Optimal. Leaf size=96 \[ \frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{x}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{c} \]
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Rubi [A]
time = 0.04, 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, 6436, 462,
222} \begin {gather*} -\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}{c x+1}} \sqrt {c x+1} \text {ArcSin}(c x)}{c}+\frac {b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 222
Rule 462
Rule 6436
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 = 0.15, size = 124, normalized size = 1.29 \begin {gather*} -\frac {a d}{x}+a e x+b d \left (c+\frac {1}{x}\right ) \sqrt {\frac {1-c x}{1+c x}}-\frac {b d \text {sech}^{-1}(c x)}{x}+b e x \text {sech}^{-1}(c x)-\frac {2 b e \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \text {ArcTan}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c-c^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 114, normalized size = 1.19
method | result | size |
derivativedivides | \(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\mathrm {arcsech}\left (c x \right ) e c x -\frac {\mathrm {arcsech}\left (c x \right ) d c}{x}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\sqrt {-c^{2} x^{2}+1}\, c^{2} d +\arcsin \left (c x \right ) e c x \right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}\right )\) | \(114\) |
default | \(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\mathrm {arcsech}\left (c x \right ) e c x -\frac {\mathrm {arcsech}\left (c x \right ) d c}{x}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\sqrt {-c^{2} x^{2}+1}\, c^{2} d +\arcsin \left (c x \right ) e c x \right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}\right )\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 68, normalized size = 0.71 \begin {gather*} {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} b d + a x e + \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} \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. 224 vs.
\(2 (56) = 112\).
time = 0.47, size = 224, normalized size = 2.33 \begin {gather*} \frac {b c^{2} d x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + a c x^{2} \cosh \left (1\right ) + a c x^{2} \sinh \left (1\right ) - a c d - 2 \, {\left (b x \cosh \left (1\right ) + b x \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 ) + {\left (b c d x - b c x \cosh \left (1\right ) - b c x \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + {\left (b c d x - b c d + {\left (b c x^{2} - b c x\right )} \cosh \left (1\right ) + {\left (b c x^{2} - b c x\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 )}{c x} \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 \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{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 [B]
time = 1.81, size = 98, normalized size = 1.02 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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