Optimal. Leaf size=111 \[ -\frac {b c \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{3 d x^3}-\frac {b c^3 \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {4771, 14}
\begin {gather*} -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{3 d x^3}-\frac {b c \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 \log (x) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 4771
Rubi steps
\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^3}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {1-c^2 x^2}{x^3} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^3}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {1}{x^3}-\frac {c^2}{x}\right ) \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^3}-\frac {b c^3 \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 134, normalized size = 1.21 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (b c x \left (1-3 c^2 x^2\right ) \sqrt {1-c^2 x^2}+2 a \left (-1+c^2 x^2\right )^2+2 b \left (-1+c^2 x^2\right )^2 \text {ArcSin}(c x)\right )}{6 x^3 \left (-1+c^2 x^2\right )}-\frac {b c^3 \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.29, size = 1117, normalized size = 10.06
method | result | size |
default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 d \,x^{3}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} c^{6}}{3 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {2 i b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c^{3}}{3 c^{2} x^{2}-3}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} \arcsin \left (c x \right ) c^{8}}{\left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{5}}{\left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} c^{8}}{6 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3}}{3 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \arcsin \left (c x \right ) c^{6}}{\left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \left (-c^{2} x^{2}+1\right ) c^{4}}{6 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,c^{4}}{6 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \sqrt {-c^{2} x^{2}+1}\, c^{5}}{2 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{4} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{7}}{\left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {10 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \arcsin \left (c x \right ) c^{4}}{3 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \left (-c^{2} x^{2}+1\right ) c^{6}}{6 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{3}}{2 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c^{2}}{3 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) x \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{6 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) x^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{3 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) x^{3} \left (c^{2} x^{2}-1\right )}+\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 )^{2}-1\right ) c^{3}}{3 c^{2} x^{2}-3}\) | \(1117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 137, normalized size = 1.23 \begin {gather*} \frac {{\left (\left (-1\right )^{-2 \, c^{2} d x^{2} + 2 \, d} c^{2} d^{\frac {3}{2}} \log \left (-2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) + c^{2} d^{\frac {3}{2}} \log \left (x^{2} - \frac {1}{c^{2}}\right ) - \frac {\sqrt {c^{4} d x^{4} - 2 \, c^{2} d x^{2} + d} d}{x^{2}}\right )} b c}{6 \, d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b \arcsin \left (c x\right )}{3 \, d x^{3}} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a}{3 \, d x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 199 vs.
\(2 (95) = 190\).
time = 2.35, size = 414, normalized size = 3.73 \begin {gather*} \left [\frac {{\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} - d}{c^{2} x^{4} - x^{2}}\right ) - \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {-c^{2} x^{2} + 1} + 2 \, {\left (a c^{4} x^{4} - 2 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \arcsin \left (c x\right ) + a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} x^{5} - x^{3}\right )}}, -\frac {2 \, {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{2} + 1\right )} \sqrt {-d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) + \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {-c^{2} x^{2} + 1} - 2 \, {\left (a c^{4} x^{4} - 2 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \arcsin \left (c x\right ) + a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} x^{5} - x^{3}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{4}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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