Optimal. Leaf size=319 \[ -\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}+\frac {4 b c^3 \sqrt {1-c^2 x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.39, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {4701, 4681, 4625, 3717, 2190, 2279, 2391, 4627, 264} \[ -\frac {2 i b^2 c^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}+\frac {4 b c^3 \sqrt {1-c^2 x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4625
Rule 4627
Rule 4681
Rule 4701
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx &=-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}+\frac {1}{3} \left (2 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx+\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x^3} \, dx}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}-\frac {\left (8 i b c^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}+\frac {4 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}+\frac {4 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}+\frac {4 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 269, normalized size = 0.84 \[ -\frac {\sqrt {1-c^2 x^2} \left (2 a^2 c^2 x^2 \sqrt {1-c^2 x^2}+a^2 \sqrt {1-c^2 x^2}-4 a b c^3 x^3 \log (c x)-b \sin ^{-1}(c x) \left (-2 a \sqrt {1-c^2 x^2} \left (2 c^2 x^2+1\right )+4 b c^3 x^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b c x\right )+a b c x+2 i b^2 c^3 x^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 x^2 \sqrt {1-c^2 x^2}+b^2 \left (2 i c^3 x^3+2 c^2 x^2 \sqrt {1-c^2 x^2}+\sqrt {1-c^2 x^2}\right ) \sin ^{-1}(c x)^2\right )}{3 x^3 \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )}}{c^{2} d x^{6} - d x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.71, size = 2320, normalized size = 7.27 \[ \text {result too large to display} \]
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 \[ \frac {1}{3} \, {\left (\frac {4 \, c^{2} \log \relax (x)}{\sqrt {d}} - \frac {1}{\sqrt {d} x^{2}}\right )} a b c - \frac {2}{3} \, a b {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} c^{2}}{d x} + \frac {\sqrt {-c^{2} d x^{2} + d}}{d x^{3}}\right )} \arcsin \left (c x\right ) - \frac {1}{3} \, a^{2} {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} c^{2}}{d x} + \frac {\sqrt {-c^{2} d x^{2} + d}}{d x^{3}}\right )} + \frac {-\frac {-\frac {1}{4} \, {\left (4 \, {\left (2 \, c^{4} x^{4} - c^{2} x^{2} - 1\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} - {\left (3 \, {\left (2 \, c^{2} x^{2} + 1\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 4 \, x^{3} \int \frac {27 \, \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} - 14 \, {\left (2 \, c^{5} x^{5} - c^{3} x^{3} - c x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{4 \, {\left (c^{2} x^{6} - x^{4}\right )}}\,{d x}\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\right )} b^{2}}{12 \, \sqrt {c x + 1} \sqrt {-c x + 1} x^{3}}}{\sqrt {d}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^4\,\sqrt {d-c^2\,d\,x^2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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