Optimal. Leaf size=66 \[ -\frac {b c \sqrt {1-c^2 x^4}}{24 x^6}-\frac {b c^3 \sqrt {1-c^2 x^4}}{12 x^2}-\frac {a+b \text {ArcSin}\left (c x^2\right )}{8 x^8} \]
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Rubi [A]
time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4926, 12, 277,
270} \begin {gather*} -\frac {a+b \text {ArcSin}\left (c x^2\right )}{8 x^8}-\frac {b c \sqrt {1-c^2 x^4}}{24 x^6}-\frac {b c^3 \sqrt {1-c^2 x^4}}{12 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 277
Rule 4926
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (c x^2\right )}{x^9} \, dx &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{8 x^8}+\frac {1}{8} b \int \frac {2 c}{x^7 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{8 x^8}+\frac {1}{4} (b c) \int \frac {1}{x^7 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {b c \sqrt {1-c^2 x^4}}{24 x^6}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{8 x^8}+\frac {1}{6} \left (b c^3\right ) \int \frac {1}{x^3 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {b c \sqrt {1-c^2 x^4}}{24 x^6}-\frac {b c^3 \sqrt {1-c^2 x^4}}{12 x^2}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{8 x^8}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 60, normalized size = 0.91 \begin {gather*} -\frac {a}{8 x^8}+\frac {1}{2} b \left (-\frac {c \sqrt {1-c^2 x^4} \left (1+2 c^2 x^4\right )}{12 x^6}-\frac {\text {ArcSin}\left (c x^2\right )}{4 x^8}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 64, normalized size = 0.97
method | result | size |
default | \(-\frac {a}{8 x^{8}}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{8 x^{8}}+\frac {c \left (c \,x^{2}-1\right ) \left (c \,x^{2}+1\right ) \left (2 c^{2} x^{4}+1\right )}{24 x^{6} \sqrt {-c^{2} x^{4}+1}}\right )\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 61, normalized size = 0.92 \begin {gather*} -\frac {1}{24} \, {\left (c {\left (\frac {3 \, \sqrt {-c^{2} x^{4} + 1} c^{2}}{x^{2}} + \frac {{\left (-c^{2} x^{4} + 1\right )}^{\frac {3}{2}}}{x^{6}}\right )} + \frac {3 \, \arcsin \left (c x^{2}\right )}{x^{8}}\right )} b - \frac {a}{8 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.84, size = 54, normalized size = 0.82 \begin {gather*} \frac {3 \, a x^{8} - 3 \, b \arcsin \left (c x^{2}\right ) - {\left (2 \, b c^{3} x^{6} + b c x^{2}\right )} \sqrt {-c^{2} x^{4} + 1} - 3 \, a}{24 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.50, size = 112, normalized size = 1.70 \begin {gather*} - \frac {a}{8 x^{8}} + \frac {b c \left (\begin {cases} - \frac {i c^{2} \sqrt {c^{2} x^{4} - 1}}{3 x^{2}} - \frac {i \sqrt {c^{2} x^{4} - 1}}{6 x^{6}} & \text {for}\: \left |{c^{2} x^{4}}\right | > 1 \\- \frac {c^{2} \sqrt {- c^{2} x^{4} + 1}}{3 x^{2}} - \frac {\sqrt {- c^{2} x^{4} + 1}}{6 x^{6}} & \text {otherwise} \end {cases}\right )}{4} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{8 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 342 vs.
\(2 (56) = 112\).
time = 0.44, size = 342, normalized size = 5.18 \begin {gather*} -\frac {\frac {3 \, b c^{9} x^{8} \arcsin \left (c x^{2}\right )}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4}} + \frac {3 \, a c^{9} x^{8}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4}} - \frac {2 \, b c^{8} x^{6}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}} + \frac {12 \, b c^{7} x^{4} \arcsin \left (c x^{2}\right )}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}} + \frac {12 \, a c^{7} x^{4}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}} - \frac {18 \, b c^{6} x^{2}}{\sqrt {-c^{2} x^{4} + 1} + 1} + 18 \, b c^{5} \arcsin \left (c x^{2}\right ) + 18 \, a c^{5} + \frac {18 \, b c^{4} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}}{x^{2}} + \frac {12 \, b c^{3} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2} \arcsin \left (c x^{2}\right )}{x^{4}} + \frac {12 \, a c^{3} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}}{x^{4}} + \frac {2 \, b c^{2} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}}{x^{6}} + \frac {3 \, b c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4} \arcsin \left (c x^{2}\right )}{x^{8}} + \frac {3 \, a c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4}}{x^{8}}}{384 \, c} \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 = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^9} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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