Optimal. Leaf size=112 \[ -\frac {b e x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^2}+d x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (6 c^2 d+e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{6 c^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6426, 12, 396,
222} \begin {gather*} d x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e x^3 \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 (6 c^2 d+e\right )}{6 c^3}-\frac {b e x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{6 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 222
Rule 396
Rule 6426
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=d x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e 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+e x^2}{3 \sqrt {1-c^2 x^2}} \, dx\\ &=d x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e 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+e x^2}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b e x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^2}+d x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (b \left (6 c^2 d+e\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 e x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^2}+d x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (6 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] Result contains complex when optimal does not.
time = 0.32, size = 186, normalized size = 1.66 \begin {gather*} a d x+\frac {1}{3} a e x^3+b e \sqrt {\frac {1-c x}{1+c x}} \left (-\frac {x}{6 c^2}-\frac {x^2}{6 c}\right )+b d x \text {sech}^{-1}(c x)+\frac {1}{3} b e x^3 \text {sech}^{-1}(c x)-\frac {2 b d \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}+\frac {i b e \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{6 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 135, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\mathrm {arcsech}\left (c x \right ) d \,c^{3} x +\frac {\mathrm {arcsech}\left (c x \right ) e \,c^{3} x^{3}}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (6 \arcsin \left (c x \right ) c^{2} d -e c x \sqrt {-c^{2} x^{2}+1}+e \arcsin \left (c x \right )\right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c}\) | \(135\) |
default | \(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\mathrm {arcsech}\left (c x \right ) d \,c^{3} x +\frac {\mathrm {arcsech}\left (c x \right ) e \,c^{3} x^{3}}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (6 \arcsin \left (c x \right ) c^{2} d -e c x \sqrt {-c^{2} x^{2}+1}+e \arcsin \left (c x \right )\right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c}\) | \(135\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 109, normalized size = 0.97 \begin {gather*} \frac {1}{3} \, a x^{3} e + a d x + \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 + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b d}{c} \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. 267 vs.
\(2 (67) = 134\).
time = 0.58, size = 267, normalized size = 2.38 \begin {gather*} \frac {2 \, a c^{3} x^{3} \cosh \left (1\right ) + 2 \, a c^{3} x^{3} \sinh \left (1\right ) + 6 \, a c^{3} d x - 2 \, {\left (6 \, b c^{2} d + b \cosh \left (1\right ) + b \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 ) - 2 \, {\left (3 \, b c^{3} d + b c^{3} \cosh \left (1\right ) + b c^{3} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 2 \, {\left (3 \, b c^{3} d x - 3 \, b c^{3} d + {\left (b c^{3} x^{3} - b c^{3}\right )} \cosh \left (1\right ) + {\left (b c^{3} x^{3} - b c^{3}\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 ) - {\left (b c^{2} x^{2} \cosh \left (1\right ) + b c^{2} x^{2} \sinh \left (1\right )\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{6 \, c^{3}} \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 \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, 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 = -0.01 \begin {gather*} \int \left (e\,x^2+d\right )\,\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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