Optimal. Leaf size=91 \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}-\frac {b c \sqrt {1-c^2 x^4}}{60 x^{10}}-\frac {2 b c^5 \sqrt {1-c^2 x^4}}{45 x^2}-\frac {b c^3 \sqrt {1-c^2 x^4}}{45 x^6} \]
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Rubi [A] time = 0.05, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4842, 12, 271, 264} \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}-\frac {2 b c^5 \sqrt {1-c^2 x^4}}{45 x^2}-\frac {b c^3 \sqrt {1-c^2 x^4}}{45 x^6}-\frac {b c \sqrt {1-c^2 x^4}}{60 x^{10}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 271
Rule 4842
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (c x^2\right )}{x^{13}} \, dx &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}+\frac {1}{12} b \int \frac {2 c}{x^{11} \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}+\frac {1}{6} (b c) \int \frac {1}{x^{11} \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {b c \sqrt {1-c^2 x^4}}{60 x^{10}}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}+\frac {1}{15} \left (2 b c^3\right ) \int \frac {1}{x^7 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {b c \sqrt {1-c^2 x^4}}{60 x^{10}}-\frac {b c^3 \sqrt {1-c^2 x^4}}{45 x^6}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}+\frac {1}{45} \left (4 b c^5\right ) \int \frac {1}{x^3 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {b c \sqrt {1-c^2 x^4}}{60 x^{10}}-\frac {b c^3 \sqrt {1-c^2 x^4}}{45 x^6}-\frac {2 b c^5 \sqrt {1-c^2 x^4}}{45 x^2}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 68, normalized size = 0.75 \[ \frac {1}{2} b \left (-\frac {c \sqrt {1-c^2 x^4} \left (8 c^4 x^8+4 c^2 x^4+3\right )}{90 x^{10}}-\frac {\sin ^{-1}\left (c x^2\right )}{6 x^{12}}\right )-\frac {a}{12 x^{12}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 64, normalized size = 0.70 \[ \frac {15 \, a x^{12} - 15 \, b \arcsin \left (c x^{2}\right ) - {\left (8 \, b c^{5} x^{10} + 4 \, b c^{3} x^{6} + 3 \, b c x^{2}\right )} \sqrt {-c^{2} x^{4} + 1} - 15 \, a}{180 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 504, normalized size = 5.54 \[ -\frac {\frac {15 \, b c^{13} x^{12} \arcsin \left (c x^{2}\right )}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{6}} + \frac {15 \, a c^{13} x^{12}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{6}} - \frac {6 \, b c^{12} x^{10}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{5}} + \frac {90 \, b c^{11} x^{8} \arcsin \left (c x^{2}\right )}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4}} + \frac {90 \, a c^{11} x^{8}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4}} - \frac {50 \, b c^{10} x^{6}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}} + \frac {225 \, b c^{9} x^{4} \arcsin \left (c x^{2}\right )}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}} + \frac {225 \, a c^{9} x^{4}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}} - \frac {300 \, b c^{8} x^{2}}{\sqrt {-c^{2} x^{4} + 1} + 1} + 300 \, b c^{7} \arcsin \left (c x^{2}\right ) + 300 \, a c^{7} + \frac {300 \, b c^{6} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}}{x^{2}} + \frac {225 \, b c^{5} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2} \arcsin \left (c x^{2}\right )}{x^{4}} + \frac {225 \, a c^{5} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}}{x^{4}} + \frac {50 \, b c^{4} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}}{x^{6}} + \frac {90 \, b c^{3} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4} \arcsin \left (c x^{2}\right )}{x^{8}} + \frac {90 \, a c^{3} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4}}{x^{8}} + \frac {6 \, b c^{2} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{5}}{x^{10}} + \frac {15 \, b c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{6} \arcsin \left (c x^{2}\right )}{x^{12}} + \frac {15 \, a c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{6}}{x^{12}}}{11520 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 72, normalized size = 0.79 \[ -\frac {a}{12 x^{12}}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{12 x^{12}}+\frac {c \left (c \,x^{2}-1\right ) \left (c \,x^{2}+1\right ) \left (8 c^{4} x^{8}+4 c^{2} x^{4}+3\right )}{180 x^{10} \sqrt {-c^{2} x^{4}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 82, normalized size = 0.90 \[ -\frac {1}{180} \, {\left ({\left (\frac {15 \, \sqrt {-c^{2} x^{4} + 1} c^{4}}{x^{2}} + \frac {10 \, {\left (-c^{2} x^{4} + 1\right )}^{\frac {3}{2}} c^{2}}{x^{6}} + \frac {3 \, {\left (-c^{2} x^{4} + 1\right )}^{\frac {5}{2}}}{x^{10}}\right )} c + \frac {15 \, \arcsin \left (c x^{2}\right )}{x^{12}}\right )} b - \frac {a}{12 \, x^{12}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^{13}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.81, size = 170, normalized size = 1.87 \[ - \frac {a}{12 x^{12}} + \frac {b c \left (\begin {cases} - \frac {4 c^{5} \sqrt {-1 + \frac {1}{c^{2} x^{4}}}}{15} - \frac {2 c^{3} \sqrt {-1 + \frac {1}{c^{2} x^{4}}}}{15 x^{4}} - \frac {c \sqrt {-1 + \frac {1}{c^{2} x^{4}}}}{10 x^{8}} & \text {for}\: \frac {1}{\left |{c^{2} x^{4}}\right |} > 1 \\- \frac {4 i c^{5} \sqrt {1 - \frac {1}{c^{2} x^{4}}}}{15} - \frac {2 i c^{3} \sqrt {1 - \frac {1}{c^{2} x^{4}}}}{15 x^{4}} - \frac {i c \sqrt {1 - \frac {1}{c^{2} x^{4}}}}{10 x^{8}} & \text {otherwise} \end {cases}\right )}{6} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{12 x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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