Optimal. Leaf size=329 \[ -\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-4 b c d^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {122}{25} b^2 c^2 d^3 x+2 i b^2 c d^3 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d^3 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right ) \]
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Rubi [A] time = 0.71, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4695, 4649, 4619, 4677, 8, 194, 4699, 4697, 4709, 4183, 2279, 2391} \[ 2 i b^2 c d^3 \text {PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d^3 \text {PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-4 b c d^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {122}{25} b^2 c^2 d^3 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 194
Rule 2279
Rule 2391
Rule 4183
Rule 4619
Rule 4649
Rule 4677
Rule 4695
Rule 4697
Rule 4699
Rule 4709
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\left (6 c^2 d\right ) \int \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (2 b c d^3\right ) \int \frac {\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx\\ &=\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {1}{5} \left (24 c^2 d^2\right ) \int \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (2 b c d^3\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac {1}{5} \left (2 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right )^2 \, dx+\frac {1}{5} \left (12 b c^3 d^3\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac {2}{3} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac {1}{5} \left (16 c^2 d^3\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{5} \left (2 b^2 c^2 d^3\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx+\frac {1}{25} \left (12 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right )^2 \, dx-\frac {1}{3} \left (2 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx+\frac {1}{5} \left (16 b c^3 d^3\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {16}{15} b^2 c^2 d^3 x+\frac {22}{45} b^2 c^4 d^3 x^3-\frac {2}{25} b^2 c^6 d^3 x^5+2 b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx+\frac {1}{25} \left (12 b^2 c^2 d^3\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx+\frac {1}{15} \left (16 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx-\left (2 b^2 c^2 d^3\right ) \int 1 \, dx+\frac {1}{5} \left (32 b c^3 d^3\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {38}{25} b^2 c^2 d^3 x-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \operatorname {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{5} \left (32 b^2 c^2 d^3\right ) \int 1 \, dx\\ &=\frac {122}{25} b^2 c^2 d^3 x-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\left (2 b^2 c d^3\right ) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\left (2 b^2 c d^3\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {122}{25} b^2 c^2 d^3 x-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+\left (2 i b^2 c d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )-\left (2 i b^2 c d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=\frac {122}{25} b^2 c^2 d^3 x-\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+2 i b^2 c d^3 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d^3 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 1.28, size = 483, normalized size = 1.47 \[ \frac {1}{720} d^3 \left (-144 a^2 c^6 x^5+720 a^2 c^4 x^3-2160 a^2 c^2 x-\frac {720 a^2}{x}-288 a b c^6 x^5 \sin ^{-1}(c x)+1440 a b c^4 x^3 \sin ^{-1}(c x)-\frac {17568}{5} a b c \sqrt {1-c^2 x^2}-1440 a b c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-4320 a b c^2 x \sin ^{-1}(c x)-\frac {288}{5} a b c^5 x^4 \sqrt {1-c^2 x^2}+\frac {2016}{5} a b c^3 x^2 \sqrt {1-c^2 x^2}-\frac {1440 a b \sin ^{-1}(c x)}{x}-3420 b^2 c \sqrt {1-c^2 x^2} \sin ^{-1}(c x)+3460 b^2 c^2 x-1890 b^2 c^2 x \sin ^{-1}(c x)^2-360 b^2 c^2 x \sin ^{-1}(c x)^2 \cos \left (2 \sin ^{-1}(c x)\right )+80 b^2 c^2 x \cos \left (2 \sin ^{-1}(c x)\right )+1440 i b^2 c \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-1440 i b^2 c \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-10 b^2 c \sin \left (3 \sin ^{-1}(c x)\right )+45 b^2 c \sin ^{-1}(c x)^2 \sin \left (3 \sin ^{-1}(c x)\right )+\frac {18}{25} b^2 c \sin \left (5 \sin ^{-1}(c x)\right )-9 b^2 c \sin ^{-1}(c x)^2 \sin \left (5 \sin ^{-1}(c x)\right )-\frac {720 b^2 \sin ^{-1}(c x)^2}{x}+1440 b^2 c \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )-1440 b^2 c \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )-90 b^2 c \sin ^{-1}(c x) \cos \left (3 \sin ^{-1}(c x)\right )-\frac {18}{5} b^2 c \sin ^{-1}(c x) \cos \left (5 \sin ^{-1}(c x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{2} c^{6} d^{3} x^{6} - 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} - a^{2} d^{3} + {\left (b^{2} c^{6} d^{3} x^{6} - 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{6} d^{3} x^{6} - 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} - a b d^{3}\right )} \arcsin \left (c x\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 524, normalized size = 1.59 \[ -\frac {2 d^{3} a b \arcsin \left (c x \right ) c^{6} x^{5}}{5}+2 d^{3} a b \arcsin \left (c x \right ) c^{4} x^{3}-6 d^{3} a b \arcsin \left (c x \right ) c^{2} x -\frac {2 d^{3} a b \,c^{5} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}+\frac {14 d^{3} a b \,c^{3} x^{2} \sqrt {-c^{2} x^{2}+1}}{25}+\frac {19 b^{2} c^{2} d^{3} x}{4}-\frac {d^{3} a^{2}}{x}-\frac {2 d^{3} a b \arcsin \left (c x \right )}{x}-\frac {19 d^{3} b^{2} \arcsin \left (c x \right )^{2} c^{2} x}{8}-\frac {122 c \,d^{3} a b \sqrt {-c^{2} x^{2}+1}}{25}-\frac {19 c \,d^{3} b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{4}-\frac {c \,d^{3} b^{2} \arcsin \left (c x \right ) \cos \left (5 \arcsin \left (c x \right )\right )}{200}-\frac {c \,d^{3} b^{2} \sin \left (5 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{80}-\frac {c \,d^{3} b^{2} \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8}-\frac {3 c \,d^{3} b^{2} \arcsin \left (c x \right )^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{16}-2 c \,d^{3} a b \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )-2 c \,d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 c \,d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i b^{2} c \,d^{3} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i b^{2} c \,d^{3} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{3} a^{2} c^{6} x^{5}}{5}+d^{3} a^{2} c^{4} x^{3}-3 d^{3} a^{2} c^{2} x -\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2}}{x}+\frac {c \,d^{3} b^{2} \sin \left (5 \arcsin \left (c x \right )\right )}{1000}+\frac {c \,d^{3} b^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{24} \]
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 \[ -\frac {1}{5} \, a^{2} c^{6} d^{3} x^{5} - \frac {2}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{6} d^{3} + a^{2} c^{4} d^{3} x^{3} + \frac {2}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{4} d^{3} - 3 \, b^{2} c^{2} d^{3} x \arcsin \left (c x\right )^{2} + 6 \, b^{2} c^{2} d^{3} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} - 3 \, a^{2} c^{2} d^{3} x - 6 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b c d^{3} - 2 \, {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} a b d^{3} - \frac {a^{2} d^{3}}{x} - \frac {{\left (b^{2} c^{6} d^{3} x^{6} - 5 \, b^{2} c^{4} d^{3} x^{4} + 5 \, b^{2} d^{3}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, x \int \frac {{\left (b^{2} c^{7} d^{3} x^{6} - 5 \, b^{2} c^{5} d^{3} x^{4} + 5 \, b^{2} c d^{3}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{c^{2} x^{3} - x}\,{d x}}{5 \, x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3}{x^2} \,d x \]
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 \[ - d^{3} \left (\int 3 a^{2} c^{2}\, dx + \int \left (- \frac {a^{2}}{x^{2}}\right )\, dx + \int \left (- 3 a^{2} c^{4} x^{2}\right )\, dx + \int a^{2} c^{6} x^{4}\, dx + \int 3 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{2}}\right )\, dx + \int 6 a b c^{2} \operatorname {asin}{\left (c x \right )}\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{2}}\right )\, dx + \int \left (- 3 b^{2} c^{4} x^{2} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{6} x^{4} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- 6 a b c^{4} x^{2} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{6} x^{4} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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