x)))\right )}{108 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1275 vs. \(2 (529 ) = 1058\).
time = 0.30, size = 1276, normalized size = 2.34
method | result | size |
default | \(\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-a^{2} d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )+a^{2} \sqrt {-c^{2} d \,x^{2}+d}\, d +\frac {68 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d}{27 \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right )^{2} x^{4} c^{4}}{3 \left (c^{2} x^{2}-1\right )}+\frac {5 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right )^{2} x^{2} c^{2}}{3 \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{4} c^{4}}{27 \left (c^{2} x^{2}-1\right )}-\frac {70 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{2} c^{2}}{27 \left (c^{2} x^{2}-1\right )}-\frac {4 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right )^{2}}{3 \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {2 i b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {2 i b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2}}{3}+\frac {8 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x c}{3 \left (c^{2} x^{2}-1\right )}-\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}}{9 \left (c^{2} x^{2}-1\right )}-\frac {8 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right )}{3 \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) x^{4} c^{4}}{3 \left (c^{2} x^{2}-1\right )}+\frac {10 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) x^{2} c^{2}}{3 \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}}{9 \left (c^{2} x^{2}-1\right )}+\frac {8 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \sqrt {-c^{2} x^{2}+1}\, x c}{3 \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}\) | \(1276\) |
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*} \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x}\, dx \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 {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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