Optimal. Leaf size=196 \[ \frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c d^3}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {3 i b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{8 c d^3}-\frac {3 i b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{8 c d^3} \]
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Rubi [A] time = 0.13, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4655, 4657, 4181, 2279, 2391, 261} \[ \frac {3 i b \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{8 c d^3}-\frac {3 i b \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{8 c d^3}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c d^3}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 261
Rule 2279
Rule 2391
Rule 4181
Rule 4655
Rule 4657
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {x}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{4 d^3}+\frac {3 \int \frac {a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d}\\ &=-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac {(3 b c) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{8 d^3}+\frac {3 \int \frac {a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx}{8 d^2}\\ &=-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {3 \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 c d^3}\\ &=-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac {3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c d^3}-\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 c d^3}+\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 c d^3}\\ &=-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac {3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c d^3}+\frac {(3 i b) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 c d^3}-\frac {(3 i b) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 c d^3}\\ &=-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac {3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c d^3}+\frac {3 i b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{8 c d^3}-\frac {3 i b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{8 c d^3}\\ \end {align*}
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Mathematica [B] time = 1.61, size = 501, normalized size = 2.56 \[ -\frac {\frac {6 a x}{c^2 x^2-1}-\frac {4 a x}{\left (c^2 x^2-1\right )^2}+\frac {3 a \log (1-c x)}{c}-\frac {3 a \log (c x+1)}{c}+\frac {3 b \sqrt {1-c^2 x^2}}{c-c^2 x}+\frac {3 b \sqrt {1-c^2 x^2}}{c^2 x+c}-\frac {b x \sqrt {1-c^2 x^2}}{3 (c x-1)^2}+\frac {2 b \sqrt {1-c^2 x^2}}{3 c (c x-1)^2}+\frac {b x \sqrt {1-c^2 x^2}}{3 (c x+1)^2}+\frac {2 b \sqrt {1-c^2 x^2}}{3 c (c x+1)^2}-\frac {3 b \sin ^{-1}(c x)}{c-c^2 x}+\frac {3 b \sin ^{-1}(c x)}{c^2 x+c}-\frac {6 i b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c}+\frac {6 i b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c}-\frac {b \sin ^{-1}(c x)}{c (c x-1)^2}+\frac {b \sin ^{-1}(c x)}{c (c x+1)^2}+\frac {3 i \pi b \sin ^{-1}(c x)}{c}-\frac {6 b \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c}-\frac {3 \pi b \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c}+\frac {6 b \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )}{c}-\frac {3 \pi b \log \left (1+i e^{i \sin ^{-1}(c x)}\right )}{c}+\frac {3 \pi b \log \left (\sin \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )}{c}+\frac {3 \pi b \log \left (-\cos \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )}{c}}{16 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \arcsin \left (c x\right ) + a}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
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 \[ \int -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 384, normalized size = 1.96 \[ -\frac {a}{16 c \,d^{3} \left (c x +1\right )^{2}}-\frac {3 a}{16 c \,d^{3} \left (c x +1\right )}+\frac {3 a \ln \left (c x +1\right )}{16 c \,d^{3}}+\frac {a}{16 c \,d^{3} \left (c x -1\right )^{2}}-\frac {3 a}{16 c \,d^{3} \left (c x -1\right )}-\frac {3 a \ln \left (c x -1\right )}{16 c \,d^{3}}-\frac {3 c^{2} b \arcsin \left (c x \right ) x^{3}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 c b \,x^{2} \sqrt {-c^{2} x^{2}+1}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {5 b \arcsin \left (c x \right ) x}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {11 b \sqrt {-c^{2} x^{2}+1}}{24 c \,d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 c \,d^{3}}+\frac {3 b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 c \,d^{3}}+\frac {3 i b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 c \,d^{3}}-\frac {3 i b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 c \,d^{3}} \]
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 \[ -\frac {1}{16} \, a {\left (\frac {2 \, {\left (3 \, c^{2} x^{3} - 5 \, x\right )}}{c^{4} d^{3} x^{4} - 2 \, c^{2} d^{3} x^{2} + d^{3}} - \frac {3 \, \log \left (c x + 1\right )}{c d^{3}} + \frac {3 \, \log \left (c x - 1\right )}{c d^{3}}\right )} + \frac {{\left (3 \, {\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) \log \left (c x + 1\right ) - 3 \, {\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) \log \left (-c x + 1\right ) - 2 \, {\left (3 \, c^{3} x^{3} - 5 \, c x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) - {\left (c^{5} d^{3} x^{4} - 2 \, c^{3} d^{3} x^{2} + c d^{3}\right )} \int \frac {{\left (6 \, c^{3} x^{3} - 10 \, c x - 3 \, {\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \log \left (-c x + 1\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}\,{d x}\right )} b}{16 \, {\left (c^{5} d^{3} x^{4} - 2 \, c^{3} d^{3} x^{2} + c d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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