Optimal. Leaf size=156 \[ -\frac {3 a^4 c \sqrt {c-a^2 c x^2}}{16 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac {4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {3}{16} a^6 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Rubi [A]
time = 0.23, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 9, 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}}{6 x^6}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}+\frac {3}{16} a^6 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {3 a^4 c \sqrt {c-a^2 c x^2}}{16 x^2}-\frac {4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3} \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^7} \, dx &=c \int \frac {(1+a x)^2 \sqrt {c-a^2 c x^2}}{x^7} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac {1}{6} \int \frac {\left (-12 a c-9 a^2 c x\right ) \sqrt {c-a^2 c x^2}}{x^6} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}+\frac {\int \frac {\left (45 a^2 c^2+24 a^3 c^2 x\right ) \sqrt {c-a^2 c x^2}}{x^5} \, dx}{30 c}\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac {\int \frac {\left (-96 a^3 c^3-45 a^4 c^3 x\right ) \sqrt {c-a^2 c x^2}}{x^4} \, dx}{120 c^2}\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac {4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{8} \left (3 a^4 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}}{6 x^6}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac {4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{16} \left (3 a^4 c\right ) \text {Subst}\left (\int \frac {\sqrt {c-a^2 c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {3 a^4 c \sqrt {c-a^2 c x^2}}{16 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac {4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}-\frac {1}{32} \left (3 a^6 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {3 a^4 c \sqrt {c-a^2 c x^2}}{16 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac {4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{16} \left (3 a^4 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 {3 a^4 c \sqrt {c-a^2 c x^2}}{16 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{6 x^6}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {3 a^2 \left (c-a^2 c x^2\right )^{3/2}}{8 x^4}-\frac {4 a^3 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {3}{16} a^6 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 = 0.16, size = 109, normalized size = 0.70 \begin {gather*} \frac {1}{240} c \left (\frac {\sqrt {c-a^2 c x^2} \left (-40-96 a x-50 a^2 x^2+32 a^3 x^3+45 a^4 x^4+64 a^5 x^5\right )}{x^6}-45 a^6 \sqrt {c} \log (x)+45 a^6 \sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right )\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. \(709\) vs.
\(2(128)=256\).
time = 0.07, size = 710, normalized size = 4.55
method | result | size |
risch | \(-\frac {\left (64 a^{7} x^{7}+45 x^{6} a^{6}-32 x^{5} a^{5}-95 a^{4} x^{4}-128 a^{3} x^{3}+10 a^{2} x^{2}+96 a x +40\right ) c^{2}}{240 x^{6} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {3 a^{6} c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{16}\) | \(113\) |
default | \(-\frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{5 c \,x^{5}}+2 a^{5} \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 )-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6 c \,x^{6}}+\frac {13 a^{2} \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 )}{6}+2 a^{3} \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^{6} \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^{6} \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^{4} \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 )\) | \(710\) |
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 = 0.36, size = 227, normalized size = 1.46 \begin {gather*} \left [\frac {45 \, a^{6} c^{\frac {3}{2}} x^{6} \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 (64 \, a^{5} c x^{5} + 45 \, a^{4} c x^{4} + 32 \, a^{3} c x^{3} - 50 \, a^{2} c x^{2} - 96 \, a c x - 40 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{480 \, x^{6}}, \frac {45 \, a^{6} \sqrt {-c} c x^{6} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (64 \, a^{5} c x^{5} + 45 \, a^{4} c x^{4} + 32 \, a^{3} c x^{3} - 50 \, a^{2} c x^{2} - 96 \, a c x - 40 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{240 \, x^{6}}\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 = 9.75, size = 636, normalized size = 4.08 \begin {gather*} a^{2} 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 ) + 2 a 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 ) + c \left (\begin {cases} \frac {a^{6} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{16} - \frac {a^{5} \sqrt {c}}{16 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {a^{3} \sqrt {c}}{48 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {5 a \sqrt {c}}{24 x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{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} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{16} + \frac {i a^{5} \sqrt {c}}{16 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {i a^{3} \sqrt {c}}{48 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {5 i a \sqrt {c}}{24 x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i \sqrt {c}}{6 a x^{7} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \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. 443 vs.
\(2 (128) = 256\).
time = 0.44, size = 443, normalized size = 2.84 \begin {gather*} -\frac {3 \, a^{6} c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{8 \, \sqrt {-c}} + \frac {45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{11} a^{6} c^{2} + 65 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{9} a^{6} c^{3} + 960 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{8} a^{5} \sqrt {-c} c^{3} {\left | a \right |} - 750 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{7} a^{6} c^{4} - 640 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{6} a^{5} \sqrt {-c} c^{4} {\left | a \right |} - 750 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{6} c^{5} + 65 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{6} c^{6} - 384 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{5} \sqrt {-c} c^{6} {\left | a \right |} + 45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{6} c^{7} + 64 \, a^{5} \sqrt {-c} c^{7} {\left | a \right |}}{120 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{6}} \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^7\,\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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