Optimal. Leaf size=144 \[ -\frac {c \sqrt {1-a^2 x^2}}{5 x^5}-\frac {3 a c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {19 a^2 c \sqrt {1-a^2 x^2}}{15 x^3}-\frac {13 a^3 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {38 a^4 c \sqrt {1-a^2 x^2}}{15 x}-\frac {13}{8} a^5 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A]
time = 0.17, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6283, 1821,
849, 821, 272, 65, 214} \begin {gather*} -\frac {c \sqrt {1-a^2 x^2}}{5 x^5}-\frac {3 a c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {19 a^2 c \sqrt {1-a^2 x^2}}{15 x^3}-\frac {13}{8} a^5 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {38 a^4 c \sqrt {1-a^2 x^2}}{15 x}-\frac {13 a^3 c \sqrt {1-a^2 x^2}}{8 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 1821
Rule 6283
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )}{x^6} \, dx &=c \int \frac {(1+a x)^3}{x^6 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{5 x^5}-\frac {1}{5} c \int \frac {-15 a-19 a^2 x-5 a^3 x^2}{x^5 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{5 x^5}-\frac {3 a c \sqrt {1-a^2 x^2}}{4 x^4}+\frac {1}{20} c \int \frac {76 a^2+65 a^3 x}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{5 x^5}-\frac {3 a c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {19 a^2 c \sqrt {1-a^2 x^2}}{15 x^3}-\frac {1}{60} c \int \frac {-195 a^3-152 a^4 x}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{5 x^5}-\frac {3 a c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {19 a^2 c \sqrt {1-a^2 x^2}}{15 x^3}-\frac {13 a^3 c \sqrt {1-a^2 x^2}}{8 x^2}+\frac {1}{120} c \int \frac {304 a^4+195 a^5 x}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{5 x^5}-\frac {3 a c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {19 a^2 c \sqrt {1-a^2 x^2}}{15 x^3}-\frac {13 a^3 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {38 a^4 c \sqrt {1-a^2 x^2}}{15 x}+\frac {1}{8} \left (13 a^5 c\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{5 x^5}-\frac {3 a c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {19 a^2 c \sqrt {1-a^2 x^2}}{15 x^3}-\frac {13 a^3 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {38 a^4 c \sqrt {1-a^2 x^2}}{15 x}+\frac {1}{16} \left (13 a^5 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {1-a^2 x^2}}{5 x^5}-\frac {3 a c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {19 a^2 c \sqrt {1-a^2 x^2}}{15 x^3}-\frac {13 a^3 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {38 a^4 c \sqrt {1-a^2 x^2}}{15 x}-\frac {1}{8} \left (13 a^3 c\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 {c \sqrt {1-a^2 x^2}}{5 x^5}-\frac {3 a c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {19 a^2 c \sqrt {1-a^2 x^2}}{15 x^3}-\frac {13 a^3 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {38 a^4 c \sqrt {1-a^2 x^2}}{15 x}-\frac {13}{8} a^5 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.09, size = 110, normalized size = 0.76 \begin {gather*} -\frac {c \sqrt {1-a^2 x^2} \left (6+38 a^2 x^2+15 a^3 x^3+76 a^4 x^4\right )}{30 x^5}-\frac {1}{2} a^5 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-3 a^5 c \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-a^2 x^2\right ) \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. \(291\) vs.
\(2(120)=240\).
time = 0.08, size = 292, normalized size = 2.03
method | result | size |
risch | \(\frac {\left (304 x^{6} a^{6}+195 x^{5} a^{5}-152 a^{4} x^{4}-105 a^{3} x^{3}-128 a^{2} x^{2}-90 a x -24\right ) c}{120 x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {13 a^{5} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{8}\) | \(85\) |
default | \(-c \left (\frac {1}{5 x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {16 a^{2} \left (-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {4 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}\right )}{5}+3 a^{4} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a \left (-\frac {1}{4 x^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {5 a^{2} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )+a^{5} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+2 a^{3} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )\right )\) | \(292\) |
meijerg | \(-\frac {2 a^{2} c \left (-8 a^{4} x^{4}+4 a^{2} x^{2}+1\right )}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {c \left (-16 x^{6} a^{6}+8 a^{4} x^{4}+2 a^{2} x^{2}+1\right )}{5 x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {a^{5} c \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}\right )}{\sqrt {\pi }}+\frac {2 a^{5} c \left (-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{\sqrt {\pi }}+\frac {3 a^{4} c \left (-2 a^{2} x^{2}+1\right )}{x \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{5} c \left (\frac {\sqrt {\pi }\, \left (-47 a^{4} x^{4}+24 a^{2} x^{2}+8\right )}{32 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (-60 a^{4} x^{4}+20 a^{2} x^{2}+8\right )}{32 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{8}+\frac {15 \left (\frac {47}{30}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{16}-\frac {\sqrt {\pi }}{4 x^{4} a^{4}}-\frac {3 \sqrt {\pi }}{4 x^{2} a^{2}}\right )}{\sqrt {\pi }}\) | \(453\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 170, normalized size = 1.18 \begin {gather*} \frac {38 \, a^{6} c x}{15 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {13}{8} \, a^{5} c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {13 \, a^{5} c}{8 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {19 \, a^{4} c}{15 \, \sqrt {-a^{2} x^{2} + 1} x} - \frac {7 \, a^{3} c}{8 \, \sqrt {-a^{2} x^{2} + 1} x^{2}} - \frac {16 \, a^{2} c}{15 \, \sqrt {-a^{2} x^{2} + 1} x^{3}} - \frac {3 \, a c}{4 \, \sqrt {-a^{2} x^{2} + 1} x^{4}} - \frac {c}{5 \, \sqrt {-a^{2} x^{2} + 1} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 84, normalized size = 0.58 \begin {gather*} \frac {195 \, a^{5} c x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (304 \, a^{4} c x^{4} + 195 \, a^{3} c x^{3} + 152 \, a^{2} c x^{2} + 90 \, a c x + 24 \, c\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 = 27.45, size = 517, normalized size = 3.59 \begin {gather*} a^{3} c \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 ) + 3 a^{2} c \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 ) + 3 a c \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 \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. 332 vs.
\(2 (120) = 240\).
time = 0.42, size = 332, normalized size = 2.31 \begin {gather*} \frac {{\left (6 \, a^{6} c + \frac {45 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c}{x} + \frac {170 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c}{x^{2}} + \frac {480 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c}{x^{3}} + \frac {1380 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c}{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 {13 \, a^{6} c \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 {1380 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{8} c}{x} + \frac {480 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{6} c}{x^{2}} + \frac {170 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a^{4} c}{x^{3}} + \frac {45 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} a^{2} c}{x^{4}} + \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c}{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 = 0.04, size = 124, normalized size = 0.86 \begin {gather*} -\frac {c\,\sqrt {1-a^2\,x^2}}{5\,x^5}-\frac {19\,a^2\,c\,\sqrt {1-a^2\,x^2}}{15\,x^3}-\frac {13\,a^3\,c\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {38\,a^4\,c\,\sqrt {1-a^2\,x^2}}{15\,x}-\frac {3\,a\,c\,\sqrt {1-a^2\,x^2}}{4\,x^4}+\frac {a^5\,c\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,13{}\mathrm {i}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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