Optimal. Leaf size=110 \[ -\frac {3 b x \sqrt {1-c x}}{40 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^3 \sqrt {1-c x}}{20 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{5} x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {3 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{40 c^5} \]
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Rubi [A]
time = 0.03, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6418, 102, 12,
92, 41, 222} \begin {gather*} \frac {1}{5} x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {3 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcSin}(c x)}{40 c^5}-\frac {3 b x \sqrt {1-c x}}{40 c^4 \sqrt {\frac {1}{c x+1}}}-\frac {b x^3 \sqrt {1-c x}}{20 c^2 \sqrt {\frac {1}{c x+1}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 41
Rule 92
Rule 102
Rule 222
Rule 6418
Rubi steps
\begin {align*} \int x^4 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {1}{5} x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^4}{\sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b x^3 \sqrt {1-c x}}{20 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{5} x^5 \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 {3 x^2}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{20 c^2}\\ &=-\frac {b x^3 \sqrt {1-c x}}{20 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{5} x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (3 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{20 c^2}\\ &=-\frac {3 b x \sqrt {1-c x}}{40 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^3 \sqrt {1-c x}}{20 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{5} x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (3 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{40 c^4}\\ &=-\frac {3 b x \sqrt {1-c x}}{40 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^3 \sqrt {1-c x}}{20 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{5} x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (3 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{40 c^4}\\ &=-\frac {3 b x \sqrt {1-c x}}{40 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^3 \sqrt {1-c x}}{20 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{5} x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {3 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{40 c^5}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 123, normalized size = 1.12 \begin {gather*} \frac {a x^5}{5}+b \sqrt {\frac {1-c x}{1+c x}} \left (-\frac {3 x}{40 c^4}-\frac {3 x^2}{40 c^3}-\frac {x^3}{20 c^2}-\frac {x^4}{20 c}\right )+\frac {1}{5} b x^5 \text {sech}^{-1}(c x)+\frac {3 i b \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{40 c^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 118, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {\frac {c^{5} x^{5} a}{5}+b \left (\frac {c^{5} x^{5} \mathrm {arcsech}\left (c x \right )}{5}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{40 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{5}}\) | \(118\) |
default | \(\frac {\frac {c^{5} x^{5} a}{5}+b \left (\frac {c^{5} x^{5} \mathrm {arcsech}\left (c x \right )}{5}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{40 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{5}}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 106, normalized size = 0.96 \begin {gather*} \frac {1}{5} \, a x^{5} + \frac {1}{40} \, {\left (8 \, x^{5} \operatorname {arsech}\left (c x\right ) - \frac {\frac {3 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} + \frac {3 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{4}}}{c}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs.
\(2 (70) = 140\).
time = 0.37, size = 174, normalized size = 1.58 \begin {gather*} \frac {8 \, a c^{5} x^{5} - 8 \, b c^{5} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 6 \, b \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) + 8 \, {\left (b c^{5} x^{5} - b c^{5}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, b c^{4} x^{4} + 3 \, b c^{2} x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{40 \, c^{5}} \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 x^{4} \left (a + b \operatorname {asech}{\left (c x \right )}\right )\, 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 x^4\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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