Optimal. Leaf size=139 \[ -a^2 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {a (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}{3 x}-\frac {(a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}-\frac {a^3 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{\sqrt {1-a x} \sqrt {a x+1}} \]
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Rubi [A] time = 0.39, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6159, 6129, 96, 94, 92, 208} \[ -a^2 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {a (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}{3 x}-\frac {(a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}-\frac {a^3 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{\sqrt {1-a x} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 92
Rule 94
Rule 96
Rule 208
Rule 6129
Rule 6159
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {1-a x} \sqrt {1+a x}}{x^4} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1+a x)^{3/2}}{x^4 \sqrt {1-a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}{3 x^2}+\frac {\left (2 a \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1+a x)^{3/2}}{x^3 \sqrt {1-a x}} \, dx}{3 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {a \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{3 x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}{3 x^2}+\frac {\left (a^2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {\sqrt {1+a x}}{x^2 \sqrt {1-a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-a^2 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{3 x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}{3 x^2}+\frac {\left (a^3 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-a^2 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{3 x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}{3 x^2}-\frac {\left (a^4 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-a^2 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{3 x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}{3 x^2}-\frac {a^3 \sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 71, normalized size = 0.51 \[ \frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-5 a^2 x^2+\frac {3 a^3 x^3 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )}{\sqrt {a^2 x^2-1}}-3 a x-1\right )}{3 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.76, size = 201, normalized size = 1.45 \[ \left [\frac {3 \, a^{2} \sqrt {-c} x^{2} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (5 \, a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, x^{2}}, \frac {3 \, a^{2} \sqrt {c} x^{2} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (5 \, a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 11.14, size = 231, normalized size = 1.66 \[ -\frac {2}{3} \, {\left (3 \, a \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x) - \frac {3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} a c \mathrm {sgn}\relax (x) - 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} c^{\frac {3}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - 12 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a c^{3} \mathrm {sgn}\relax (x) - 5 \, c^{\frac {7}{2}} {\left | a \right |} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{3}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 378, normalized size = 2.72 \[ \frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, a \left (-6 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{4} a^{3} c +6 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{3}+6 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{3} a -6 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{3} a -6 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{3} a^{2} c +3 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{3} a^{2} c +3 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x \,a^{2}+3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{3} c^{2}+a \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\right )}{3 x^{2} \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c} \]
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 \[ -\frac {{\left (2 \, a^{2} \sqrt {c} x^{2} + \sqrt {c}\right )} \sqrt {a x + 1} \sqrt {a x - 1}}{3 \, a x^{3}} - \int \frac {{\left (a \sqrt {c} x + 2 \, \sqrt {c}\right )} \sqrt {a x + 1} \sqrt {a x - 1}}{a^{2} x^{5} - x^{3}}\,{d x} \]
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 \[ -\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^2}{x^3\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{4} - x^{3}}\, dx - \int \frac {a x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{4} - x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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