Optimal. Leaf size=158 \[ \frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^2 \sqrt {1-c x}}{144 x^4 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^4 \sqrt {1-c x}}{96 x^2 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}+\frac {5}{96} b c^6 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c x} \sqrt {1+c x}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6418, 105, 12,
94, 214} \begin {gather*} -\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}+\frac {5}{96} b c^6 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c x} \sqrt {c x+1}\right )+\frac {5 b c^4 \sqrt {1-c x}}{96 x^2 \sqrt {\frac {1}{c x+1}}}+\frac {5 b c^2 \sqrt {1-c x}}{144 x^4 \sqrt {\frac {1}{c x+1}}}+\frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{c x+1}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 94
Rule 105
Rule 214
Rule 6418
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{x^7} \, dx &=-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}-\frac {1}{6} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^7 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}+\frac {1}{36} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int -\frac {5 c^2}{x^5 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}-\frac {1}{36} \left (5 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^5 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^2 \sqrt {1-c x}}{144 x^4 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}+\frac {1}{144} \left (5 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int -\frac {3 c^2}{x^3 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^2 \sqrt {1-c x}}{144 x^4 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}-\frac {1}{48} \left (5 b c^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^3 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^2 \sqrt {1-c x}}{144 x^4 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^4 \sqrt {1-c x}}{96 x^2 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}-\frac {1}{96} \left (5 b c^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {c^2}{x \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^2 \sqrt {1-c x}}{144 x^4 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^4 \sqrt {1-c x}}{96 x^2 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}-\frac {1}{96} \left (5 b c^6 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^2 \sqrt {1-c x}}{144 x^4 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^4 \sqrt {1-c x}}{96 x^2 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}+\frac {1}{96} \left (5 b c^7 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{c-c x^2} \, dx,x,\sqrt {1-c x} \sqrt {1+c x}\right )\\ &=\frac {b \sqrt {1-c x}}{36 x^6 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^2 \sqrt {1-c x}}{144 x^4 \sqrt {\frac {1}{1+c x}}}+\frac {5 b c^4 \sqrt {1-c x}}{96 x^2 \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}+\frac {5}{96} b c^6 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c x} \sqrt {1+c x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 157, normalized size = 0.99 \begin {gather*} -\frac {a}{6 x^6}+b \left (\frac {1}{36 x^6}+\frac {c}{36 x^5}+\frac {5 c^2}{144 x^4}+\frac {5 c^3}{144 x^3}+\frac {5 c^4}{96 x^2}+\frac {5 c^5}{96 x}\right ) \sqrt {\frac {1-c x}{1+c x}}-\frac {b \text {sech}^{-1}(c x)}{6 x^6}-\frac {5}{96} b c^6 \log (x)+\frac {5}{96} b c^6 \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 155, normalized size = 0.98
method | result | size |
derivativedivides | \(c^{6} \left (-\frac {a}{6 c^{6} x^{6}}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{6 c^{6} x^{6}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (15 \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{6} x^{6}+15 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+10 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+8 \sqrt {-c^{2} x^{2}+1}\right )}{288 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(155\) |
default | \(c^{6} \left (-\frac {a}{6 c^{6} x^{6}}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{6 c^{6} x^{6}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (15 \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{6} x^{6}+15 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+10 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+8 \sqrt {-c^{2} x^{2}+1}\right )}{288 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 185, normalized size = 1.17 \begin {gather*} \frac {1}{576} \, b {\left (\frac {15 \, c^{7} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} + 1\right ) - 15 \, c^{7} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} - 1\right ) - \frac {2 \, {\left (15 \, c^{12} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 40 \, c^{10} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 33 \, c^{8} x \sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}}{c^{6} x^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} - 3 \, c^{4} x^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 1}}{c} - \frac {96 \, \operatorname {arsech}\left (c x\right )}{x^{6}}\right )} - \frac {a}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 100, normalized size = 0.63 \begin {gather*} \frac {3 \, {\left (5 \, b c^{6} x^{6} - 16 \, b\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (15 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 8 \, b c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 48 \, a}{288 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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