Optimal. Leaf size=129 \[ \frac {a^3 c^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {2 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {1}{8} a^5 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6263, 849, 821,
272, 43, 65, 214} \begin {gather*} -\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {2 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {1}{8} a^5 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {a^3 c^2 \sqrt {1-a^2 x^2}}{8 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 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^2}{x^6} \, dx &=c \int \frac {(c-a c x) \sqrt {1-a^2 x^2}}{x^6} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac {1}{5} c \int \frac {\left (5 a c-2 a^2 c x\right ) \sqrt {1-a^2 x^2}}{x^5} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {1}{20} c \int \frac {\left (8 a^2 c-5 a^3 c x\right ) \sqrt {1-a^2 x^2}}{x^4} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {2 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {1}{4} \left (a^3 c^2\right ) \int \frac {\sqrt {1-a^2 x^2}}{x^3} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {2 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {1}{8} \left (a^3 c^2\right ) \text {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x^2} \, dx,x,x^2\right )\\ &=\frac {a^3 c^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {2 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac {1}{16} \left (a^5 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {a^3 c^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {2 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {1}{8} \left (a^3 c^2\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 )\\ &=\frac {a^3 c^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {2 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {1}{8} a^5 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 107, normalized size = 0.83 \begin {gather*} -\frac {c^2 \left (24-30 a x-32 a^2 x^2+45 a^3 x^3-8 a^4 x^4-15 a^5 x^5+16 a^6 x^6+15 a^5 x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\right )}{120 x^5 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.11, size = 168, normalized size = 1.30
method | result | size |
risch | \(-\frac {\left (16 a^{6} x^{6}-15 a^{5} x^{5}-8 a^{4} x^{4}+45 a^{3} x^{3}-32 a^{2} x^{2}-30 a x +24\right ) c^{2}}{120 x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {a^{5} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{8}\) | \(89\) |
default | \(c^{2} \left (-\frac {a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )}{5}-a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}+\frac {3 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4}\right )+a^{3} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )-\frac {\sqrt {-a^{2} x^{2}+1}}{5 x^{5}}\right )\) | \(168\) |
meijerg | \(-\frac {a^{5} c^{2} \left (-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{2} a^{2}}\right )}{2 \sqrt {\pi }}+\frac {a^{2} c^{2} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {a^{5} c^{2} \left (\frac {\sqrt {\pi }\, \left (-7 a^{4} x^{4}+8 a^{2} x^{2}+8\right )}{16 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (12 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{16 a^{4} x^{4}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{4}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{8}-\frac {\sqrt {\pi }}{2 x^{4} a^{4}}-\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{2 \sqrt {\pi }}-\frac {c^{2} \left (\frac {8}{3} a^{4} x^{4}+\frac {4}{3} a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{5 x^{5}}\) | \(319\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 145, normalized size = 1.12 \begin {gather*} -\frac {1}{8} \, a^{5} c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}}{15 \, x} - \frac {\sqrt {-a^{2} x^{2} + 1} a^{3} c^{2}}{8 \, x^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} a^{2} c^{2}}{15 \, x^{3}} + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{2}}{4 \, x^{4}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 95, normalized size = 0.74 \begin {gather*} \frac {15 \, a^{5} c^{2} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (16 \, a^{4} c^{2} x^{4} - 15 \, a^{3} c^{2} x^{3} + 8 \, a^{2} c^{2} x^{2} + 30 \, a c^{2} x - 24 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, 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 = 6.03, size = 520, normalized size = 4.03 \begin {gather*} a^{3} c^{2} \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^{2} c^{2} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {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 {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {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 ) + c^{2} \left (\begin {cases} - \frac {8 a^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {8 i a^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \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. 297 vs.
\(2 (109) = 218\).
time = 0.42, size = 297, normalized size = 2.30 \begin {gather*} \frac {{\left (6 \, a^{6} c^{2} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{2}}{x} + \frac {10 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{2}}{x^{2}} - \frac {60 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{2}}{a^{2} x^{4}}\right )} a^{10} x^{5}}{960 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} {\left | a \right |}} - \frac {a^{6} c^{2} \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 {\frac {60 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{8} c^{2}}{x} - \frac {10 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a^{4} c^{2}}{x^{3}} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} a^{2} c^{2}}{x^{4}} - \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{2}}{x^{5}}}{960 \, a^{4} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.79, size = 136, normalized size = 1.05 \begin {gather*} \frac {a\,c^2\,\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{5\,x^5}+\frac {a^2\,c^2\,\sqrt {1-a^2\,x^2}}{15\,x^3}-\frac {a^3\,c^2\,\sqrt {1-a^2\,x^2}}{8\,x^2}+\frac {2\,a^4\,c^2\,\sqrt {1-a^2\,x^2}}{15\,x}+\frac {a^5\,c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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