Integrand size = 22, antiderivative size = 75 \[ \int \frac {e^{\text {sech}^{-1}(c x)} x^3}{1-c^2 x^2} \, dx=-\frac {x}{c^3}-\frac {x \sqrt {1-c x}}{2 c^3 \sqrt {\frac {1}{1+c x}}}+\frac {\sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arcsin (c x)}{2 c^4}+\frac {\text {arctanh}(c x)}{c^4} \]
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Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6476, 1972, 92, 41, 222, 327, 212} \[ \int \frac {e^{\text {sech}^{-1}(c x)} x^3}{1-c^2 x^2} \, dx=\frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \arcsin (c x)}{2 c^4}+\frac {\text {arctanh}(c x)}{c^4}-\frac {x \sqrt {1-c x}}{2 c^3 \sqrt {\frac {1}{c x+1}}}-\frac {x}{c^3} \]
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Rule 41
Rule 92
Rule 212
Rule 222
Rule 327
Rule 1972
Rule 6476
Rubi steps \begin{align*} \text {integral}& = \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 {\text {arctanh}(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 {\text {arctanh}(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} \arcsin (c x)}{2 c^4}+\frac {\text {arctanh}(c x)}{c^4} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.47 \[ \int \frac {e^{\text {sech}^{-1}(c x)} x^3}{1-c^2 x^2} \, dx=-\frac {2 c x+c x \sqrt {\frac {1-c x}{1+c x}}+c^2 x^2 \sqrt {\frac {1-c x}{1+c x}}+\log (1-c x)-\log (1+c x)-i \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{2 c^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.68 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.63
method | result | size |
default | \(-\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (x \sqrt {-c^{2} x^{2}+1}\, \operatorname {csgn}\left (c \right ) c -\arctan \left (\frac {\operatorname {csgn}\left (c \right ) c x}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \operatorname {csgn}\left (c \right )}{2 c^{3} \sqrt {-c^{2} x^{2}+1}}+\frac {-\frac {x}{c^{2}}+\frac {\ln \left (c x +1\right )}{2 c^{3}}-\frac {\ln \left (c x -1\right )}{2 c^{3}}}{c}\) | \(122\) |
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Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (45) = 90\).
Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\text {sech}^{-1}(c x)} x^3}{1-c^2 x^2} \, dx=-\frac {c^{2} x^{2} \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} + 2 \, c x + \arctan \left (\sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}}\right ) - \log \left (c x + 1\right ) + \log \left (c x - 1\right )}{2 \, c^{4}} \]
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\[ \int \frac {e^{\text {sech}^{-1}(c x)} x^3}{1-c^2 x^2} \, dx=- \frac {\int \frac {x^{2}}{c^{2} x^{2} - 1}\, dx + \int \frac {c x^{3} \sqrt {-1 + \frac {1}{c x}} \sqrt {1 + \frac {1}{c x}}}{c^{2} x^{2} - 1}\, dx}{c} \]
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\[ \int \frac {e^{\text {sech}^{-1}(c x)} x^3}{1-c^2 x^2} \, dx=\int { -\frac {x^{3} {\left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}}{c^{2} x^{2} - 1} \,d x } \]
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\[ \int \frac {e^{\text {sech}^{-1}(c x)} x^3}{1-c^2 x^2} \, dx=\int { -\frac {x^{3} {\left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}}{c^{2} x^{2} - 1} \,d x } \]
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Time = 11.19 (sec) , antiderivative size = 340, normalized size of antiderivative = 4.53 \[ \int \frac {e^{\text {sech}^{-1}(c x)} x^3}{1-c^2 x^2} \, dx=\frac {\mathrm {atanh}\left (c\,x\right )-c\,x}{c^4}-\frac {\ln \left (\frac {\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{c\,x}+1}-1}\right )\,1{}\mathrm {i}}{2\,c^4}-\frac {\frac {1{}\mathrm {i}}{32\,c^4}+\frac {{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{16\,c^4\,{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{32\,c^4\,{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^4}}{\frac {{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^6}}+\frac {\ln \left (\frac {2\,c\,\sqrt {\frac {c+\frac {1}{x}}{c}}-\frac {2}{x}+c\,\sqrt {-\frac {c-\frac {1}{x}}{c}}\,2{}\mathrm {i}}{2\,c+\frac {1}{x}-2\,c\,\sqrt {\frac {c+\frac {1}{x}}{c}}}\right )\,1{}\mathrm {i}}{2\,c^4}-\frac {{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{32\,c^4\,{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^2} \]
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