Optimal. Leaf size=111 \[ -\frac {c^2 \sqrt {1-a^2 x^2}}{a}-\frac {c^2 \sqrt {1-a^2 x^2}}{a^2 x}-\frac {16 c^2 (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {5 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {5 c^2 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.30, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {6131, 6128, 1805, 1807, 1809, 844, 216, 266, 63, 208} \[ -\frac {c^2 \sqrt {1-a^2 x^2}}{a}-\frac {c^2 \sqrt {1-a^2 x^2}}{a^2 x}-\frac {16 c^2 (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {5 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {5 c^2 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1805
Rule 1807
Rule 1809
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx &=\frac {c^2 \int \frac {e^{-3 \tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \frac {(1-a x)^5}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2}\\ &=-\frac {16 c^2 (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c^2 \int \frac {-1+5 a x+5 a^2 x^2-a^3 x^3}{x^2 \sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac {16 c^2 (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c^2 \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c^2 \int \frac {-5 a-5 a^2 x+a^3 x^2}{x \sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac {16 c^2 (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c^2 \sqrt {1-a^2 x^2}}{a}-\frac {c^2 \sqrt {1-a^2 x^2}}{a^2 x}-\frac {c^2 \int \frac {5 a^3+5 a^4 x}{x \sqrt {1-a^2 x^2}} \, dx}{a^4}\\ &=-\frac {16 c^2 (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c^2 \sqrt {1-a^2 x^2}}{a}-\frac {c^2 \sqrt {1-a^2 x^2}}{a^2 x}-\left (5 c^2\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\frac {\left (5 c^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{a}\\ &=-\frac {16 c^2 (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c^2 \sqrt {1-a^2 x^2}}{a}-\frac {c^2 \sqrt {1-a^2 x^2}}{a^2 x}-\frac {5 c^2 \sin ^{-1}(a x)}{a}-\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {16 c^2 (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c^2 \sqrt {1-a^2 x^2}}{a}-\frac {c^2 \sqrt {1-a^2 x^2}}{a^2 x}-\frac {5 c^2 \sin ^{-1}(a x)}{a}+\frac {\left (5 c^2\right ) \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 {16 c^2 (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c^2 \sqrt {1-a^2 x^2}}{a}-\frac {c^2 \sqrt {1-a^2 x^2}}{a^2 x}-\frac {5 c^2 \sin ^{-1}(a x)}{a}+\frac {5 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 81, normalized size = 0.73 \[ \frac {c^2 \left (-\frac {\sqrt {1-a^2 x^2} \left (a^2 x^2+18 a x+1\right )}{a x (a x+1)}+5 \log \left (\sqrt {1-a^2 x^2}+1\right )-5 \log (a x)-5 \sin ^{-1}(a x)\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.60, size = 149, normalized size = 1.34 \[ -\frac {17 \, a^{2} c^{2} x^{2} + 17 \, a c^{2} x - 10 \, {\left (a^{2} c^{2} x^{2} + a c^{2} x\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 5 \, {\left (a^{2} c^{2} x^{2} + a c^{2} x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (a^{2} c^{2} x^{2} + 18 \, a c^{2} x + c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{a^{3} x^{2} + a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 197, normalized size = 1.77 \[ -\frac {5 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {5 \, c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} + \frac {{\left (c^{2} + \frac {65 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{a^{2} x}\right )} a^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{2 \, a^{2} x {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 329, normalized size = 2.96 \[ -\frac {5 c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a}-\frac {5 c^{2} \sqrt {-a^{2} x^{2}+1}}{a}+\frac {5 c^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a}-\frac {c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{a^{2} x}-c^{2} x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}-\frac {3 c^{2} x \sqrt {-a^{2} x^{2}+1}}{2}-\frac {3 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}-\frac {4 c^{2} \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{4} \left (x +\frac {1}{a}\right )^{3}}-\frac {4 c^{2} \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{3} \left (x +\frac {1}{a}\right )^{2}}-\frac {7 c^{2} \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a}-\frac {7 c^{2} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{2}-\frac {7 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{2}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 138, normalized size = 1.24 \[ \frac {16\,c^2\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{a}-\frac {5\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{a^2\,x}-\frac {c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{a} \]
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 \[ \frac {c^{2} \left (\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\, dx + \int \left (- \frac {2 a x \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\right )\, dx + \int \frac {2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\, dx + \int \left (- \frac {a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\right )\, dx\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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