Optimal. Leaf size=176 \[ -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b d^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{9 x^3}+\frac {2 b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (c^2 d+9 e\right )}{9 x}+\frac {b e^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sin ^{-1}(c x)}{c} \]
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Rubi [A] time = 0.14, antiderivative size = 176, 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, 451, 216} \[ -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b d^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{9 x^3}+\frac {2 b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (c^2 d+9 e\right )}{9 x}+\frac {b e^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sin ^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 270
Rule 451
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^4} \, dx &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-d^2-6 d e x^2+3 e^2 x^4}{3 x^4 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e^2 x \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 {-d^2-6 d e x^2+3 e^2 x^4}{x^4 \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}}{9 x^3}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {sech}^{-1}(c x)\right )-\frac {1}{9} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {2 d \left (c^2 d+9 e\right )-9 e^2 x^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}}{9 x^3}+\frac {2 b d \left (c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{9 x}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\left (b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\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}}{9 x^3}+\frac {2 b d \left (c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{9 x}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{c}\\ \end {align*}
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Mathematica [C] time = 0.31, size = 149, normalized size = 0.85 \[ \frac {-3 a c \left (d^2+6 d e x^2-3 e^2 x^4\right )+b c d \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (2 c^2 d x^2+d+18 e x^2\right )-3 b c \text {sech}^{-1}(c x) \left (d^2+6 d e x^2-3 e^2 x^4\right )+9 i b e^2 x^3 \log \left (2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)-2 i c x\right )}{9 c x^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 267, normalized size = 1.52 \[ \frac {9 \, a c e^{2} x^{4} - 18 \, b e^{2} x^{3} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 18 \, a c d e x^{2} + 3 \, {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 3 \, a c d^{2} + 3 \, {\left (3 \, b c e^{2} x^{4} - 6 \, b c d e x^{2} - b c d^{2} + {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (b c^{2} d^{2} x + 2 \, {\left (b c^{4} d^{2} + 9 \, b c^{2} d e\right )} x^{3}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{9 \, c x^{3}} \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 205, normalized size = 1.16 \[ c^{3} \left (\frac {a \left (c x \,e^{2}-\frac {2 c d e}{x}-\frac {d^{2} c}{3 x^{3}}\right )}{c^{4}}+\frac {b \left (\mathrm {arcsech}\left (c x \right ) c x \,e^{2}-\frac {2 \,\mathrm {arcsech}\left (c x \right ) c d e}{x}-\frac {\mathrm {arcsech}\left (c x \right ) d^{2} c}{3 x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 \sqrt {-c^{2} x^{2}+1}\, c^{6} x^{2} d^{2}+18 c^{4} d e \,x^{2} \sqrt {-c^{2} x^{2}+1}+\sqrt {-c^{2} x^{2}+1}\, c^{4} d^{2}+9 \arcsin \left (c x \right ) c^{3} x^{3} e^{2}\right )}{9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 134, normalized size = 0.76 \[ 2 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} b d e + a e^{2} x + \frac {1}{9} \, b d^{2} {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {3 \, \operatorname {arsech}\left (c x\right )}{x^{3}}\right )} + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b e^{2}}{c} - \frac {2 \, a d e}{x} - \frac {a d^{2}}{3 \, x^{3}} \]
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 \[ \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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