Optimal. Leaf size=141 \[ \frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c d^2}-\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {i b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{2 c d^2}-\frac {i b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{2 c d^2} \]
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Rubi [A] time = 0.10, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {i b \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{2 c d^2}-\frac {i b \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{2 c d^2}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c d^2}-\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}} \]
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 )^2} \, dx &=\frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(b c) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 d}\\ &=-\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c d^2}\\ &=-\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c d^2}-\frac {b \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c d^2}+\frac {b \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c d^2}\\ &=-\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c d^2}+\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 c d^2}-\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 c d^2}\\ &=-\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c d^2}+\frac {i b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{2 c d^2}-\frac {i b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{2 c d^2}\\ \end {align*}
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Mathematica [B] time = 0.84, size = 334, normalized size = 2.37 \[ -\frac {\frac {2 a x}{c^2 x^2-1}+\frac {a \log (1-c x)}{c}-\frac {a \log (c x+1)}{c}+\frac {b \sqrt {1-c^2 x^2}}{c-c^2 x}+\frac {b \sqrt {1-c^2 x^2}}{c^2 x+c}+\frac {b \sin ^{-1}(c x)}{c^2 x+c}-\frac {2 i b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c}+\frac {2 i b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c}+\frac {b \sin ^{-1}(c x)}{c (c x-1)}+\frac {i \pi b \sin ^{-1}(c x)}{c}-\frac {2 b \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c}-\frac {\pi b \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c}+\frac {2 b \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )}{c}-\frac {\pi b \log \left (1+i e^{i \sin ^{-1}(c x)}\right )}{c}+\frac {\pi b \log \left (\sin \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )}{c}+\frac {\pi b \log \left (-\cos \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )}{c}}{4 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arcsin \left (c x\right ) + a}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 260, normalized size = 1.84 \[ -\frac {a}{4 c \,d^{2} \left (c x +1\right )}+\frac {a \ln \left (c x +1\right )}{4 c \,d^{2}}-\frac {a}{4 c \,d^{2} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{4 c \,d^{2}}-\frac {b \arcsin \left (c x \right ) x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-c^{2} x^{2}+1}}{2 c \,d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 c \,d^{2}}+\frac {b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 c \,d^{2}}+\frac {i b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 c \,d^{2}}-\frac {i b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 c \,d^{2}} \]
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 \[ -\frac {1}{4} \, a {\left (\frac {2 \, x}{c^{2} d^{2} x^{2} - d^{2}} - \frac {\log \left (c x + 1\right )}{c d^{2}} + \frac {\log \left (c x - 1\right )}{c d^{2}}\right )} - \frac {{\left (2 \, c x \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) - {\left (c^{2} x^{2} - 1\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) \log \left (c x + 1\right ) + {\left (c^{2} x^{2} - 1\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) \log \left (-c x + 1\right ) + {\left (c^{3} d^{2} x^{2} - c d^{2}\right )} \int \frac {{\left (2 \, c x - {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + {\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}\,{d x}\right )} b}{4 \, {\left (c^{3} d^{2} x^{2} - c d^{2}\right )}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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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