Optimal. Leaf size=169 \[ \frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \text {ArcSin}(a x)}{a}-\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a} \]
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Rubi [A]
time = 0.16, antiderivative size = 169, 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 = {6292, 6284,
825, 827, 858, 222, 272, 65, 214} \begin {gather*} \frac {c^4 (35 a x+16) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {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 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 )^4 \, dx &=\frac {c^4 \int \frac {e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=\frac {c^4 \int \frac {(1-a x) \left (1-a^2 x^2\right )^{7/2}}{x^8} \, dx}{a^8}\\ &=-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}-\frac {c^4 \int \frac {\left (12 a^2-14 a^3 x\right ) \left (1-a^2 x^2\right )^{5/2}}{x^6} \, dx}{12 a^8}\\ &=\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \int \frac {\left (96 a^4-140 a^5 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{96 a^8}\\ &=-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}-\frac {c^4 \int \frac {\left (384 a^6-840 a^7 x\right ) \sqrt {1-a^2 x^2}}{x^2} \, dx}{384 a^8}\\ &=\frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \int \frac {1680 a^7+768 a^8 x}{x \sqrt {1-a^2 x^2}} \, dx}{768 a^8}\\ &=\frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+c^4 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx+\frac {\left (35 c^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{16 a}\\ &=\frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \sin ^{-1}(a x)}{a}+\frac {\left (35 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{32 a}\\ &=\frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \sin ^{-1}(a x)}{a}-\frac {\left (35 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 )}{16 a^3}\\ &=\frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \sin ^{-1}(a x)}{a}-\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 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 = 0.02, size = 70, normalized size = 0.41 \begin {gather*} \frac {c^4 \left (-\frac {9 \, _2F_1\left (-\frac {7}{2},-\frac {7}{2};-\frac {5}{2};a^2 x^2\right )}{x^7}+7 a^7 \left (1-a^2 x^2\right )^{9/2} \, _2F_1\left (4,\frac {9}{2};\frac {11}{2};1-a^2 x^2\right )\right )}{63 a^8} \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. \(426\) vs.
\(2(149)=298\).
time = 0.82, size = 427, normalized size = 2.53
method | result | size |
risch | \(-\frac {\left (2816 a^{8} x^{8}+3045 a^{7} x^{7}-4768 a^{6} x^{6}-4375 a^{5} x^{5}+3008 a^{4} x^{4}+1610 a^{3} x^{3}-1296 a^{2} x^{2}-280 a x +240\right ) c^{4}}{1680 \sqrt {-a^{2} x^{2}+1}\, x^{7} a^{8}}+\frac {\left (a^{7} \sqrt {-a^{2} x^{2}+1}+\frac {a^{8} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {35 a^{7} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}\right ) c^{4}}{a^{8}}\) | \(158\) |
default | \(\frac {c^{4} \left (-\frac {a^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x^{3}}+3 a^{3} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 x^{4}}+\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )-3 a^{5} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )-a^{6} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )-\frac {17 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{5 x^{5}}-\frac {2 a^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{15 x^{3}}\right )}{7}-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{7 x^{7}}-a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{6 x^{6}}+\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 x^{4}}+\frac {a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )}{2}\right )+a^{7} \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\right )}{a^{8}}\) | \(427\) |
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 = 0.46, size = 176, normalized size = 1.04 \begin {gather*} -\frac {3360 \, a^{7} c^{4} x^{7} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 3675 \, a^{7} c^{4} x^{7} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 1680 \, a^{7} c^{4} x^{7} - {\left (1680 \, a^{7} c^{4} x^{7} + 2816 \, a^{6} c^{4} x^{6} + 3045 \, a^{5} c^{4} x^{5} - 1952 \, a^{4} c^{4} x^{4} - 1330 \, a^{3} c^{4} x^{3} + 1056 \, a^{2} c^{4} x^{2} + 280 \, a c^{4} x - 240 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{1680 \, a^{8} x^{7}} \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 = 14.03, size = 1110, normalized size = 6.57 \begin {gather*} \frac {c^{4} \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^{4} \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 {3 c^{4} \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 {3 c^{4} \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^{4} \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 {3 c^{4} \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}} - \frac {c^{4} \left (\begin {cases} \frac {a^{6} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{16} - \frac {a^{5}}{16 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {a^{3}}{48 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {5 a}{24 x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{6 a x^{7} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{6} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{16} + \frac {i a^{5}}{16 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {i a^{3}}{48 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {5 i a}{24 x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{6 a x^{7} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{7}} + \frac {c^{4} \left (\begin {cases} \frac {8 a^{7} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{105} + \frac {4 a^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{105 x^{2}} + \frac {a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{35 x^{4}} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{7 x^{6}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {8 i a^{7} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{105} + \frac {4 i a^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{105 x^{2}} + \frac {i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{35 x^{4}} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{7 x^{6}} & \text {otherwise} \end {cases}\right )}{a^{8}} \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. 504 vs.
\(2 (149) = 298\).
time = 0.44, size = 504, normalized size = 2.98 \begin {gather*} \frac {{\left (15 \, c^{4} - \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} - \frac {189 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}} + \frac {525 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{6} x^{3}} + \frac {1295 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{8} x^{4}} - \frac {4935 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{10} x^{5}} - \frac {9765 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{12} x^{6}}\right )} a^{14} x^{7}}{13440 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} {\left | a \right |}} + \frac {c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {35 \, 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 )}{16 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} + \frac {\frac {9765 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{4}}{x} + \frac {4935 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac {1295 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} - \frac {525 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{2} x^{4}} + \frac {189 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{4} x^{5}} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{6} x^{6}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} c^{4}}{a^{8} x^{7}}}{13440 \, a^{6} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 227, normalized size = 1.34 \begin {gather*} \frac {c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{a}+\frac {176\,c^4\,\sqrt {1-a^2\,x^2}}{105\,a^2\,x}+\frac {29\,c^4\,\sqrt {1-a^2\,x^2}}{16\,a^3\,x^2}-\frac {122\,c^4\,\sqrt {1-a^2\,x^2}}{105\,a^4\,x^3}-\frac {19\,c^4\,\sqrt {1-a^2\,x^2}}{24\,a^5\,x^4}+\frac {22\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^6\,x^5}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{6\,a^7\,x^6}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{7\,a^8\,x^7}+\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,35{}\mathrm {i}}{16\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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