Optimal. Leaf size=77 \[ -\frac {c \sqrt {1-a^2 x^2}}{a}-\frac {8 c (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {4 c \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.21, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6131, 6128, 1805, 1809, 844, 216, 266, 63, 208} \[ -\frac {c \sqrt {1-a^2 x^2}}{a}-\frac {8 c (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {4 c \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1805
Rule 1809
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx &=-\frac {c \int \frac {e^{-3 \tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=-\frac {c \int \frac {(1-a x)^4}{x \left (1-a^2 x^2\right )^{3/2}} \, dx}{a}\\ &=-\frac {8 c (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {c \int \frac {-1-4 a x+a^2 x^2}{x \sqrt {1-a^2 x^2}} \, dx}{a}\\ &=-\frac {8 c (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {1-a^2 x^2}}{a}-\frac {c \int \frac {a^2+4 a^3 x}{x \sqrt {1-a^2 x^2}} \, dx}{a^3}\\ &=-\frac {8 c (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {1-a^2 x^2}}{a}-(4 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\frac {c \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{a}\\ &=-\frac {8 c (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {1-a^2 x^2}}{a}-\frac {4 c \sin ^{-1}(a x)}{a}-\frac {c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {8 c (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {1-a^2 x^2}}{a}-\frac {4 c \sin ^{-1}(a x)}{a}+\frac {c \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a^3}\\ &=-\frac {8 c (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {1-a^2 x^2}}{a}-\frac {4 c \sin ^{-1}(a x)}{a}+\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 61, normalized size = 0.79 \[ \frac {c \left (-\frac {\sqrt {1-a^2 x^2} (a x+9)}{a x+1}+\log \left (\sqrt {1-a^2 x^2}+1\right )-4 \sin ^{-1}(a x)-\log (x)\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.43, size = 97, normalized size = 1.26 \[ -\frac {9 \, a c x - 8 \, {\left (a c x + c\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (a c x + c\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a c x + 9 \, c\right )} + 9 \, c}{a^{2} x + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.04, size = 104, normalized size = 1.35 \[ -\frac {4 \, c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c}{a} + \frac {16 \, c}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 223, normalized size = 2.90 \[ -\frac {c \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a}-\frac {c \sqrt {-a^{2} x^{2}+1}}{a}+\frac {c \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a}-\frac {2 c \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{4} \left (x +\frac {1}{a}\right )^{3}}-\frac {3 c \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{3} \left (x +\frac {1}{a}\right )^{2}}-\frac {8 c \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a}-4 c \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x -\frac {4 c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 102, normalized size = 1.32 \[ \frac {c\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{a}-\frac {c\,\sqrt {1-a^2\,x^2}}{a}-\frac {4\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {8\,c\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
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 \[ \frac {c \left (\int \left (- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\right )\, dx + \int \frac {a x \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\, dx + \int \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\, dx + \int \left (- \frac {a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\right )\, dx\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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