Optimal. Leaf size=103 \[ \frac {c^2 (2+3 a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac {c^2 \text {ArcSin}(a x)}{a}-\frac {3 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6292, 6284,
825, 827, 858, 222, 272, 65, 214} \begin {gather*} \frac {c^2 (3 a x+2) \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 (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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 214
Rule 222
Rule 272
Rule 825
Rule 827
Rule 858
Rule 6284
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*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.79, size = 183, normalized size = 1.78
method | result | size |
risch | \(-\frac {\left (8 a^{4} x^{4}+3 a^{3} x^{3}-10 a^{2} x^{2}-3 a x +2\right ) c^{2}}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{4}}+\frac {\left (a^{3} \sqrt {-a^{2} x^{2}+1}+\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {3 a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right ) c^{2}}{a^{4}}\) | \(126\) |
default | \(\frac {c^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 x^{3}}-a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )-a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )+a^{3} \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\right )}{a^{4}}\) | \(183\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 132, normalized size = 1.28 \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 3.93, size = 381, normalized size = 3.70 \begin {gather*} \frac {c^{2} \left (\begin {cases} i \sqrt {a^{2} x^{2} - 1} - \log {\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} + i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} - \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right )}{a} - \frac {c^{2} \left (\begin {cases} - \frac {i a^{2} x}{\sqrt {a^{2} x^{2} - 1}} + i a \operatorname {acosh}{\left (a x \right )} + \frac {i}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} x}{\sqrt {- a^{2} x^{2} + 1}} - a \operatorname {asin}{\left (a x \right )} - \frac {1}{x \sqrt {- a^{2} x^{2} + 1}} & \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 {a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right )}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 262 vs.
\(2 (91) = 182\).
time = 0.42, size = 262, normalized size = 2.54 \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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 135, normalized size = 1.31 \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.
[In]
[Out]
________________________________________________________________________________________