Optimal. Leaf size=248 \[ \frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}-\frac {6 c^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^3}+\frac {3 i b c^2 \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac {3 i b c^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.35, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4701, 4705, 4679, 4419, 4183, 2279, 2391, 191, 192, 271} \[ \frac {3 i b c^2 \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac {3 i b c^2 \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}-\frac {6 c^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^3}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 271
Rule 2279
Rule 2391
Rule 4183
Rule 4419
Rule 4679
Rule 4701
Rule 4705
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx &=-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\left (3 c^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2}} \, dx}{2 d^3}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}-\frac {\left (3 b c^3\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{4 d^3}+\frac {\left (2 b c^3\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^3}+\frac {\left (3 c^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx}{d}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac {\left (b c^3\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^3}+\frac {\left (4 b c^3\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}-\frac {\left (3 b c^3\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^3}+\frac {\left (3 c^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d^2}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac {\left (3 c^2\right ) \operatorname {Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac {\left (6 c^2\right ) \operatorname {Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac {\left (3 b c^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}+\frac {\left (3 b c^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac {\left (3 i b c^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac {\left (3 i b c^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac {3 i b c^2 \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac {3 i b c^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}\\ \end {align*}
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Mathematica [A] time = 1.69, size = 256, normalized size = 1.03 \[ -\frac {\frac {12 a c^2}{c^2 x^2-1}-\frac {3 a c^2}{\left (c^2 x^2-1\right )^2}+18 a c^2 \log \left (1-c^2 x^2\right )-36 a c^2 \log (x)+\frac {6 a}{x^2}+b c^2 \left (\frac {14 c x}{\sqrt {1-c^2 x^2}}+\frac {c x}{\left (1-c^2 x^2\right )^{3/2}}+\frac {6 \sqrt {1-c^2 x^2}}{c x}+\frac {12 \sin ^{-1}(c x)}{c^2 x^2-1}-\frac {3 \sin ^{-1}(c x)}{\left (c^2 x^2-1\right )^2}+\frac {6 \sin ^{-1}(c x)}{c^2 x^2}-18 i \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )+18 i \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-36 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+36 \sin ^{-1}(c x) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )\right )}{12 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \arcsin \left (c x\right ) + a}{c^{6} d^{3} x^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}, x\right ) \]
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 \[ \int -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 635, normalized size = 2.56 \[ \frac {c^{2} a}{16 d^{3} \left (c x +1\right )^{2}}+\frac {9 c^{2} a}{16 d^{3} \left (c x +1\right )}-\frac {3 c^{2} a \ln \left (c x +1\right )}{2 d^{3}}-\frac {a}{2 d^{3} x^{2}}+\frac {3 c^{2} a \ln \left (c x \right )}{d^{3}}+\frac {c^{2} a}{16 d^{3} \left (c x -1\right )^{2}}-\frac {9 c^{2} a}{16 d^{3} \left (c x -1\right )}-\frac {3 c^{2} a \ln \left (c x -1\right )}{2 d^{3}}+\frac {3 i b \,c^{2} \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d^{3}}+\frac {2 c^{5} b \,x^{3} \sqrt {-c^{2} x^{2}+1}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 c^{4} b \arcsin \left (c x \right ) x^{2}}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {2 i c^{6} b \,x^{4}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {c^{3} b x \sqrt {-c^{2} x^{2}+1}}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {9 c^{2} b \arcsin \left (c x \right )}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {2 i c^{2} b}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {c b \sqrt {-c^{2} x^{2}+1}}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x}-\frac {b \arcsin \left (c x \right )}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}+\frac {3 c^{2} b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}}-\frac {3 i c^{2} b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}}-\frac {3 c^{2} b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{3}}+\frac {4 i c^{4} b \,x^{2}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 c^{2} b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}}-\frac {3 i c^{2} b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}} \]
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 \[ -\frac {1}{4} \, a {\left (\frac {6 \, c^{4} x^{4} - 9 \, c^{2} x^{2} + 2}{c^{4} d^{3} x^{6} - 2 \, c^{2} d^{3} x^{4} + d^{3} x^{2}} + \frac {6 \, c^{2} \log \left (c x + 1\right )}{d^{3}} + \frac {6 \, c^{2} \log \left (c x - 1\right )}{d^{3}} - \frac {12 \, c^{2} \log \relax (x)}{d^{3}}\right )} - b \int \frac {\arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{c^{6} d^{3} x^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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 \[ - \frac {\int \frac {a}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx + \int \frac {b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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