Optimal. Leaf size=333 \[ -\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 d x^2}-\frac {(a+b \text {ArcSin}(c x))^2}{3 d x^3}-\frac {c^2 (a+b \text {ArcSin}(c x))^2}{d x}-\frac {2 i c^3 (a+b \text {ArcSin}(c x))^2 \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{d}-\frac {14 b c^3 (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{3 d}+\frac {7 i b^2 c^3 \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{3 d}+\frac {2 i b c^3 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{d}-\frac {2 i b c^3 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{d}-\frac {7 i b^2 c^3 \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{3 d}-\frac {2 b^2 c^3 \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(c x)}\right )}{d}+\frac {2 b^2 c^3 \text {PolyLog}\left (3,i e^{i \text {ArcSin}(c x)}\right )}{d} \]
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Rubi [A]
time = 0.45, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {4789, 4749,
4266, 2611, 2320, 6724, 4803, 4268, 2317, 2438, 30} \begin {gather*} -\frac {2 i c^3 \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{d}+\frac {2 i b c^3 \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d}-\frac {2 i b c^3 \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d}-\frac {14 b c^3 \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 d}-\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 d x^2}-\frac {c^2 (a+b \text {ArcSin}(c x))^2}{d x}-\frac {(a+b \text {ArcSin}(c x))^2}{3 d x^3}+\frac {7 i b^2 c^3 \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right )}{3 d}-\frac {7 i b^2 c^3 \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right )}{3 d}-\frac {2 b^2 c^3 \text {Li}_3\left (-i e^{i \text {ArcSin}(c x)}\right )}{d}+\frac {2 b^2 c^3 \text {Li}_3\left (i e^{i \text {ArcSin}(c x)}\right )}{d}-\frac {b^2 c^2}{3 d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4266
Rule 4268
Rule 4749
Rule 4789
Rule 4803
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}+c^2 \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )} \, dx+\frac {(2 b c) \int \frac {a+b \sin ^{-1}(c x)}{x^3 \sqrt {1-c^2 x^2}} \, dx}{3 d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}+c^4 \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2} \, dx}{3 d}+\frac {\left (b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx}{3 d}+\frac {\left (2 b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx}{d}\\ &=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}+\frac {c^3 \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac {\left (b c^3\right ) \text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d}+\frac {\left (2 b c^3\right ) \text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}-\frac {2 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d}-\frac {14 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac {\left (2 b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac {\left (2 b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}-\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d}+\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d}-\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}-\frac {2 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d}-\frac {14 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac {2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac {2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d}+\frac {\left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac {\left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d}-\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d}-\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}-\frac {2 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d}-\frac {14 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac {7 i b^2 c^3 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac {2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac {2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac {7 i b^2 c^3 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d}+\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d}\\ &=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}-\frac {2 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d}-\frac {14 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac {7 i b^2 c^3 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac {2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac {2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac {7 i b^2 c^3 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac {2 b^2 c^3 \text {Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{d}+\frac {2 b^2 c^3 \text {Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{d}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(849\) vs. \(2(333)=666\).
