Optimal. Leaf size=111 \[ \frac {c^3 \sqrt {1-a^2 x^2}}{a}-\frac {4 c^3 \sqrt {1-a^2 x^2}}{a^2 x}+\frac {13 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}+\frac {c^3 \sqrt {1-a^2 x^2}}{2 a^3 x^2}+\frac {4 c^3 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.31, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6131, 6128, 1807, 1809, 844, 216, 266, 63, 208} \[ \frac {c^3 \sqrt {1-a^2 x^2}}{a}-\frac {4 c^3 \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c^3 \sqrt {1-a^2 x^2}}{2 a^3 x^2}+\frac {13 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}+\frac {4 c^3 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1807
Rule 1809
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{-\tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=-\frac {c^3 \int \frac {(1-a x)^4}{x^3 \sqrt {1-a^2 x^2}} \, dx}{a^3}\\ &=\frac {c^3 \sqrt {1-a^2 x^2}}{2 a^3 x^2}+\frac {c^3 \int \frac {8 a-13 a^2 x+8 a^3 x^2-2 a^4 x^3}{x^2 \sqrt {1-a^2 x^2}} \, dx}{2 a^3}\\ &=\frac {c^3 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {4 c^3 \sqrt {1-a^2 x^2}}{a^2 x}-\frac {c^3 \int \frac {13 a^2-8 a^3 x+2 a^4 x^2}{x \sqrt {1-a^2 x^2}} \, dx}{2 a^3}\\ &=\frac {c^3 \sqrt {1-a^2 x^2}}{a}+\frac {c^3 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {4 c^3 \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c^3 \int \frac {-13 a^4+8 a^5 x}{x \sqrt {1-a^2 x^2}} \, dx}{2 a^5}\\ &=\frac {c^3 \sqrt {1-a^2 x^2}}{a}+\frac {c^3 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {4 c^3 \sqrt {1-a^2 x^2}}{a^2 x}+\left (4 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\frac {\left (13 c^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{2 a}\\ &=\frac {c^3 \sqrt {1-a^2 x^2}}{a}+\frac {c^3 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {4 c^3 \sqrt {1-a^2 x^2}}{a^2 x}+\frac {4 c^3 \sin ^{-1}(a x)}{a}-\frac {\left (13 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 a}\\ &=\frac {c^3 \sqrt {1-a^2 x^2}}{a}+\frac {c^3 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {4 c^3 \sqrt {1-a^2 x^2}}{a^2 x}+\frac {4 c^3 \sin ^{-1}(a x)}{a}+\frac {\left (13 c^3\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 )}{2 a^3}\\ &=\frac {c^3 \sqrt {1-a^2 x^2}}{a}+\frac {c^3 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {4 c^3 \sqrt {1-a^2 x^2}}{a^2 x}+\frac {4 c^3 \sin ^{-1}(a x)}{a}+\frac {13 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 77, normalized size = 0.69 \[ \frac {c^3 \left (\frac {\sqrt {1-a^2 x^2} \left (2 a^2 x^2-8 a x+1\right )}{a^2 x^2}+13 \log \left (\sqrt {1-a^2 x^2}+1\right )-13 \log (a x)+8 \sin ^{-1}(a x)\right )}{2 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.43, size = 119, normalized size = 1.07 \[ -\frac {16 \, a^{2} c^{3} x^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 13 \, a^{2} c^{3} x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 2 \, a^{2} c^{3} x^{2} - {\left (2 \, a^{2} c^{3} x^{2} - 8 \, a c^{3} x + c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 206, normalized size = 1.86 \[ -\frac {{\left (c^{3} - \frac {16 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{a^{2} x}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}} + \frac {4 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {13 \, c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} - \frac {\frac {16 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3} {\left | a \right |}}{a^{2} x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3} {\left | a \right |}}{a^{4} x^{2}}}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 209, normalized size = 1.88 \[ -\frac {13 c^{3} \sqrt {-a^{2} x^{2}+1}}{2 a}+\frac {13 c^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a}-\frac {4 c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{a^{2} x}-4 c^{3} x \sqrt {-a^{2} x^{2}+1}-\frac {4 c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2} a^{3}}+\frac {8 c^{3} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a}+\frac {8 c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}} \]
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 \[ 3 \, a c^{3} {\left (\frac {\arcsin \left (a x\right )}{a^{2}} + \frac {\log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a^{2}}\right )} + c^{3} {\left (\frac {\arcsin \left (a x\right )}{a} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a}\right )} + \int \frac {{\left (3 \, a c^{3} x - c^{3}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{a^{4} x^{4} + a^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 113, normalized size = 1.02 \[ \frac {4\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{a}-\frac {4\,c^3\,\sqrt {1-a^2\,x^2}}{a^2\,x}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,13{}\mathrm {i}}{2\,a} \]
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 \[ \frac {c^{3} \left (\int \left (- \frac {\sqrt {- a^{2} x^{2} + 1}}{a x^{4} + x^{3}}\right )\, dx + \int \frac {3 a x \sqrt {- a^{2} x^{2} + 1}}{a x^{4} + x^{3}}\, dx + \int \left (- \frac {3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a x^{4} + x^{3}}\right )\, dx + \int \frac {a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a x^{4} + x^{3}}\, dx\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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