Optimal. Leaf size=275 \[ -\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1680 c^6}-\frac {b e \left (84 c^2 d+25 e\right ) x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{1680 c^7} \]
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Rubi [A]
time = 0.16, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 6436, 12,
1281, 470, 327, 222} \begin {gather*} \frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcSin}(c x) \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^7}-\frac {b e^2 x^5 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{42 c^2}-\frac {b e x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (84 c^2 d+25 e\right )}{840 c^4}-\frac {b x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 222
Rule 276
Rule 327
Rule 470
Rule 1281
Rule 6436
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{105} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \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 {x^2 \left (-210 c^2 d^2-3 e \left (84 c^2 d+25 e\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{630 c^2}\\ &=-\frac {b e \left (84 c^2 d+25 e\right ) x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+-\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2520 c^4}\\ &=-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1680 c^6}-\frac {b e \left (84 c^2 d+25 e\right ) x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+-\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{5040 c^6}\\ &=-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1680 c^6}-\frac {b e \left (84 c^2 d+25 e\right ) x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{1680 c^7}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 207, normalized size = 0.75 \begin {gather*} \frac {16 a c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-b c x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (75 e^2+2 c^2 e \left (126 d+25 e x^2\right )+8 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )\right )+16 b c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right ) \text {sech}^{-1}(c x)+i b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{1680 c^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 300, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \,\mathrm {arcsech}\left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\mathrm {arcsech}\left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (280 d^{2} c^{5} x \sqrt {-c^{2} x^{2}+1}+168 c^{5} d e \sqrt {-c^{2} x^{2}+1}\, x^{3}+40 e^{2} \sqrt {-c^{2} x^{2}+1}\, c^{5} x^{5}-280 d^{2} c^{4} \arcsin \left (c x \right )+252 \sqrt {-c^{2} x^{2}+1}\, c^{3} d e x +50 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-252 \arcsin \left (c x \right ) c^{2} d e +75 e^{2} c x \sqrt {-c^{2} x^{2}+1}-75 e^{2} \arcsin \left (c x \right )\right )}{1680 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c^{3}}\) | \(300\) |
default | \(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \,\mathrm {arcsech}\left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\mathrm {arcsech}\left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (280 d^{2} c^{5} x \sqrt {-c^{2} x^{2}+1}+168 c^{5} d e \sqrt {-c^{2} x^{2}+1}\, x^{3}+40 e^{2} \sqrt {-c^{2} x^{2}+1}\, c^{5} x^{5}-280 d^{2} c^{4} \arcsin \left (c x \right )+252 \sqrt {-c^{2} x^{2}+1}\, c^{3} d e x +50 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-252 \arcsin \left (c x \right ) c^{2} d e +75 e^{2} c x \sqrt {-c^{2} x^{2}+1}-75 e^{2} \arcsin \left (c x \right )\right )}{1680 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c^{3}}\) | \(300\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 328, normalized size = 1.19 \begin {gather*} \frac {1}{7} \, a x^{7} e^{2} + \frac {2}{5} \, a d x^{5} e + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{6} \, {\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^{2} + \frac {1}{20} \, {\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 d e + \frac {1}{336} \, {\left (48 \, x^{7} \operatorname {arsech}\left (c x\right ) - \frac {\frac {15 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{6}}}{c}\right )} b e^{2} \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. 600 vs.
\(2 (176) = 352\).
time = 0.57, size = 600, normalized size = 2.18 \begin {gather*} \frac {240 \, a c^{7} x^{7} \cosh \left (1\right )^{2} + 240 \, a c^{7} x^{7} \sinh \left (1\right )^{2} + 672 \, a c^{7} d x^{5} \cosh \left (1\right ) + 560 \, a c^{7} d^{2} x^{3} - 2 \, {\left (280 \, b c^{4} d^{2} + 252 \, b c^{2} d \cosh \left (1\right ) + 75 \, b \cosh \left (1\right )^{2} + 75 \, b \sinh \left (1\right )^{2} + 6 \, {\left (42 \, b c^{2} d + 25 \, b \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 16 \, {\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d \cosh \left (1\right ) + 15 \, b c^{7} \cosh \left (1\right )^{2} + 15 \, b c^{7} \sinh \left (1\right )^{2} + 6 \, {\left (7 \, b c^{7} d + 5 \, b c^{7} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 16 \, {\left (35 \, b c^{7} d^{2} x^{3} - 35 \, b c^{7} d^{2} + 15 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \cosh \left (1\right )^{2} + 15 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \sinh \left (1\right )^{2} + 42 \, {\left (b c^{7} d x^{5} - b c^{7} d\right )} \cosh \left (1\right ) + 6 \, {\left (7 \, b c^{7} d x^{5} - 7 \, b c^{7} d + 5 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 96 \, {\left (5 \, a c^{7} x^{7} \cosh \left (1\right ) + 7 \, a c^{7} d x^{5}\right )} \sinh \left (1\right ) - {\left (280 \, b c^{6} d^{2} x^{2} + 5 \, {\left (8 \, b c^{6} x^{6} + 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \cosh \left (1\right )^{2} + 5 \, {\left (8 \, b c^{6} x^{6} + 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \sinh \left (1\right )^{2} + 84 \, {\left (2 \, b c^{6} d x^{4} + 3 \, b c^{4} d x^{2}\right )} \cosh \left (1\right ) + 2 \, {\left (84 \, b c^{6} d x^{4} + 126 \, b c^{4} d x^{2} + 5 \, {\left (8 \, b c^{6} x^{6} + 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{1680 \, c^{7}} \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^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, 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 = -0.00 \begin {gather*} \int x^2\,{\left (e\,x^2+d\right )}^2\,\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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