time = 7.16, size = 849, normalized size = 2.55 \begin {gather*} -\frac {a^2}{3 d x^3}-\frac {a^2 c^2}{d x}-\frac {a^2 c^3 \log (1-c x)}{2 d}+\frac {a^2 c^3 \log (1+c x)}{2 d}-\frac {2 a b \left (-c^2 \left (-\frac {\text {ArcSin}(c x)}{x}-c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\right )+\frac {c x \sqrt {1-c^2 x^2}+2 \text {ArcSin}(c x)+c^3 x^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{6 x^3}+\frac {1}{2} c^4 \left (\frac {3 i \pi \text {ArcSin}(c x)}{2 c}-\frac {i \text {ArcSin}(c x)^2}{2 c}+\frac {2 \pi \log \left (1+e^{-i \text {ArcSin}(c x)}\right )}{c}-\frac {\pi \log \left (1+i e^{i \text {ArcSin}(c x)}\right )}{c}+\frac {2 \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )}{c}-\frac {2 \pi \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{c}+\frac {\pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )}{c}-\frac {2 i \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{c}\right )-\frac {1}{2} c^4 \left (\frac {i \pi \text {ArcSin}(c x)}{2 c}-\frac {i \text {ArcSin}(c x)^2}{2 c}+\frac {2 \pi \log \left (1+e^{-i \text {ArcSin}(c x)}\right )}{c}+\frac {\pi \log \left (1-i e^{i \text {ArcSin}(c x)}\right )}{c}+\frac {2 \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )}{c}-\frac {2 \pi \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{c}-\frac {\pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )}{c}-\frac {2 i \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{c}\right )\right )}{d}-\frac {b^2 c^3 \left (4 \cot \left (\frac {1}{2} \text {ArcSin}(c x)\right )+14 \text {ArcSin}(c x)^2 \cot \left (\frac {1}{2} \text {ArcSin}(c x)\right )+2 \text {ArcSin}(c x) \csc ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )+\frac {1}{2} c x \text {ArcSin}(c x)^2 \csc ^4\left (\frac {1}{2} \text {ArcSin}(c x)\right )-56 \text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )-24 \text {ArcSin}(c x)^2 \log \left (1-i e^{i \text {ArcSin}(c x)}\right )+24 \text {ArcSin}(c x)^2 \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+56 \text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )-56 i \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-48 i \text {ArcSin}(c x) \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )+48 i \text {ArcSin}(c x) \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )+56 i \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )+48 \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(c x)}\right )-48 \text {PolyLog}\left (3,i e^{i \text {ArcSin}(c x)}\right )-2 \text {ArcSin}(c x) \sec ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )+\frac {8 \text {ArcSin}(c x)^2 \sin ^4\left (\frac {1}{2} \text {ArcSin}(c x)\right )}{c^3 x^3}+4 \tan \left (\frac {1}{2} \text {ArcSin}(c x)\right )+14 \text {ArcSin}(c x)^2 \tan \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{24 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 691, normalized size = 2.08
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {2 a b \arcsin \left (c x \right )}{d c x}-\frac {b^{2}}{3 d c x}-\frac {b^{2} \arcsin \left (c x \right )^{2}}{d c x}-\frac {a^{2}}{3 d \,c^{3} x^{3}}-\frac {a b \sqrt {-c^{2} x^{2}+1}}{3 d \,c^{2} x^{2}}-\frac {2 a b \arcsin \left (c x \right )}{3 d \,c^{3} x^{3}}-\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 d \,c^{2} x^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2}}{3 d \,c^{3} x^{3}}-\frac {7 b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3 d}+\frac {7 i b^{2} \dilog \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3 d}+\frac {7 i b^{2} \dilog \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3 d}+\frac {7 a b \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{3 d}-\frac {7 a b \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3 d}-\frac {a^{2}}{d c x}-\frac {a^{2} \ln \left (c x -1\right )}{2 d}+\frac {a^{2} \ln \left (c x +1\right )}{2 d}+\frac {2 i a b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 i a b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 b^{2} \polylog \left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 b^{2} \polylog \left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\) | \(691\) |
default | \(c^{3} \left (-\frac {2 a b \arcsin \left (c x \right )}{d c x}-\frac {b^{2}}{3 d c x}-\frac {b^{2} \arcsin \left (c x \right )^{2}}{d c x}-\frac {a^{2}}{3 d \,c^{3} x^{3}}-\frac {a b \sqrt {-c^{2} x^{2}+1}}{3 d \,c^{2} x^{2}}-\frac {2 a b \arcsin \left (c x \right )}{3 d \,c^{3} x^{3}}-\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 d \,c^{2} x^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2}}{3 d \,c^{3} x^{3}}-\frac {7 b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3 d}+\frac {7 i b^{2} \dilog \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3 d}+\frac {7 i b^{2} \dilog \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3 d}+\frac {7 a b \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{3 d}-\frac {7 a b \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3 d}-\frac {a^{2}}{d c x}-\frac {a^{2} \ln \left (c x -1\right )}{2 d}+\frac {a^{2} \ln \left (c x +1\right )}{2 d}+\frac {2 i a b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 i a b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 b^{2} \polylog \left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 b^{2} \polylog \left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\) | \(691\) |
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^{2} x^{6} - x^{4}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{6} - x^{4}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{2} x^{6} - x^{4}}\, dx}{d} \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^4\,\left (d-c^2\,d\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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