Optimal. Leaf size=99 \[ -\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}+\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}+a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Rubi [A] time = 0.27, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6152, 1807, 835, 807, 266, 63, 208} \[ -\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}+\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}+a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 1807
Rule 6152
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^4} \, dx &=c \int \frac {(1-a x)^2}{x^4 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {1}{3} \int \frac {6 a c-5 a^2 c x}{x^3 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}+\frac {a \sqrt {c-a^2 c x^2}}{x^2}+\frac {\int \frac {10 a^2 c^2-6 a^3 c^2 x}{x^2 \sqrt {c-a^2 c x^2}} \, dx}{6 c}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}+\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}-\left (a^3 c\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}+\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}-\frac {1}{2} \left (a^3 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}+\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}+a \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}+\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}+a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.15, size = 82, normalized size = 0.83 \[ a^3 \left (-\sqrt {c}\right ) \log (x)+\frac {\left (-5 a^2 x^2+3 a x-1\right ) \sqrt {c-a^2 c x^2}}{3 x^3}+a^3 \sqrt {c} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 165, normalized size = 1.67 \[ \left [\frac {3 \, a^{3} \sqrt {c} x^{3} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (5 \, a^{2} x^{2} - 3 \, a x + 1\right )}}{6 \, x^{3}}, \frac {3 \, a^{3} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (5 \, a^{2} x^{2} - 3 \, a x + 1\right )}}{3 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 210, normalized size = 2.12 \[ -\frac {1}{12} \, {\left (\frac {24 \, a^{2} c \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {-c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{\sqrt {-c}} - \frac {2 \, {\left (3 \, \pi a^{2} c - 10 \, a^{2} c\right )} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{\sqrt {-c}} + \frac {9 \, a^{2} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 3 \, a^{2} c^{3} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) - 8 \, a^{2} c^{2} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{{\left (c - \frac {c}{a x + 1}\right )}^{3}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 253, normalized size = 2.56 \[ \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right ) a^{3}-\sqrt {-a^{2} c \,x^{2}+c}\, a^{3}-\frac {2 a^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}-2 a^{4} x \sqrt {-a^{2} c \,x^{2}+c}-\frac {2 a^{4} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 c \,x^{3}}+\frac {a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c \,x^{2}}+2 a^{3} \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}+\frac {2 a^{4} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )}}{{\left (a x + 1\right )}^{2} x^{4}}\,{d x} \]
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 \[ -\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )}{x^4\,{\left (a\,x+1\right )}^2} \,d x \]
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 \[ - \int \left (- \frac {\sqrt {- a^{2} c x^{2} + c}}{a x^{5} + x^{4}}\right )\, dx - \int \frac {a x \sqrt {- a^{2} c x^{2} + c}}{a x^{5} + x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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