Optimal. Leaf size=86 \[ \frac {1}{6} x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x^5 \sqrt {\frac {1}{c^2 x^2}+1}}{30 c}+\frac {4 b x \sqrt {\frac {1}{c^2 x^2}+1}}{45 c^5}-\frac {2 b x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{45 c^3} \]
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Rubi [A] time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6284, 271, 191} \[ \frac {1}{6} x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x^5 \sqrt {\frac {1}{c^2 x^2}+1}}{30 c}-\frac {2 b x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{45 c^3}+\frac {4 b x \sqrt {\frac {1}{c^2 x^2}+1}}{45 c^5} \]
Antiderivative was successfully verified.
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Rule 191
Rule 271
Rule 6284
Rubi steps
\begin {align*} \int x^5 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {1}{6} x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b \int \frac {x^4}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{6 c}\\ &=\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^5}{30 c}+\frac {1}{6} x^6 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(2 b) \int \frac {x^2}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{15 c^3}\\ &=-\frac {2 b \sqrt {1+\frac {1}{c^2 x^2}} x^3}{45 c^3}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^5}{30 c}+\frac {1}{6} x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {(4 b) \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{45 c^5}\\ &=\frac {4 b \sqrt {1+\frac {1}{c^2 x^2}} x}{45 c^5}-\frac {2 b \sqrt {1+\frac {1}{c^2 x^2}} x^3}{45 c^3}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^5}{30 c}+\frac {1}{6} x^6 \left (a+b \text {csch}^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 72, normalized size = 0.84 \[ \frac {a x^6}{6}+b \sqrt {\frac {c^2 x^2+1}{c^2 x^2}} \left (\frac {4 x}{45 c^5}-\frac {2 x^3}{45 c^3}+\frac {x^5}{30 c}\right )+\frac {1}{6} b x^6 \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [A] time = 2.01, size = 97, normalized size = 1.13 \[ \frac {15 \, b c^{5} x^{6} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 15 \, a c^{5} x^{6} + {\left (3 \, b c^{4} x^{5} - 4 \, b c^{2} x^{3} + 8 \, b x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{90 \, c^{5}} \]
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 \[ \int {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 83, normalized size = 0.97 \[ \frac {\frac {c^{6} x^{6} a}{6}+b \left (\frac {c^{6} x^{6} \mathrm {arccsch}\left (c x \right )}{6}+\frac {\left (c^{2} x^{2}+1\right ) \left (3 c^{4} x^{4}-4 c^{2} x^{2}+8\right )}{90 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 77, normalized size = 0.90 \[ \frac {1}{6} \, a x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arcsch}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b \]
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 \[ \int x^5\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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 \[ \int x^{5} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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