Optimal. Leaf size=131 \[ -\frac {a^3 c \sqrt {c-a^2 c x^2}}{4 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{4} a^5 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Rubi [A]
time = 0.20, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6286, 1821,
849, 821, 272, 43, 65, 214} \begin {gather*} -\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{4} a^5 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {a^3 c \sqrt {c-a^2 c x^2}}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 1821
Rule 6286
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^6} \, dx &=c \int \frac {(1+a x)^2 \sqrt {c-a^2 c x^2}}{x^6} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {1}{5} \int \frac {\left (-10 a c-7 a^2 c x\right ) \sqrt {c-a^2 c x^2}}{x^5} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}+\frac {\int \frac {\left (28 a^2 c^2+10 a^3 c^2 x\right ) \sqrt {c-a^2 c x^2}}{x^4} \, dx}{20 c}\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{2} \left (a^3 c\right ) \int \frac {\sqrt {c-a^2 c x^2}}{x^3} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{4} \left (a^3 c\right ) \text {Subst}\left (\int \frac {\sqrt {c-a^2 c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {a^3 c \sqrt {c-a^2 c x^2}}{4 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}-\frac {1}{8} \left (a^5 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {a^3 c \sqrt {c-a^2 c x^2}}{4 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{4} \left (a^3 c\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )\\ &=-\frac {a^3 c \sqrt {c-a^2 c x^2}}{4 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{4} a^5 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 104, normalized size = 0.79 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (-12-30 a x-16 a^2 x^2+15 a^3 x^3+28 a^4 x^4\right )}{60 x^5}-\frac {1}{4} a^5 c^{3/2} \log (x)+\frac {1}{4} a^5 c^{3/2} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(685\) vs.
\(2(107)=214\).
time = 0.07, size = 686, normalized size = 5.24
method | result | size |
risch | \(-\frac {\left (28 x^{6} a^{6}+15 x^{5} a^{5}-44 a^{4} x^{4}-45 a^{3} x^{3}+4 a^{2} x^{2}+30 a x +12\right ) c^{2}}{60 x^{5} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {a^{5} c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{4}\) | \(105\) |
default | \(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{5 c \,x^{5}}+2 a^{4} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )\right )+2 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 c \,x^{3}}-\frac {2 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )\right )}{3}\right )+2 a \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{4 c \,x^{4}}-\frac {a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )}{2}\right )}{4}\right )+2 a^{5} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )-2 a^{5} \left (\frac {\left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {c \,a^{2}}}\right )\right )+2 a^{3} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )}{2}\right )\) | \(686\) |
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. 221 vs.
\(2 (107) = 214\).
time = 0.47, size = 221, normalized size = 1.69 \begin {gather*} \frac {{\left (a^{2} c^{\frac {3}{2}} x^{2} - c^{\frac {3}{2}}\right )} \sqrt {a x + 1} \sqrt {-a x + 1} a^{2}}{3 \, x^{3}} - \frac {a^{6} c^{\frac {5}{2}} \log \left (\frac {\sqrt {-a^{2} c x^{2} + c} - \sqrt {c}}{\sqrt {-a^{2} c x^{2} + c} + \sqrt {c}}\right ) + \frac {2 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{6} c^{3} + \sqrt {-a^{2} c x^{2} + c} a^{6} c^{4}\right )}}{{\left (a^{2} c x^{2} - c\right )}^{2} + 2 \, {\left (a^{2} c x^{2} - c\right )} c + c^{2}}}{8 \, a c} + \frac {{\left (2 \, a^{4} c^{\frac {3}{2}} x^{4} + a^{2} c^{\frac {3}{2}} x^{2} - 3 \, c^{\frac {3}{2}}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 209, normalized size = 1.60 \begin {gather*} \left [\frac {15 \, a^{5} c^{\frac {3}{2}} x^{5} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (28 \, a^{4} c x^{4} + 15 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} - 30 \, a c x - 12 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{120 \, x^{5}}, \frac {15 \, a^{5} \sqrt {-c} c x^{5} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (28 \, a^{4} c x^{4} + 15 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} - 30 \, a c x - 12 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{60 \, x^{5}}\right ] \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 = 11.64, size = 484, normalized size = 3.69 \begin {gather*} a^{2} c \left (\begin {cases} \frac {a^{3} \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {c} \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 {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {a^{4} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {a^{3} \sqrt {c}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 a \sqrt {c}}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{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} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {i a^{3} \sqrt {c}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 i a \sqrt {c}}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i \sqrt {c}}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {2 i a^{4} \sqrt {c} \sqrt {a^{2} x^{2} - 1}}{15 x} + \frac {i a^{2} \sqrt {c} \sqrt {a^{2} x^{2} - 1}}{15 x^{3}} - \frac {i \sqrt {c} \sqrt {a^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {2 a^{4} \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{15 x} + \frac {a^{2} \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{15 x^{3}} - \frac {\sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right ) \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. 414 vs.
\(2 (107) = 214\).
time = 0.43, size = 414, normalized size = 3.16 \begin {gather*} -\frac {a^{5} c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{2 \, \sqrt {-c}} + \frac {15 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{9} a^{5} c^{2} - 60 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{8} a^{4} \sqrt {-c} c^{2} {\left | a \right |} + 90 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{7} a^{5} c^{3} + 240 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{6} a^{4} \sqrt {-c} c^{3} {\left | a \right |} - 40 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{4} \sqrt {-c} c^{4} {\left | a \right |} - 90 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{5} c^{5} + 80 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{4} \sqrt {-c} c^{5} {\left | a \right |} - 15 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{5} c^{6} - 28 \, a^{4} \sqrt {-c} c^{6} {\left | a \right |}}{30 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{x^6\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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