Integrand size = 18, antiderivative size = 204 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {b e \left (40 c^2 d+9 e\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{120 c^4}-\frac {b e^2 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arcsin (c x)}{120 c^5} \]
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Time = 0.11 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {200, 6426, 12, 1173, 396, 222} \[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \arcsin (c x) \left (120 c^4 d^2+40 c^2 d e+9 e^2\right )}{120 c^5}-\frac {b e^2 x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{20 c^2}-\frac {b e x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (40 c^2 d+9 e\right )}{120 c^4} \]
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Rule 12
Rule 200
Rule 222
Rule 396
Rule 1173
Rule 6426
Rubi steps \begin{align*} \text {integral}& = d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {15 d^2+10 d e x^2+3 e^2 x^4}{15 \sqrt {1-c^2 x^2}} \, dx \\ & = d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{15} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {15 d^2+10 d e x^2+3 e^2 x^4}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {b e^2 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 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 {-60 c^2 d^2-e \left (40 c^2 d+9 e\right ) x^2}{\sqrt {1-c^2 x^2}} \, dx}{60 c^2} \\ & = -\frac {b e \left (40 c^2 d+9 e\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{120 c^4}-\frac {b e^2 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {\left (b \left (-120 c^4 d^2-e \left (40 c^2 d+9 e\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{120 c^4} \\ & = -\frac {b e \left (40 c^2 d+9 e\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{120 c^4}-\frac {b e^2 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arcsin (c x)}{120 c^5} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.85 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {8 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-b c e x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (9 e+c^2 \left (40 d+6 e x^2\right )\right )+8 b c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \text {sech}^{-1}(c x)+i b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{120 c^5} \]
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Time = 0.62 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.02
method | result | size |
parts | \(a \left (\frac {1}{5} e^{2} x^{5}+\frac {2}{3} d e \,x^{3}+x \,d^{2}\right )+\frac {b \left (\frac {c \,\operatorname {arcsech}\left (c x \right ) e^{2} x^{5}}{5}+\frac {2 c \,\operatorname {arcsech}\left (c x \right ) d e \,x^{3}}{3}+\operatorname {arcsech}\left (c x \right ) x c \,d^{2}+\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (120 d^{2} c^{4} \arcsin \left (c x \right )-40 d \,c^{3} e x \sqrt {-c^{2} x^{2}+1}-6 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+40 d \,c^{2} e \arcsin \left (c x \right )-9 e^{2} c x \sqrt {-c^{2} x^{2}+1}+9 e^{2} \arcsin \left (c x \right )\right )}{120 c^{3} \sqrt {-c^{2} x^{2}+1}}\right )}{c}\) | \(208\) |
derivativedivides | \(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\operatorname {arcsech}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arcsech}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arcsech}\left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (120 d^{2} c^{4} \arcsin \left (c x \right )-40 d \,c^{3} e x \sqrt {-c^{2} x^{2}+1}-6 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+40 d \,c^{2} e \arcsin \left (c x \right )-9 e^{2} c x \sqrt {-c^{2} x^{2}+1}+9 e^{2} \arcsin \left (c x \right )\right )}{120 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c}\) | \(228\) |
default | \(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\operatorname {arcsech}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arcsech}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arcsech}\left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (120 d^{2} c^{4} \arcsin \left (c x \right )-40 d \,c^{3} e x \sqrt {-c^{2} x^{2}+1}-6 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+40 d \,c^{2} e \arcsin \left (c x \right )-9 e^{2} c x \sqrt {-c^{2} x^{2}+1}+9 e^{2} \arcsin \left (c x \right )\right )}{120 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c}\) | \(228\) |
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Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (130) = 260\).
Time = 0.37 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.50 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {24 \, a c^{5} e^{2} x^{5} + 80 \, a c^{5} d e x^{3} + 120 \, a c^{5} d^{2} x - 2 \, {\left (120 \, b c^{4} d^{2} + 40 \, b c^{2} d e + 9 \, b e^{2}\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 8 \, {\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 8 \, {\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x - 15 \, b c^{5} d^{2} - 10 \, b c^{5} d e - 3 \, b c^{5} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (6 \, b c^{4} e^{2} x^{4} + {\left (40 \, b c^{4} d e + 9 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{120 \, c^{5}} \]
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\[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b d e + \frac {1}{40} \, {\left (8 \, x^{5} \operatorname {arsech}\left (c x\right ) - \frac {\frac {3 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} + \frac {3 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{4}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b d^{2}}{c} \]
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\[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \,d x } \]
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Timed out. \[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int {\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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