Optimal. Leaf size=157 \[ -\frac {3 c^3 (8+a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8-a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {3 c^3 \text {ArcSin}(a x)}{a}+\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 a} \]
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Rubi [A]
time = 0.22, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {6292, 6284,
1821, 825, 827, 858, 222, 272, 65, 214} \begin {gather*} -\frac {3 c^3 (a x+8) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 a}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}+\frac {c^3 (8-a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}-\frac {3 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 825
Rule 827
Rule 858
Rule 1821
Rule 6284
Rule 6292
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{-3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6}\\ &=-\frac {c^3 \int \frac {(1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{x^6} \, dx}{a^6}\\ &=\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2} \left (15 a-15 a^2 x+5 a^3 x^2\right )}{x^5} \, dx}{5 a^6}\\ &=\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {c^3 \int \frac {\left (60 a^2-5 a^3 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{20 a^6}\\ &=\frac {c^3 (8-a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}+\frac {c^3 \int \frac {\left (240 a^4-30 a^5 x\right ) \sqrt {1-a^2 x^2}}{x^2} \, dx}{80 a^6}\\ &=-\frac {3 c^3 (8+a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8-a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {c^3 \int \frac {60 a^5+480 a^6 x}{x \sqrt {1-a^2 x^2}} \, dx}{160 a^6}\\ &=-\frac {3 c^3 (8+a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8-a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\left (3 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\frac {\left (3 c^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac {3 c^3 (8+a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8-a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {3 c^3 \sin ^{-1}(a x)}{a}-\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{16 a}\\ &=-\frac {3 c^3 (8+a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8-a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {3 c^3 \sin ^{-1}(a x)}{a}+\frac {\left (3 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 )}{8 a^3}\\ &=-\frac {3 c^3 (8+a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8-a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {3 c^3 \sin ^{-1}(a x)}{a}+\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 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 = 0.07, size = 186, normalized size = 1.18 \begin {gather*} \frac {c^3 \left (8-30 a x-24 a^2 x^2+105 a^3 x^3+24 a^4 x^4-75 a^5 x^5-8 a^6 x^6-45 a^5 x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+40 a^2 x^2 \sqrt {1-a^2 x^2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};a^2 x^2\right )+8 a^5 x^5 \left (-1+a^2 x^2\right )^3 \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};1-a^2 x^2\right )\right )}{40 a^6 x^5 \sqrt {1-a^2 x^2}} \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. \(294\) vs.
\(2(137)=274\).
time = 0.76, size = 295, normalized size = 1.88
method | result | size |
risch | \(\frac {\left (152 a^{6} x^{6}-55 a^{5} x^{5}-176 a^{4} x^{4}+85 a^{3} x^{3}+16 a^{2} x^{2}-30 a x +8\right ) c^{3}}{40 x^{5} \sqrt {-a^{2} x^{2}+1}\, a^{6}}+\frac {\left (-a^{5} \sqrt {-a^{2} x^{2}+1}-\frac {3 a^{6} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {3 a^{5} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}\right ) c^{3}}{a^{6}}\) | \(144\) |
default | \(\frac {c^{3} \left (-3 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{3 x^{3}}-\frac {2 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-4 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )\right )}{3}\right )+3 a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{4 x^{4}}-\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )+a^{3} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )+\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{5 x^{5}}\right )}{a^{6}}\) | \(295\) |
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 = 0.39, size = 154, normalized size = 0.98 \begin {gather*} \frac {240 \, a^{5} c^{3} x^{5} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 15 \, a^{5} c^{3} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 40 \, a^{5} c^{3} x^{5} - {\left (40 \, a^{5} c^{3} x^{5} + 152 \, a^{4} c^{3} x^{4} - 55 \, a^{3} c^{3} x^{3} - 24 \, a^{2} c^{3} x^{2} + 30 \, a c^{3} x - 8 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{40 \, a^{6} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 7.87, size = 695, normalized size = 4.43 \begin {gather*} - \frac {c^{3} \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 {3 c^{3} \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 {2 c^{3} \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 {2 c^{3} \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}} + \frac {3 c^{3} \left (\begin {cases} \frac {a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{5}} - \frac {c^{3} \left (\begin {cases} \frac {2 i a^{4} \sqrt {a^{2} x^{2} - 1}}{15 x} + \frac {i a^{2} \sqrt {a^{2} x^{2} - 1}}{15 x^{3}} - \frac {i \sqrt {a^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {2 a^{4} \sqrt {- a^{2} x^{2} + 1}}{15 x} + \frac {a^{2} \sqrt {- a^{2} x^{2} + 1}}{15 x^{3}} - \frac {\sqrt {- a^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right )}{a^{6}} \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. 386 vs.
\(2 (136) = 272\).
time = 0.42, size = 386, normalized size = 2.46 \begin {gather*} -\frac {{\left (2 \, c^{3} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{a^{2} x} + \frac {30 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{a^{4} x^{2}} + \frac {80 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{6} x^{3}} - \frac {580 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{8} x^{4}}\right )} a^{10} x^{5}}{320 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} {\left | a \right |}} - \frac {3 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {3 \, 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 )}{8 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} - \frac {\frac {580 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c^{3}}{x} - \frac {80 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}} - \frac {30 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{2} x^{3}} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{4} x^{4}} - \frac {2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{3}}{a^{6} x^{5}}}{320 \, a^{4} {\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 = 183, normalized size = 1.17 \begin {gather*} \frac {11\,c^3\,\sqrt {1-a^2\,x^2}}{8\,a^3\,x^2}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{a}-\frac {19\,c^3\,\sqrt {1-a^2\,x^2}}{5\,a^2\,x}-\frac {3\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {3\,c^3\,\sqrt {1-a^2\,x^2}}{5\,a^4\,x^3}-\frac {3\,c^3\,\sqrt {1-a^2\,x^2}}{4\,a^5\,x^4}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{5\,a^6\,x^5}-\frac {c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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