Optimal. Leaf size=150 \[ \frac {2 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {a+b \sin ^{-1}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{d^2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.16, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {4701, 4653, 260, 266, 36, 29, 31} \[ \frac {2 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {a+b \sin ^{-1}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \log (x)}{d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{2 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 260
Rule 266
Rule 4653
Rule 4701
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {a+b \sin ^{-1}(c x)}{d x \sqrt {d-c^2 d x^2}}+\left (2 c^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \sin ^{-1}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \sin ^{-1}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {\left (b c^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )}{2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \sin ^{-1}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \log (x)}{d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{2 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 117, normalized size = 0.78 \[ -\frac {\sqrt {d-c^2 d x^2} \left (4 a c^2 x^2-2 a+b c x \sqrt {1-c^2 x^2} \log \left (x^2\right )+b c x \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )+2 b \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )}{2 d^2 x \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 15.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{c^{4} d^{2} x^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 239, normalized size = 1.59 \[ -\frac {a}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 a \,c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x \,c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{\left (c^{2} x^{2}-1\right ) d^{2} x}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{4}-1\right ) c}{d^{2} \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 129, normalized size = 0.86 \[ \frac {1}{2} \, b c {\left (\frac {\log \left (c x + 1\right )}{d^{\frac {3}{2}}} + \frac {\log \left (c x - 1\right )}{d^{\frac {3}{2}}} + \frac {2 \, \log \relax (x)}{d^{\frac {3}{2}}}\right )} + {\left (\frac {2 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x}\right )} b \arcsin \left (c x\right ) + {\left (\frac {2 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x}\right )} a \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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