Optimal. Leaf size=109 \[ \frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {4 b \sqrt {1-c x}}{45 c^6 \sqrt {\frac {1}{c x+1}}}-\frac {2 b x^2 \sqrt {1-c x}}{45 c^4 \sqrt {\frac {1}{c x+1}}}-\frac {b x^4 \sqrt {1-c x}}{30 c^2 \sqrt {\frac {1}{c x+1}}} \]
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Rubi [A] time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6283, 100, 12, 74} \[ \frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b x^4 \sqrt {1-c x}}{30 c^2 \sqrt {\frac {1}{c x+1}}}-\frac {2 b x^2 \sqrt {1-c x}}{45 c^4 \sqrt {\frac {1}{c x+1}}}-\frac {4 b \sqrt {1-c x}}{45 c^6 \sqrt {\frac {1}{c x+1}}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 100
Rule 6283
Rubi steps
\begin {align*} \int x^5 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^5}{\sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b x^4 \sqrt {1-c x}}{30 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int -\frac {4 x^3}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{30 c^2}\\ &=-\frac {b x^4 \sqrt {1-c x}}{30 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{15 c^2}\\ &=-\frac {2 b x^2 \sqrt {1-c x}}{45 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^4 \sqrt {1-c x}}{30 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int -\frac {2 x}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{45 c^4}\\ &=-\frac {2 b x^2 \sqrt {1-c x}}{45 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^4 \sqrt {1-c x}}{30 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{45 c^4}\\ &=-\frac {4 b \sqrt {1-c x}}{45 c^6 \sqrt {\frac {1}{1+c x}}}-\frac {2 b x^2 \sqrt {1-c x}}{45 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^4 \sqrt {1-c x}}{30 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{6} x^6 \left (a+b \text {sech}^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 97, normalized size = 0.89 \[ \frac {a x^6}{6}+b \sqrt {\frac {1-c x}{c x+1}} \left (-\frac {4}{45 c^6}-\frac {4 x}{45 c^5}-\frac {2 x^2}{45 c^4}-\frac {2 x^3}{45 c^3}-\frac {x^4}{30 c^2}-\frac {x^5}{30 c}\right )+\frac {1}{6} b x^6 \text {sech}^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 100, normalized size = 0.92 \[ \frac {15 \, b c^{5} x^{6} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 15 \, a c^{5} x^{6} - {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{90 \, c^{5}} \]
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 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 81, normalized size = 0.74 \[ \frac {\frac {c^{6} x^{6} a}{6}+b \left (\frac {c^{6} x^{6} \mathrm {arcsech}\left (c x \right )}{6}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right )}{90}\right )}{c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 78, normalized size = 0.72 \[ \frac {1}{6} \, a x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arsech}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b \]
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 \[ \int x^5\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.57, size = 94, normalized size = 0.86 \[ \begin {cases} \frac {a x^{6}}{6} + \frac {b x^{6} \operatorname {asech}{\left (c x \right )}}{6} - \frac {b x^{4} \sqrt {- c^{2} x^{2} + 1}}{30 c^{2}} - \frac {2 b x^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{4}} - \frac {4 b \sqrt {- c^{2} x^{2} + 1}}{45 c^{6}} & \text {for}\: c \neq 0 \\\frac {x^{6} \left (a + \infty b\right )}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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