Optimal. Leaf size=89 \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )}{10 x^{10}}-\frac {b c \sqrt {1-c^2 x^4}}{40 x^8}-\frac {3}{80} b c^5 \tanh ^{-1}\left (\sqrt {1-c^2 x^4}\right )-\frac {3 b c^3 \sqrt {1-c^2 x^4}}{80 x^4} \]
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Rubi [A] time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4842, 12, 266, 51, 63, 208} \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )}{10 x^{10}}-\frac {3 b c^3 \sqrt {1-c^2 x^4}}{80 x^4}-\frac {b c \sqrt {1-c^2 x^4}}{40 x^8}-\frac {3}{80} b c^5 \tanh ^{-1}\left (\sqrt {1-c^2 x^4}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 208
Rule 266
Rule 4842
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (c x^2\right )}{x^{11}} \, dx &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{10 x^{10}}+\frac {1}{10} b \int \frac {2 c}{x^9 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{10 x^{10}}+\frac {1}{5} (b c) \int \frac {1}{x^9 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{10 x^{10}}+\frac {1}{20} (b c) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-c^2 x}} \, dx,x,x^4\right )\\ &=-\frac {b c \sqrt {1-c^2 x^4}}{40 x^8}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{10 x^{10}}+\frac {1}{80} \left (3 b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-c^2 x}} \, dx,x,x^4\right )\\ &=-\frac {b c \sqrt {1-c^2 x^4}}{40 x^8}-\frac {3 b c^3 \sqrt {1-c^2 x^4}}{80 x^4}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{10 x^{10}}+\frac {1}{160} \left (3 b c^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^4\right )\\ &=-\frac {b c \sqrt {1-c^2 x^4}}{40 x^8}-\frac {3 b c^3 \sqrt {1-c^2 x^4}}{80 x^4}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{10 x^{10}}-\frac {1}{80} \left (3 b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^4}\right )\\ &=-\frac {b c \sqrt {1-c^2 x^4}}{40 x^8}-\frac {3 b c^3 \sqrt {1-c^2 x^4}}{80 x^4}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{10 x^{10}}-\frac {3}{80} b c^5 \tanh ^{-1}\left (\sqrt {1-c^2 x^4}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 63, normalized size = 0.71 \[ -\frac {a}{10 x^{10}}-\frac {1}{10} b c^5 \sqrt {1-c^2 x^4} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-c^2 x^4\right )-\frac {b \sin ^{-1}\left (c x^2\right )}{10 x^{10}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 97, normalized size = 1.09 \[ -\frac {3 \, b c^{5} x^{10} \log \left (\sqrt {-c^{2} x^{4} + 1} + 1\right ) - 3 \, b c^{5} x^{10} \log \left (\sqrt {-c^{2} x^{4} + 1} - 1\right ) + 16 \, b \arcsin \left (c x^{2}\right ) + 2 \, {\left (3 \, b c^{3} x^{6} + 2 \, b c x^{2}\right )} \sqrt {-c^{2} x^{4} + 1} + 16 \, a}{160 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.55, size = 467, normalized size = 5.25 \[ -\frac {\frac {2 \, b c^{11} x^{10} \arcsin \left (c x^{2}\right )}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{5}} + \frac {2 \, a c^{11} x^{10}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{5}} - \frac {b c^{10} x^{8}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4}} + \frac {10 \, b c^{9} x^{6} \arcsin \left (c x^{2}\right )}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}} + \frac {10 \, a c^{9} x^{6}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}} - \frac {8 \, b c^{8} x^{4}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}} + \frac {20 \, b c^{7} x^{2} \arcsin \left (c x^{2}\right )}{\sqrt {-c^{2} x^{4} + 1} + 1} + \frac {20 \, a c^{7} x^{2}}{\sqrt {-c^{2} x^{4} + 1} + 1} - 24 \, b c^{6} \log \left (x^{2} {\left | c \right |}\right ) + 24 \, b c^{6} \log \left (\sqrt {-c^{2} x^{4} + 1} + 1\right ) + \frac {20 \, b c^{5} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )} \arcsin \left (c x^{2}\right )}{x^{2}} + \frac {20 \, a c^{5} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}}{x^{2}} + \frac {8 \, b c^{4} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}}{x^{4}} + \frac {10 \, b c^{3} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3} \arcsin \left (c x^{2}\right )}{x^{6}} + \frac {10 \, a c^{3} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}}{x^{6}} + \frac {b c^{2} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4}}{x^{8}} + \frac {2 \, b c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{5} \arcsin \left (c x^{2}\right )}{x^{10}} + \frac {2 \, a c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{5}}{x^{10}}}{640 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 0.94 \[ -\frac {a}{10 x^{10}}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{10 x^{10}}+\frac {c \left (-\frac {\sqrt {-c^{2} x^{4}+1}}{8 x^{8}}+\frac {3 c^{2} \left (-\frac {\sqrt {-c^{2} x^{4}+1}}{2 x^{4}}-\frac {c^{2} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{4}+1}}\right )}{2}\right )}{8}\right )}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 125, normalized size = 1.40 \[ -\frac {1}{160} \, {\left ({\left (3 \, c^{4} \log \left (\sqrt {-c^{2} x^{4} + 1} + 1\right ) - 3 \, c^{4} \log \left (\sqrt {-c^{2} x^{4} + 1} - 1\right ) - \frac {2 \, {\left (3 \, {\left (-c^{2} x^{4} + 1\right )}^{\frac {3}{2}} c^{4} - 5 \, \sqrt {-c^{2} x^{4} + 1} c^{4}\right )}}{2 \, c^{2} x^{4} + {\left (c^{2} x^{4} - 1\right )}^{2} - 1}\right )} c + \frac {16 \, \arcsin \left (c x^{2}\right )}{x^{10}}\right )} b - \frac {a}{10 \, x^{10}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^{11}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.58, size = 201, normalized size = 2.26 \[ - \frac {a}{10 x^{10}} + \frac {b c \left (\begin {cases} - \frac {3 c^{4} \operatorname {acosh}{\left (\frac {1}{c x^{2}} \right )}}{16} + \frac {3 c^{3}}{16 x^{2} \sqrt {-1 + \frac {1}{c^{2} x^{4}}}} - \frac {c}{16 x^{6} \sqrt {-1 + \frac {1}{c^{2} x^{4}}}} - \frac {1}{8 c x^{10} \sqrt {-1 + \frac {1}{c^{2} x^{4}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{4}}\right |} > 1 \\\frac {3 i c^{4} \operatorname {asin}{\left (\frac {1}{c x^{2}} \right )}}{16} - \frac {3 i c^{3}}{16 x^{2} \sqrt {1 - \frac {1}{c^{2} x^{4}}}} + \frac {i c}{16 x^{6} \sqrt {1 - \frac {1}{c^{2} x^{4}}}} + \frac {i}{8 c x^{10} \sqrt {1 - \frac {1}{c^{2} x^{4}}}} & \text {otherwise} \end {cases}\right )}{5} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{10 x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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