Optimal. Leaf size=140 \[ \frac {c^4 \sqrt {1-a^2 x^2}}{a}-\frac {c^4 \sqrt {1-a^2 x^2}}{3 a^4 x^3}+\frac {5 c^4 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {32 c^4 \sqrt {1-a^2 x^2}}{3 a^2 x}+\frac {5 c^4 \text {ArcSin}(a x)}{a}+\frac {25 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \]
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Rubi [A]
time = 0.25, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 11, 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^4 \sqrt {1-a^2 x^2}}{a}-\frac {32 c^4 \sqrt {1-a^2 x^2}}{3 a^2 x}+\frac {25 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}-\frac {c^4 \sqrt {1-a^2 x^2}}{3 a^4 x^3}+\frac {5 c^4 \sqrt {1-a^2 x^2}}{2 a^3 x^2}+\frac {5 c^4 \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 )^4 \, dx &=\frac {c^4 \int \frac {e^{-\tanh ^{-1}(a x)} (1-a x)^4}{x^4} \, dx}{a^4}\\ &=\frac {c^4 \int \frac {(1-a x)^5}{x^4 \sqrt {1-a^2 x^2}} \, dx}{a^4}\\ &=-\frac {c^4 \sqrt {1-a^2 x^2}}{3 a^4 x^3}-\frac {c^4 \int \frac {15 a-32 a^2 x+30 a^3 x^2-15 a^4 x^3+3 a^5 x^4}{x^3 \sqrt {1-a^2 x^2}} \, dx}{3 a^4}\\ &=-\frac {c^4 \sqrt {1-a^2 x^2}}{3 a^4 x^3}+\frac {5 c^4 \sqrt {1-a^2 x^2}}{2 a^3 x^2}+\frac {c^4 \int \frac {64 a^2-75 a^3 x+30 a^4 x^2-6 a^5 x^3}{x^2 \sqrt {1-a^2 x^2}} \, dx}{6 a^4}\\ &=-\frac {c^4 \sqrt {1-a^2 x^2}}{3 a^4 x^3}+\frac {5 c^4 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {32 c^4 \sqrt {1-a^2 x^2}}{3 a^2 x}-\frac {c^4 \int \frac {75 a^3-30 a^4 x+6 a^5 x^2}{x \sqrt {1-a^2 x^2}} \, dx}{6 a^4}\\ &=\frac {c^4 \sqrt {1-a^2 x^2}}{a}-\frac {c^4 \sqrt {1-a^2 x^2}}{3 a^4 x^3}+\frac {5 c^4 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {32 c^4 \sqrt {1-a^2 x^2}}{3 a^2 x}+\frac {c^4 \int \frac {-75 a^5+30 a^6 x}{x \sqrt {1-a^2 x^2}} \, dx}{6 a^6}\\ &=\frac {c^4 \sqrt {1-a^2 x^2}}{a}-\frac {c^4 \sqrt {1-a^2 x^2}}{3 a^4 x^3}+\frac {5 c^4 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {32 c^4 \sqrt {1-a^2 x^2}}{3 a^2 x}+\left (5 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\frac {\left (25 c^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{2 a}\\ &=\frac {c^4 \sqrt {1-a^2 x^2}}{a}-\frac {c^4 \sqrt {1-a^2 x^2}}{3 a^4 x^3}+\frac {5 c^4 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {32 c^4 \sqrt {1-a^2 x^2}}{3 a^2 x}+\frac {5 c^4 \sin ^{-1}(a x)}{a}-\frac {\left (25 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 a}\\ &=\frac {c^4 \sqrt {1-a^2 x^2}}{a}-\frac {c^4 \sqrt {1-a^2 x^2}}{3 a^4 x^3}+\frac {5 c^4 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {32 c^4 \sqrt {1-a^2 x^2}}{3 a^2 x}+\frac {5 c^4 \sin ^{-1}(a x)}{a}+\frac {\left (25 c^4\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^4 \sqrt {1-a^2 x^2}}{a}-\frac {c^4 \sqrt {1-a^2 x^2}}{3 a^4 x^3}+\frac {5 c^4 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {32 c^4 \sqrt {1-a^2 x^2}}{3 a^2 x}+\frac {5 c^4 \sin ^{-1}(a x)}{a}+\frac {25 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 85, normalized size = 0.61 \begin {gather*} \frac {c^4 \left (\frac {\sqrt {1-a^2 x^2} \left (-2+15 a x-64 a^2 x^2+6 a^3 x^3\right )}{a^3 x^3}+30 \text {ArcSin}(a x)-75 \log (a x)+75 \log \left (1+\sqrt {1-a^2 x^2}\right )\right )}{6 a} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(250\) vs.
\(2(122)=244\).
time = 0.90, size = 251, normalized size = 1.79
method | result | size |
risch | \(\frac {\left (64 a^{4} x^{4}-15 a^{3} x^{3}-62 a^{2} x^{2}+15 a x -2\right ) c^{4}}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{4}}+\frac {\left (a^{3} \sqrt {-a^{2} x^{2}+1}+\frac {5 a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {25 a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right ) c^{4}}{a^{4}}\) | \(127\) |
default | \(\frac {c^{4} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 x^{3}}+16 a^{3} \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )-5 a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )+11 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )-15 a^{3} \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\right )}{a^{4}}\) | \(251\) |
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 = 132, normalized size = 0.94 \begin {gather*} -\frac {60 \, a^{3} c^{4} x^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 75 \, a^{3} c^{4} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 6 \, a^{3} c^{4} x^{3} - {\left (6 \, a^{3} c^{4} x^{3} - 64 \, a^{2} c^{4} x^{2} + 15 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}} \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^{4} \left (\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx + \int \left (- \frac {4 a x \sqrt {- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\right )\, dx + \int \frac {6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx + \int \left (- \frac {4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\right )\, dx + \int \frac {a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx\right )}{a^{4}} \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. 262 vs.
\(2 (122) = 244\).
time = 0.42, size = 262, normalized size = 1.87 \begin {gather*} \frac {{\left (c^{4} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} + \frac {129 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} + \frac {5 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {25 \, c^{4} \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^{4}}{a} - \frac {\frac {129 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{x} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{2} x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{4} x^{3}}}{24 \, a^{2} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.85, size = 136, normalized size = 0.97 \begin {gather*} \frac {5\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{a}-\frac {32\,c^4\,\sqrt {1-a^2\,x^2}}{3\,a^2\,x}+\frac {5\,c^4\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{3\,a^4\,x^3}-\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,25{}\mathrm {i}}{2\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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