Optimal. Leaf size=103 \[ \frac {c^2 (2-3 a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 (2+3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac {c^2 \text {ArcSin}(a x)}{a}+\frac {3 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \]
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Rubi [A]
time = 0.11, antiderivative size = 103, 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 = {6292, 6283,
825, 827, 858, 222, 272, 65, 214} \begin {gather*} \frac {c^2 (2-3 a x) \sqrt {1-a^2 x^2}}{2 a^2 x}+\frac {3 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}-\frac {c^2 (3 a x+2) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac {c^2 \text {ArcSin}(a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 825
Rule 827
Rule 858
Rule 6283
Rule 6292
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx &=\frac {c^2 \int \frac {e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4}\\ &=\frac {c^2 \int \frac {(1+a x) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{a^4}\\ &=-\frac {c^2 (2+3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}-\frac {c^2 \int \frac {\left (4 a^2+6 a^3 x\right ) \sqrt {1-a^2 x^2}}{x^2} \, dx}{4 a^4}\\ &=\frac {c^2 (2-3 a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 (2+3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac {c^2 \int \frac {-12 a^3+8 a^4 x}{x \sqrt {1-a^2 x^2}} \, dx}{8 a^4}\\ &=\frac {c^2 (2-3 a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 (2+3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+c^2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\frac {\left (3 c^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{2 a}\\ &=\frac {c^2 (2-3 a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 (2+3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac {c^2 \sin ^{-1}(a x)}{a}-\frac {\left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 a}\\ &=\frac {c^2 (2-3 a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 (2+3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac {c^2 \sin ^{-1}(a x)}{a}+\frac {\left (3 c^2\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^2 (2-3 a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 (2+3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac {c^2 \sin ^{-1}(a x)}{a}+\frac {3 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 70, normalized size = 0.68 \begin {gather*} \frac {c^2 \left (-\frac {5 \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};a^2 x^2\right )}{x^3}-3 a^3 \left (1-a^2 x^2\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};1-a^2 x^2\right )\right )}{15 a^4} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.68, size = 148, normalized size = 1.44
method | result | size |
risch | \(-\frac {\left (8 a^{4} x^{4}-3 a^{3} x^{3}-10 a^{2} x^{2}+3 a x +2\right ) c^{2}}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{4}}+\frac {\left (-a^{3} \sqrt {-a^{2} x^{2}+1}+\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {3 a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right ) c^{2}}{a^{4}}\) | \(127\) |
default | \(\frac {c^{2} \left (-a^{3} \sqrt {-a^{2} x^{2}+1}+\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+\frac {4 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}+a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )+2 a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{a^{4}}\) | \(148\) |
meijerg | \(-\frac {c^{2} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}-\frac {c^{2} \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }\right )}{a \sqrt {\pi }}-\frac {c^{2} \left (-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{2} a^{2}}\right )}{2 a \sqrt {\pi }}+\frac {c^{2} \arcsin \left (a x \right )}{a}+\frac {2 c^{2} \sqrt {-a^{2} x^{2}+1}}{a^{2} x}-\frac {c^{2} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 a^{4} x^{3}}\) | \(263\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs.
\(2 (91) = 182\).
time = 0.47, size = 189, normalized size = 1.83 \begin {gather*} \frac {c^{2} \arcsin \left (a x\right )}{a} + \frac {2 \, c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} - \frac {{\left (a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{2}}\right )} c^{2}}{2 \, a^{3}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a^{2} x} - \frac {{\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x} + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{3}}\right )} c^{2}}{3 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 131, normalized size = 1.27 \begin {gather*} -\frac {12 \, a^{3} c^{2} x^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 9 \, a^{3} c^{2} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 6 \, a^{3} c^{2} x^{3} + {\left (6 \, a^{3} c^{2} x^{3} - 8 \, a^{2} c^{2} x^{2} + 3 \, a c^{2} x + 2 \, c^{2}\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 [A]
time = 6.62, size = 352, normalized size = 3.42 \begin {gather*} a c^{2} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) - \frac {2 c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right )}{a} - \frac {2 c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} + \frac {c^{2} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{3}} + \frac {c^{2} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\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. 263 vs.
\(2 (91) = 182\).
time = 0.42, size = 263, normalized size = 2.55 \begin {gather*} \frac {{\left (c^{2} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{a^{2} x} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{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 {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {3 \, 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 )}{2 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} + \frac {\frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{x} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{a^{2} x^{2}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{2}}{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 = 0.04, size = 136, normalized size = 1.32 \begin {gather*} \frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{a}+\frac {4\,c^2\,\sqrt {1-a^2\,x^2}}{3\,a^2\,x}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,a^4\,x^3}-\frac {c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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