Optimal. Leaf size=249 \[ -\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-4 b c d^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{27} b^2 c^4 d^2 x^3+\frac {32}{9} b^2 c^2 d^2 x+2 i b^2 c d^2 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d^2 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right ) \]
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Rubi [A] time = 0.49, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {4695, 4649, 4619, 4677, 8, 4699, 4697, 4709, 4183, 2279, 2391} \[ 2 i b^2 c d^2 \text {PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d^2 \text {PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-4 b c d^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{27} b^2 c^4 d^2 x^3+\frac {32}{9} b^2 c^2 d^2 x \]
Antiderivative was successfully verified.
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Rule 8
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 )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\left (4 c^2 d\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^2\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx\\ &=\frac {2}{3} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac {1}{3} \left (8 c^2 d^2\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{3} \left (2 b^2 c^2 d^2\right ) \int \left (1-c^2 x^2\right ) \, dx+\frac {1}{3} \left (8 b c^3 d^2\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {2}{3} b^2 c^2 d^2 x+\frac {2}{9} b^2 c^4 d^2 x^3+2 b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx+\frac {1}{9} \left (8 b^2 c^2 d^2\right ) \int \left (1-c^2 x^2\right ) \, dx-\left (2 b^2 c^2 d^2\right ) \int 1 \, dx+\frac {1}{3} \left (16 b c^3 d^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {16}{9} b^2 c^2 d^2 x-\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \operatorname {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{3} \left (16 b^2 c^2 d^2\right ) \int 1 \, dx\\ &=\frac {32}{9} b^2 c^2 d^2 x-\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\left (2 b^2 c d^2\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^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {32}{9} b^2 c^2 d^2 x-\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^2 \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^2\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^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=\frac {32}{9} b^2 c^2 d^2 x-\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+2 i b^2 c d^2 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d^2 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.97, size = 322, normalized size = 1.29 \[ \frac {1}{54} d^2 \left (18 a^2 c^4 x^3-108 a^2 c^2 x-\frac {54 a^2}{x}+36 a b c^4 x^3 \sin ^{-1}(c x)+12 a b c \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )-216 a b c \left (\sqrt {1-c^2 x^2}+c x \sin ^{-1}(c x)\right )-\frac {108 a b \left (c x \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )+\sin ^{-1}(c x)\right )}{x}+2 b^2 c^2 x \left (9 c^2 x^2 \sin ^{-1}(c x)^2-2 \left (c^2 x^2+6\right )\right )-189 b^2 c \sqrt {1-c^2 x^2} \sin ^{-1}(c x)-108 b^2 c^2 x \left (\sin ^{-1}(c x)^2-2\right )+108 i b^2 c \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-108 i b^2 c \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-\frac {54 b^2 \sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 c x \left (\log \left (1+e^{i \sin ^{-1}(c x)}\right )-\log \left (1-e^{i \sin ^{-1}(c x)}\right )\right )\right )}{x}-3 b^2 c \sin ^{-1}(c x) \cos \left (3 \sin ^{-1}(c x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arcsin \left (c x\right )}{x^{2}}, x\right ) \]
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 = 0.43, size = 411, normalized size = 1.65 \[ \frac {d^{2} a^{2} c^{4} x^{3}}{3}-2 d^{2} a^{2} c^{2} x -\frac {d^{2} a^{2}}{x}-\frac {7 c \,d^{2} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{2}-\frac {7 d^{2} b^{2} \arcsin \left (c x \right )^{2} c^{2} x}{4}+\frac {7 b^{2} c^{2} d^{2} x}{2}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2}}{x}-2 c \,d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 c \,d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i b^{2} c \,d^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i b^{2} c \,d^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {c \,d^{2} b^{2} \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{18}-\frac {c \,d^{2} b^{2} \sin \left (3 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{12}+\frac {c \,d^{2} b^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{54}+\frac {2 d^{2} a b \arcsin \left (c x \right ) c^{4} x^{3}}{3}-4 d^{2} a b \arcsin \left (c x \right ) c^{2} x -\frac {2 d^{2} a b \arcsin \left (c x \right )}{x}+\frac {2 d^{2} a b \,c^{3} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {32 c \,d^{2} a b \sqrt {-c^{2} x^{2}+1}}{9}-2 c \,d^{2} a b \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) \]
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 \[ \frac {1}{3} \, a^{2} c^{4} d^{2} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{4} d^{2} - 2 \, b^{2} c^{2} d^{2} x \arcsin \left (c x\right )^{2} + 4 \, b^{2} c^{2} d^{2} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} - 2 \, a^{2} c^{2} d^{2} x - 4 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b c d^{2} - 2 \, {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} a b d^{2} - \frac {a^{2} d^{2}}{x} + \frac {{\left (b^{2} c^{4} d^{2} x^{4} - 3 \, b^{2} d^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, x \int \frac {{\left (b^{2} c^{5} d^{2} x^{4} - 3 \, b^{2} c d^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{c^{2} x^{3} - x}\,{d x}}{3 \, x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2}{x^2} \,d x \]
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 \[ d^{2} \left (\int \left (- 2 a^{2} c^{2}\right )\, dx + \int \frac {a^{2}}{x^{2}}\, dx + \int a^{2} c^{4} x^{2}\, dx + \int \left (- 2 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \left (- 4 a b c^{2} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{2}}\, dx + \int b^{2} c^{4} x^{2} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{4} x^{2} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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