Optimal. Leaf size=75 \[ -\frac {x}{c^3}-\frac {x \sqrt {1-c x}}{2 c^3 \sqrt {\frac {1}{1+c x}}}+\frac {\sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{2 c^4}+\frac {\tanh ^{-1}(c x)}{c^4} \]
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Rubi [A]
time = 0.11, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6476, 1972, 92,
41, 222, 327, 212} \begin {gather*} \frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcSin}(c x)}{2 c^4}+\frac {\tanh ^{-1}(c x)}{c^4}-\frac {x \sqrt {1-c x}}{2 c^3 \sqrt {\frac {1}{c x+1}}}-\frac {x}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 92
Rule 212
Rule 222
Rule 327
Rule 1972
Rule 6476
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}(c x)} x^3}{1-c^2 x^2} \, dx &=\frac {\int \frac {x^2 \sqrt {\frac {1}{1+c x}}}{\sqrt {1-c x}} \, dx}{c}+\frac {\int \frac {x^2}{1-c^2 x^2} \, dx}{c}\\ &=-\frac {x}{c^3}+\frac {\int \frac {1}{1-c^2 x^2} \, dx}{c^3}+\frac {\left (\sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{c}\\ &=-\frac {x}{c^3}-\frac {x \sqrt {1-c x}}{2 c^3 \sqrt {\frac {1}{1+c x}}}+\frac {\tanh ^{-1}(c x)}{c^4}+\frac {\left (\sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{2 c^3}\\ &=-\frac {x}{c^3}-\frac {x \sqrt {1-c x}}{2 c^3 \sqrt {\frac {1}{1+c x}}}+\frac {\tanh ^{-1}(c x)}{c^4}+\frac {\left (\sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 c^3}\\ &=-\frac {x}{c^3}-\frac {x \sqrt {1-c x}}{2 c^3 \sqrt {\frac {1}{1+c x}}}+\frac {\sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{2 c^4}+\frac {\tanh ^{-1}(c x)}{c^4}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.13, size = 110, normalized size = 1.47 \begin {gather*} -\frac {2 c x+c x \sqrt {\frac {1-c x}{1+c x}}+c^2 x^2 \sqrt {\frac {1-c x}{1+c x}}+\log (1-c x)-\log (1+c x)-i \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{2 c^4} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.32, size = 122, normalized size = 1.63
method | result | size |
default | \(-\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (x \sqrt {-c^{2} x^{2}+1}\, \mathrm {csgn}\left (c \right ) c -\arctan \left (\frac {\mathrm {csgn}\left (c \right ) c x}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \mathrm {csgn}\left (c \right )}{2 c^{3} \sqrt {-c^{2} x^{2}+1}}+\frac {-\frac {x}{c^{2}}+\frac {\ln \left (c x +1\right )}{2 c^{3}}-\frac {\ln \left (c x -1\right )}{2 c^{3}}}{c}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 91 vs.
\(2 (45) = 90\).
time = 0.44, size = 91, normalized size = 1.21 \begin {gather*} -\frac {c^{2} x^{2} \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} + 2 \, c x + \arctan \left (\sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}}\right ) - \log \left (c x + 1\right ) + \log \left (c x - 1\right )}{2 \, c^{4}} \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*} - \frac {\int \frac {x^{2}}{c^{2} x^{2} - 1}\, dx + \int \frac {c x^{3} \sqrt {-1 + \frac {1}{c x}} \sqrt {1 + \frac {1}{c x}}}{c^{2} x^{2} - 1}\, dx}{c} \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 [B]
time = 8.08, size = 340, normalized size = 4.53 \begin {gather*} \frac {\mathrm {atanh}\left (c\,x\right )-c\,x}{c^4}-\frac {\ln \left (\frac {\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{c\,x}+1}-1}\right )\,1{}\mathrm {i}}{2\,c^4}-\frac {\frac {1{}\mathrm {i}}{32\,c^4}+\frac {{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{16\,c^4\,{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{32\,c^4\,{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^4}}{\frac {{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^6}}+\frac {\ln \left (\frac {2\,c\,\sqrt {\frac {c+\frac {1}{x}}{c}}-\frac {2}{x}+c\,\sqrt {-\frac {c-\frac {1}{x}}{c}}\,2{}\mathrm {i}}{2\,c+\frac {1}{x}-2\,c\,\sqrt {\frac {c+\frac {1}{x}}{c}}}\right )\,1{}\mathrm {i}}{2\,c^4}-\frac {{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{32\,c^4\,{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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