Optimal. Leaf size=130 \[ -\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}+\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7}{8} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )+\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x} \]
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Rubi [A] time = 0.31, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}+\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {7}{8} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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^5} \, dx &=c \int \frac {(1-a x)^2}{x^5 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {1}{4} \int \frac {8 a c-7 a^2 c x}{x^4 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}+\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}+\frac {\int \frac {21 a^2 c^2-16 a^3 c^2 x}{x^3 \sqrt {c-a^2 c x^2}} \, dx}{12 c}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}+\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\int \frac {32 a^3 c^3-21 a^4 c^3 x}{x^2 \sqrt {c-a^2 c x^2}} \, dx}{24 c^2}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}+\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}+\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}+\frac {1}{8} \left (7 a^4 c\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}+\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}+\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}+\frac {1}{16} \left (7 a^4 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}}{4 x^4}+\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}+\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}-\frac {1}{8} \left (7 a^2\right ) \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}}{4 x^4}+\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}+\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}-\frac {7}{8} a^4 \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 = 0.16, size = 95, normalized size = 0.73 \[ \frac {7}{8} a^4 \sqrt {c} \log (x)-\frac {7}{8} a^4 \sqrt {c} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )+\frac {\left (32 a^3 x^3-21 a^2 x^2+16 a x-6\right ) \sqrt {c-a^2 c x^2}}{24 x^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.70, size = 181, normalized size = 1.39 \[ \left [\frac {21 \, a^{4} \sqrt {c} x^{4} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, x^{4}}, -\frac {21 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - {\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 258, normalized size = 1.98 \[ \frac {1}{192} \, {\left (\frac {336 \, a^{3} c \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {-c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{\sqrt {-c}} - \frac {4 \, {\left (21 \, \pi a^{3} c - 64 \, a^{3} c\right )} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{\sqrt {-c}} + \frac {75 \, a^{3} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 83 \, a^{3} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{2} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 21 \, a^{3} c^{4} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) - 77 \, a^{3} c^{3} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{{\left (c - \frac {c}{a x + 1}\right )}^{4}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 279, normalized size = 2.15 \[ -\frac {7 \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right ) a^{4}}{8}+\frac {7 \sqrt {-a^{2} c \,x^{2}+c}\, a^{4}}{8}+\frac {2 a^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}+2 a^{5} x \sqrt {-a^{2} c \,x^{2}+c}+\frac {2 a^{5} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}+\frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 c \,x^{3}}-\frac {9 a^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{8 c \,x^{2}}-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 c \,x^{4}}-2 a^{4} \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}-\frac {2 a^{5} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2} c}} \]
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 \[ -\int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )}}{{\left (a x + 1\right )}^{2} x^{5}}\,{d x} \]
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 \[ -\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )}{x^5\,{\left (a\,x+1\right )}^2} \,d x \]
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 \[ - \int \left (- \frac {\sqrt {- a^{2} c x^{2} + c}}{a x^{6} + x^{5}}\right )\, dx - \int \frac {a x \sqrt {- a^{2} c x^{2} + c}}{a x^{6} + x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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