Optimal. Leaf size=270 \[ -\frac {b c (a+b \text {ArcSin}(c x))}{d^2 x \sqrt {1-c^2 x^2}}+\frac {c^2 (a+b \text {ArcSin}(c x))^2}{d^2 \left (1-c^2 x^2\right )}-\frac {(a+b \text {ArcSin}(c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {4 c^2 (a+b \text {ArcSin}(c x))^2 \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {2 i b c^2 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{d^2}-\frac {2 i b c^2 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{d^2}-\frac {b^2 c^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )}{d^2}+\frac {b^2 c^2 \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c x)}\right )}{d^2} \]
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Rubi [A]
time = 0.38, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 15, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used =
{4789, 4793, 4769, 4504, 4268, 2611, 2320, 6724, 4745, 266, 277, 197, 4779, 457, 78}
\begin {gather*} \frac {2 i b c^2 \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^2}-\frac {2 i b c^2 \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^2}+\frac {c^2 (a+b \text {ArcSin}(c x))^2}{d^2 \left (1-c^2 x^2\right )}-\frac {b c (a+b \text {ArcSin}(c x))}{d^2 x \sqrt {1-c^2 x^2}}-\frac {(a+b \text {ArcSin}(c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {4 c^2 \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{d^2}-\frac {b^2 c^2 \text {Li}_3\left (-e^{2 i \text {ArcSin}(c x)}\right )}{d^2}+\frac {b^2 c^2 \text {Li}_3\left (e^{2 i \text {ArcSin}(c x)}\right )}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {b^2 c^2 \log (x)}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 197
Rule 266
Rule 277
Rule 457
Rule 2320
Rule 2611
Rule 4268
Rule 4504
Rule 4745
Rule 4769
Rule 4779
Rule 4789
Rule 4793
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\left (2 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^2} \, dx+\frac {(b c) \int \frac {a+b \sin ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt {1-c^2 x^2}}+\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {\left (b^2 c^2\right ) \int \frac {-1+2 c^2 x^2}{x \left (1-c^2 x^2\right )} \, dx}{d^2}-\frac {\left (2 b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac {\left (2 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )} \, dx}{d}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2}}+\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {\left (2 c^2\right ) \text {Subst}\left (\int (a+b x)^2 \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}-\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {-1+2 c^2 x}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d^2}+\frac {\left (2 b^2 c^4\right ) \int \frac {x}{1-c^2 x^2} \, dx}{d^2}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2}}+\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{d^2}+\frac {\left (4 c^2\right ) \text {Subst}\left (\int (a+b x)^2 \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}-\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \left (-\frac {1}{x}-\frac {c^2}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{2 d^2}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2}}+\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}-\frac {\left (4 b c^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}+\frac {\left (4 b c^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2}}+\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {2 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}-\frac {2 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2}-\frac {\left (2 i b^2 c^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}+\frac {\left (2 i b^2 c^2\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2}}+\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {2 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}-\frac {2 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2}-\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{d^2}+\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{d^2}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2}}+\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {2 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}-\frac {2 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2}-\frac {b^2 c^2 \text {Li}_3\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}+\frac {b^2 c^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2}\\ \end {align*}
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Mathematica [A]
time = 1.10, size = 430, normalized size = 1.59 \begin {gather*} \frac {-\frac {a^2}{x^2}+\frac {a^2 c^2}{1-c^2 x^2}+4 a^2 c^2 \log (x)-2 a^2 c^2 \log \left (1-c^2 x^2\right )+2 a b \left (-\frac {c^3 x}{\sqrt {1-c^2 x^2}}-\frac {c \sqrt {1-c^2 x^2}}{x}-\frac {\text {ArcSin}(c x)}{x^2}+\frac {c^2 \text {ArcSin}(c x)}{1-c^2 x^2}+2 c^2 \left (2 \text {ArcSin}(c x) \left (\log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-\log \left (1+e^{2 i \text {ArcSin}(c x)}\right )\right )+i \left (\text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )\right )\right )+b^2 c^2 \left (-\frac {2 c x \text {ArcSin}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c x}-\frac {\text {ArcSin}(c x)^2}{c^2 x^2}+\frac {\text {ArcSin}(c x)^2}{1-c^2 x^2}+4 \text {ArcSin}(c x)^2 \left (\log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-\log \left (1+e^{2 i \text {ArcSin}(c x)}\right )\right )+2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+4 i \text {ArcSin}(c x) \left (\text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )+2 \left (-\text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )+\text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c x)}\right )\right )\right )}{2 d^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. 844 vs. \(2 (312 ) = 624\).
time = 0.28, size = 845, normalized size = 3.13
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {a^{2}}{4 d^{2} \left (c x -1\right )}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}-\frac {b^{2} \polylog \left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {a^{2} \ln \left (c x +1\right )}{d^{2}}-\frac {a^{2} \ln \left (c x -1\right )}{d^{2}}+\frac {a^{2}}{4 d^{2} \left (c x +1\right )}+\frac {2 a^{2} \ln \left (c x \right )}{d^{2}}+\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {4 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {2 i a b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {a^{2}}{2 d^{2} c^{2} x^{2}}+\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{d^{2}}+\frac {b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {4 i a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 i a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} c x \left (c^{2} x^{2}-1\right )}+\frac {a b \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right ) c x}-\frac {b^{2} \arcsin \left (c x \right )^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {4 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}\right )\) | \(845\) |
default | \(c^{2} \left (-\frac {a^{2}}{4 d^{2} \left (c x -1\right )}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}-\frac {b^{2} \polylog \left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {a^{2} \ln \left (c x +1\right )}{d^{2}}-\frac {a^{2} \ln \left (c x -1\right )}{d^{2}}+\frac {a^{2}}{4 d^{2} \left (c x +1\right )}+\frac {2 a^{2} \ln \left (c x \right )}{d^{2}}+\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {4 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {2 i a b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {a^{2}}{2 d^{2} c^{2} x^{2}}+\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{d^{2}}+\frac {b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {4 i a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 i a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} c x \left (c^{2} x^{2}-1\right )}+\frac {a b \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right ) c x}-\frac {b^{2} \arcsin \left (c x \right )^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {4 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}\right )\) | \(845\) |
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*} \frac {\int \frac {a^{2}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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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}{x^3\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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