Optimal. Leaf size=129 \[ \frac {7 a^3 c^4 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {17 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {7}{8} a^5 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A]
time = 0.16, 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, 1821,
821, 272, 43, 65, 214} \begin {gather*} -\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {17 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {7}{8} a^5 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {7 a^3 c^4 \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 1821
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^4}{x^6} \, dx &=c \int \frac {(c-a c x)^3 \sqrt {1-a^2 x^2}}{x^6} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac {1}{5} c \int \frac {\sqrt {1-a^2 x^2} \left (15 a c^3-17 a^2 c^3 x+5 a^3 c^3 x^2\right )}{x^5} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {1}{20} c \int \frac {\left (68 a^2 c^3-35 a^3 c^3 x\right ) \sqrt {1-a^2 x^2}}{x^4} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {17 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {1}{4} \left (7 a^3 c^4\right ) \int \frac {\sqrt {1-a^2 x^2}}{x^3} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {17 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {1}{8} \left (7 a^3 c^4\right ) \text {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x^2} \, dx,x,x^2\right )\\ &=\frac {7 a^3 c^4 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {17 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac {1}{16} \left (7 a^5 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {7 a^3 c^4 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {17 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {1}{8} \left (7 a^3 c^4\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 {7 a^3 c^4 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {17 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {7}{8} a^5 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 107, normalized size = 0.83 \begin {gather*} -\frac {c^4 \left (24-90 a x+88 a^2 x^2+75 a^3 x^3-248 a^4 x^4+15 a^5 x^5+136 a^6 x^6+105 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 = 1.30, size = 207, normalized size = 1.60
method | result | size |
risch | \(-\frac {\left (136 a^{6} x^{6}+15 a^{5} x^{5}-248 a^{4} x^{4}+75 a^{3} x^{3}+88 a^{2} x^{2}-90 a x +24\right ) c^{4}}{120 x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {7 a^{5} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{8}\) | \(89\) |
default | \(c^{4} \left (\frac {14 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )}{5}-3 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}+\frac {3 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4}\right )+2 a^{3} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )+\frac {3 a^{4} \sqrt {-a^{2} x^{2}+1}}{x}-\frac {\sqrt {-a^{2} x^{2}+1}}{5 x^{5}}-a^{5} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(207\) |
meijerg | \(\frac {a^{5} c^{4} \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }\right )}{2 \sqrt {\pi }}+\frac {3 a^{4} c^{4} \sqrt {-a^{2} x^{2}+1}}{x}-\frac {a^{5} c^{4} \left (-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{2} a^{2}}\right )}{\sqrt {\pi }}-\frac {2 a^{2} c^{4} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {3 a^{5} c^{4} \left (\frac {\sqrt {\pi }\, \left (-7 a^{4} x^{4}+8 a^{2} x^{2}+8\right )}{16 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (12 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{16 a^{4} x^{4}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{4}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{8}-\frac {\sqrt {\pi }}{2 x^{4} a^{4}}-\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{2 \sqrt {\pi }}-\frac {c^{4} \left (\frac {8}{3} a^{4} x^{4}+\frac {4}{3} a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{5 x^{5}}\) | \(395\) |
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 {7}{8} \, a^{5} c^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {17 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{4}}{15 \, x} + \frac {\sqrt {-a^{2} x^{2} + 1} a^{3} c^{4}}{8 \, x^{2}} - \frac {14 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4}}{15 \, x^{3}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a c^{4}}{4 \, x^{4}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{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 {105 \, a^{5} c^{4} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (136 \, a^{4} c^{4} x^{4} + 15 \, a^{3} c^{4} x^{3} - 112 \, a^{2} c^{4} x^{2} + 90 \, a c^{4} x - 24 \, c^{4}\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 = 7.70, size = 605, normalized size = 4.69 \begin {gather*} a^{5} c^{4} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) - 3 a^{4} c^{4} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{4} \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 ) + 2 a^{2} c^{4} \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^{4} \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^{4} \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. 354 vs.
\(2 (109) = 218\).
time = 0.42, size = 354, normalized size = 2.74 \begin {gather*} \frac {{\left (6 \, a^{6} c^{4} - \frac {45 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{4}}{x} + \frac {130 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac {120 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} - \frac {420 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{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 {7 \, a^{6} c^{4} \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 {420 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{8} c^{4}}{x} + \frac {120 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{6} c^{4}}{x^{2}} - \frac {130 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a^{4} c^{4}}{x^{3}} + \frac {45 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} a^{2} c^{4}}{x^{4}} - \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{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 = 0.05, size = 136, normalized size = 1.05 \begin {gather*} \frac {3\,a\,c^4\,\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{5\,x^5}-\frac {14\,a^2\,c^4\,\sqrt {1-a^2\,x^2}}{15\,x^3}+\frac {a^3\,c^4\,\sqrt {1-a^2\,x^2}}{8\,x^2}+\frac {17\,a^4\,c^4\,\sqrt {1-a^2\,x^2}}{15\,x}+\frac {a^5\,c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,7{}\mathrm {i}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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