Optimal. Leaf size=96 \[ -\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{x}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sin ^{-1}(c x)}{c} \]
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Rubi [A] time = 0.07, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {14, 6301, 451, 216} \[ -\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{x}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sin ^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Rule 14
Rule 216
Rule 451
Rule 6301
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e 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+e x^2}{x^2 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{x}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e x \left (a+b \text {sech}^{-1}(c x)\right )+\left (b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{x}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{c}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 107, normalized size = 1.11 \[ -\frac {a d}{x}+a e x-\frac {b e \sqrt {\frac {1-c x}{c x+1}} \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c (c x-1)}+b d \left (c+\frac {1}{x}\right ) \sqrt {\frac {1-c x}{c x+1}}-\frac {b d \text {sech}^{-1}(c x)}{x}+b e x \text {sech}^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 182, normalized size = 1.90 \[ \frac {b c^{2} d x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + a c e x^{2} - 2 \, b e x \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - a c d + {\left (b c d - b c e\right )} x \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + {\left (b c e x^{2} - b c d + {\left (b c d - b c e\right )} x\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c x} \]
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 \[ \int \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 114, normalized size = 1.19 \[ c \left (\frac {a \left (c x e -\frac {c d}{x}\right )}{c^{2}}+\frac {b \left (\mathrm {arcsech}\left (c x \right ) c x e -\frac {\mathrm {arcsech}\left (c x \right ) c d}{x}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\sqrt {-c^{2} x^{2}+1}\, c^{2} d +\arcsin \left (c x \right ) c x e \right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 66, normalized size = 0.69 \[ {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} b d + a e x + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b e}{c} - \frac {a d}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.81, size = 98, normalized size = 1.02 \[ a\,e\,x-\frac {a\,d}{x}+b\,c\,d\,\left (\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}-\frac {\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{c\,x}\right )+\frac {b\,e\,\mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}}\right )}{c}+b\,e\,x\,\mathrm {acosh}\left (\frac {1}{c\,x}\right ) \]
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 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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