Optimal. Leaf size=64 \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )}{6 x^6}-\frac {b c \sqrt {1-c^2 x^4}}{12 x^4}-\frac {1}{12} b c^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^4}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )}{6 x^6}-\frac {b c \sqrt {1-c^2 x^4}}{12 x^4}-\frac {1}{12} b c^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^4}\right ) \]
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^7} \, dx &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{6 x^6}+\frac {1}{6} b \int \frac {2 c}{x^5 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{6 x^6}+\frac {1}{3} (b c) \int \frac {1}{x^5 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{6 x^6}+\frac {1}{12} (b c) \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}}{12 x^4}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{6 x^6}+\frac {1}{24} \left (b c^3\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}}{12 x^4}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{6 x^6}-\frac {1}{12} (b c) \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}}{12 x^4}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{6 x^6}-\frac {1}{12} b c^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^4}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 69, normalized size = 1.08 \[ -\frac {a}{6 x^6}-\frac {b c \sqrt {1-c^2 x^4}}{12 x^4}-\frac {1}{12} b c^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^4}\right )-\frac {b \sin ^{-1}\left (c x^2\right )}{6 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 84, normalized size = 1.31 \[ -\frac {b c^{3} x^{6} \log \left (\sqrt {-c^{2} x^{4} + 1} + 1\right ) - b c^{3} x^{6} \log \left (\sqrt {-c^{2} x^{4} + 1} - 1\right ) + 2 \, \sqrt {-c^{2} x^{4} + 1} b c x^{2} + 4 \, b \arcsin \left (c x^{2}\right ) + 4 \, a}{24 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 301, normalized size = 4.70 \[ -\frac {\frac {b c^{7} x^{6} \arcsin \left (c x^{2}\right )}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}} + \frac {a c^{7} x^{6}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}} - \frac {b c^{6} x^{4}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}} + \frac {3 \, b c^{5} x^{2} \arcsin \left (c x^{2}\right )}{\sqrt {-c^{2} x^{4} + 1} + 1} + \frac {3 \, a c^{5} x^{2}}{\sqrt {-c^{2} x^{4} + 1} + 1} - 4 \, b c^{4} \log \left (x^{2} {\left | c \right |}\right ) + 4 \, b c^{4} \log \left (\sqrt {-c^{2} x^{4} + 1} + 1\right ) + \frac {3 \, b c^{3} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )} \arcsin \left (c x^{2}\right )}{x^{2}} + \frac {3 \, a c^{3} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}}{x^{2}} + \frac {b c^{2} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}}{x^{4}} + \frac {b c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3} \arcsin \left (c x^{2}\right )}{x^{6}} + \frac {a c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}}{x^{6}}}{48 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 61, normalized size = 0.95 \[ -\frac {a}{6 x^{6}}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{6 x^{6}}+\frac {c \left (-\frac {\sqrt {-c^{2} x^{4}+1}}{4 x^{4}}-\frac {c^{2} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{4}+1}}\right )}{4}\right )}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 81, normalized size = 1.27 \[ -\frac {1}{24} \, {\left ({\left (c^{2} \log \left (\sqrt {-c^{2} x^{4} + 1} + 1\right ) - c^{2} \log \left (\sqrt {-c^{2} x^{4} + 1} - 1\right ) + \frac {2 \, \sqrt {-c^{2} x^{4} + 1}}{x^{4}}\right )} c + \frac {4 \, \arcsin \left (c x^{2}\right )}{x^{6}}\right )} b - \frac {a}{6 \, x^{6}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.35, size = 128, normalized size = 2.00 \[ - \frac {a}{6 x^{6}} + \frac {b c \left (\begin {cases} - \frac {c^{2} \operatorname {acosh}{\left (\frac {1}{c x^{2}} \right )}}{4} - \frac {c \sqrt {-1 + \frac {1}{c^{2} x^{4}}}}{4 x^{2}} & \text {for}\: \frac {1}{\left |{c^{2} x^{4}}\right |} > 1 \\\frac {i c^{2} \operatorname {asin}{\left (\frac {1}{c x^{2}} \right )}}{4} - \frac {i c}{4 x^{2} \sqrt {1 - \frac {1}{c^{2} x^{4}}}} + \frac {i}{4 c x^{6} \sqrt {1 - \frac {1}{c^{2} x^{4}}}} & \text {otherwise} \end {cases}\right )}{3} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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