Optimal. Leaf size=111 \[ \frac {c^3 \sqrt {1-a^2 x^2}}{a}+\frac {c^3 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {4 c^3 \sqrt {1-a^2 x^2}}{a^2 x}+\frac {4 c^3 \text {ArcSin}(a x)}{a}+\frac {13 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \]
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Rubi [A]
time = 0.20, 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 = {6266, 6263,
1821, 1823, 858, 222, 272, 65, 214} \begin {gather*} \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 \text {ArcSin}(a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 1821
Rule 1823
Rule 6263
Rule 6266
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 ) \text {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 ) \text {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 = 0.16, size = 77, normalized size = 0.69 \begin {gather*} \frac {c^3 \left (\frac {\sqrt {1-a^2 x^2} \left (1-8 a x+2 a^2 x^2\right )}{a^2 x^2}+8 \text {ArcSin}(a x)-13 \log (a x)+13 \log \left (1+\sqrt {1-a^2 x^2}\right )\right )}{2 a} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.78, size = 195, normalized size = 1.76
method | result | size |
risch | \(\frac {\left (8 a^{3} x^{3}-a^{2} x^{2}-8 a x +1\right ) c^{3}}{2 x^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3}}+\frac {\left (a^{2} \sqrt {-a^{2} x^{2}+1}+\frac {4 a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {13 a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right ) c^{3}}{a^{3}}\) | \(119\) |
default | \(\frac {c^{3} \left (8 a^{2} \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )+\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {13 a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}+4 a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{3}}\) | \(195\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 119, normalized size = 1.07 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 206 vs.
\(2 (99) = 198\).
time = 0.42, size = 206, normalized size = 1.86 \begin {gather*} -\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}\left (a\right )}{{\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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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