Optimal. Leaf size=293 \[ -\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{6 (a x+1)}+\frac {x (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 (a x+1)}-\frac {7 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{24 (1-a x) (a x+1)}-\frac {7 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 (a x+1)^2}+\frac {2 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \sin ^{-1}(a x)}{(1-a x)^{5/2} (a x+1)^{5/2}}-\frac {9 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{8 (1-a x)^{5/2} (a x+1)^{5/2}}+\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{(1-a x)^2 (a x+1)} \]
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Rubi [A] time = 0.48, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6167, 6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} \[ -\frac {7 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 (a x+1)^2}+\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{(1-a x)^2 (a x+1)}-\frac {7 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{24 (1-a x) (a x+1)}-\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{6 (a x+1)}+\frac {x (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 (a x+1)}+\frac {2 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \sin ^{-1}(a x)}{(1-a x)^{5/2} (a x+1)^{5/2}}-\frac {9 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{8 (1-a x)^{5/2} (a x+1)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 92
Rule 97
Rule 149
Rule 154
Rule 157
Rule 208
Rule 216
Rule 6129
Rule 6159
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx\\ &=-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {e^{-2 \tanh ^{-1}(a x)} (1-a x)^{5/2} (1+a x)^{5/2}}{x^5} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1-a x)^{7/2} (1+a x)^{3/2}}{x^5} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1-a x)^{5/2} \sqrt {1+a x} \left (-2 a-5 a^2 x\right )}{x^4} \, dx}{4 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1-a x)^{3/2} \sqrt {1+a x} \left (-7 a^2+17 a^3 x\right )}{x^3} \, dx}{12 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {\sqrt {1-a x} \sqrt {1+a x} \left (48 a^3-27 a^4 x\right )}{x^2} \, dx}{24 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {\sqrt {1+a x} \left (-27 a^4-21 a^5 x\right )}{x \sqrt {1-a x}} \, dx}{24 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac {7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {27 a^5+48 a^6 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{24 a (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac {7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}+\frac {\left (9 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{8 (1-a x)^{5/2} (1+a x)^{5/2}}+\frac {\left (2 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac {7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}-\frac {\left (9 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}}+\frac {\left (2 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac {7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{6 (1+a x)}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{(1-a x)^2 (1+a x)}-\frac {7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{24 (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1-a x)}{4 (1+a x)}+\frac {2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \sin ^{-1}(a x)}{(1-a x)^{5/2} (1+a x)^{5/2}}-\frac {9 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 134, normalized size = 0.46 \[ \frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \left (-48 a^4 x^4 \log \left (\sqrt {a^2 x^2-1}+a x\right )+27 a^4 x^4 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )+\sqrt {a^2 x^2-1} \left (24 a^4 x^4+64 a^3 x^3-3 a^2 x^2-16 a x+6\right )\right )}{24 a^4 x^3 \sqrt {a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 394, normalized size = 1.34 \[ \left [\frac {96 \, a^{3} \sqrt {-c} c^{2} x^{3} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 27 \, a^{3} \sqrt {-c} c^{2} x^{3} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (24 \, a^{4} c^{2} x^{4} + 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, a^{4} x^{3}}, \frac {27 \, a^{3} c^{\frac {5}{2}} x^{3} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 24 \, a^{3} c^{\frac {5}{2}} x^{3} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (24 \, a^{4} c^{2} x^{4} + 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, a^{4} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 7.13, size = 416, normalized size = 1.42 \[ -\frac {1}{12} \, {\left (\frac {27 \, c^{\frac {5}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x)}{a^{2}} - \frac {24 \, c^{\frac {5}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a {\left | a \right |}} - \frac {12 \, \sqrt {a^{2} c x^{2} - c} c^{2} \mathrm {sgn}\relax (x)}{a^{2}} - \frac {3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{3} {\left | a \right |} \mathrm {sgn}\relax (x) + 96 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {7}{2}} \mathrm {sgn}\relax (x) - 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{4} {\left | a \right |} \mathrm {sgn}\relax (x) + 192 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {9}{2}} \mathrm {sgn}\relax (x) + 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{5} {\left | a \right |} \mathrm {sgn}\relax (x) + 160 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {11}{2}} \mathrm {sgn}\relax (x) - 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{6} {\left | a \right |} \mathrm {sgn}\relax (x) + 64 \, a c^{\frac {13}{2}} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{4} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 625, normalized size = 2.13 \[ -\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {5}{2}} x \left (-80 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} x^{5} a^{7} c +80 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} x^{3} a^{7}-48 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{4} a^{6} c -27 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} x^{4} a^{6} c +60 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x^{5} a^{5} c^{2}+75 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} x^{2} a^{6}+100 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{5} a^{5} c^{2}-80 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} x \,a^{5}+45 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{4} a^{4} c^{2}-90 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{5} a^{3} c^{3}-150 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{5} a^{3} c^{3}+30 a^{4} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}+150 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {7}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{4} a +90 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {7}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{4} a -135 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{4} a^{2} c^{3}-135 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{4} c^{4}\right )}{120 a^{2} \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c} \]
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 \[ \int \frac {{\left (a x - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 19.52, size = 500, normalized size = 1.71 \[ c^{2} \left (\begin {cases} \frac {\sqrt {c} \sqrt {a^{2} x^{2} - 1}}{a} - \frac {i \sqrt {c} \log {\left (a x \right )}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} + \frac {\sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} - \frac {i \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )}}{a} & \text {otherwise} \end {cases}\right ) - \frac {2 c^{2} \left (\begin {cases} - \frac {a \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {\sqrt {c}}{a x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i a \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - i \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {i \sqrt {c}}{a x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a} + \frac {2 c^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {a^{2} \left (c - \frac {c}{a^{2} x^{2}}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right )}{a^{3}} - \frac {c^{2} \left (\begin {cases} \frac {i a^{3} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {i a^{2} \sqrt {c}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 i \sqrt {c}}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {i \sqrt {c}}{4 a^{2} x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {a^{3} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {a^{2} \sqrt {c}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 \sqrt {c}}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {\sqrt {c}}{4 a^{2} x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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