Optimal. Leaf size=177 \[ -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b d^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{x}-\frac {b e^2 x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{6 c^2}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (12 c^2 d+e\right ) \sin ^{-1}(c x)}{6 c^3} \]
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Rubi [A] time = 0.13, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {270, 6301, 12, 1265, 388, 216} \[ -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b d^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{x}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (12 c^2 d+e\right ) \sin ^{-1}(c x)}{6 c^3}-\frac {b e^2 x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{6 c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 270
Rule 388
Rule 1265
Rule 6301
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-3 d^2+6 d e x^2+e^2 x^4}{3 x^2 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-3 d^2+6 d e x^2+e^2 x^4}{x^2 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{x}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {1}{3} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-6 d e-e^2 x^2}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{x}-\frac {b e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^2}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+-\frac {\left (b \left (-12 c^2 d e-e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{6 c^2}\\ &=\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{x}-\frac {b e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^2}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b e \left (12 c^2 d+e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{6 c^3}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 158, normalized size = 0.89 \[ \frac {2 a c^3 \left (-3 d^2+6 d e x^2+e^2 x^4\right )+2 b c^3 \text {sech}^{-1}(c x) \left (-3 d^2+6 d e x^2+e^2 x^4\right )-b c \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (e^2 x^2-6 c^2 d^2\right )+i b e x \left (12 c^2 d+e\right ) \log \left (2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)-2 i c x\right )}{6 c^3 x} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.84, size = 287, normalized size = 1.62 \[ \frac {2 \, a c^{3} e^{2} x^{4} + 12 \, a c^{3} d e x^{2} - 6 \, a c^{3} d^{2} - 2 \, {\left (12 \, b c^{2} d e + b e^{2}\right )} x \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) + 2 \, {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 2 \, {\left (b c^{3} e^{2} x^{4} + 6 \, b c^{3} d e x^{2} - 3 \, b c^{3} d^{2} + {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (6 \, b c^{4} d^{2} x - b c^{2} e^{2} x^{3}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{6 \, c^{3} x} \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 197, normalized size = 1.11 \[ c \left (\frac {a \left (\frac {c^{3} x^{3} e^{2}}{3}+2 c^{3} d e x -\frac {d^{2} c^{3}}{x}\right )}{c^{4}}+\frac {b \left (\frac {e^{2} \mathrm {arcsech}\left (c x \right ) c^{3} x^{3}}{3}+2 \,\mathrm {arcsech}\left (c x \right ) c^{3} d e x -\frac {\mathrm {arcsech}\left (c x \right ) d^{2} c^{3}}{x}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (6 \sqrt {-c^{2} x^{2}+1}\, c^{4} d^{2}+12 \arcsin \left (c x \right ) c^{3} x d e -c^{2} x^{2} e^{2} \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right ) c x \,e^{2}\right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 152, normalized size = 0.86 \[ \frac {1}{3} \, a e^{2} x^{3} + {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} b d^{2} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b e^{2} + 2 \, a d e x + \frac {2 \, {\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b d e}{c} - \frac {a d^{2}}{x} \]
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 \[ \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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