Optimal. Leaf size=201 \[ -\frac {b d e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2}-\frac {b e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^2}+\frac {(d+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{6 c^3}-\frac {b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{3 e} \]
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Rubi [A]
time = 0.15, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6423, 1823,
858, 222, 272, 65, 214} \begin {gather*} \frac {(d+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcSin}(c x) \left (6 c^2 d^2+e^2\right )}{6 c^3}-\frac {b d^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{3 e}-\frac {b d e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{c^2}-\frac {b e^2 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 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 1823
Rule 6423
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(d+e x)^3}{x \sqrt {1-c^2 x^2}} \, dx}{3 e}\\ &=-\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+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-2 c^2 d^3-e \left (6 c^2 d^2+e^2\right ) x-6 c^2 d e^2 x^2}{x \sqrt {1-c^2 x^2}} \, dx}{6 c^2 e}\\ &=-\frac {b d e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2}-\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+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {2 c^4 d^3+c^2 e \left (6 c^2 d^2+e^2\right ) x}{x \sqrt {1-c^2 x^2}} \, dx}{6 c^4 e}\\ &=-\frac {b d e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2}-\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+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}+\frac {\left (b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx}{3 e}+\frac {\left (b \left (6 c^2 d^2+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 e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2}-\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+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{6 c^3}+\frac {\left (b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{6 e}\\ &=-\frac {b d e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2}-\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+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{6 c^3}-\frac {\left (b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{3 c^2 e}\\ &=-\frac {b d e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2}-\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+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{6 c^3}-\frac {b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{3 e}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.14, size = 147, normalized size = 0.73 \begin {gather*} \frac {-b c e \sqrt {\frac {1-c x}{1+c x}} (1+c x) (6 d+e x)+2 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+2 b c^3 x \left (3 d^2+3 d e x+e^2 x^2\right ) \text {sech}^{-1}(c x)+i b \left (6 c^2 d^2+e^2\right ) \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 = 0.28, size = 215, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {\frac {\left (c e x +c d \right )^{3} a}{3 c^{2} e}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{3} d^{3}}{3 e}+\mathrm {arcsech}\left (c x \right ) c^{3} d^{2} x +e \,\mathrm {arcsech}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \mathrm {arcsech}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-2 c^{3} d^{3} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+6 c^{2} d^{2} e \arcsin \left (c x \right )-6 c d \,e^{2} \sqrt {-c^{2} x^{2}+1}-e^{3} c x \sqrt {-c^{2} x^{2}+1}+e^{3} \arcsin \left (c x \right )\right )}{6 e \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c}\) | \(215\) |
default | \(\frac {\frac {\left (c e x +c d \right )^{3} a}{3 c^{2} e}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{3} d^{3}}{3 e}+\mathrm {arcsech}\left (c x \right ) c^{3} d^{2} x +e \,\mathrm {arcsech}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \mathrm {arcsech}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-2 c^{3} d^{3} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+6 c^{2} d^{2} e \arcsin \left (c x \right )-6 c d \,e^{2} \sqrt {-c^{2} x^{2}+1}-e^{3} c x \sqrt {-c^{2} x^{2}+1}+e^{3} \arcsin \left (c x \right )\right )}{6 e \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c}\) | \(215\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 152, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x + {\left (x^{2} \operatorname {arsech}\left (c x\right ) - \frac {x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c}\right )} b d 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^{2}}{c} + \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} \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. 464 vs.
\(2 (105) = 210\).
time = 0.42, size = 464, normalized size = 2.31 \begin {gather*} \frac {2 \, a c^{3} x^{3} \cosh \left (1\right )^{2} + 2 \, a c^{3} x^{3} \sinh \left (1\right )^{2} + 6 \, a c^{3} d x^{2} \cosh \left (1\right ) + 6 \, a c^{3} d^{2} x - 2 \, {\left (6 \, b c^{2} d^{2} + b \cosh \left (1\right )^{2} + 2 \, b \cosh \left (1\right ) \sinh \left (1\right ) + b \sinh \left (1\right )^{2}\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^{2} + 3 \, b c^{3} d \cosh \left (1\right ) + b c^{3} \cosh \left (1\right )^{2} + b c^{3} \sinh \left (1\right )^{2} + {\left (3 \, b c^{3} d + 2 \, b c^{3} \cosh \left (1\right )\right )} \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^{2} x - 3 \, b c^{3} d^{2} + {\left (b c^{3} x^{3} - b c^{3}\right )} \cosh \left (1\right )^{2} + {\left (b c^{3} x^{3} - b c^{3}\right )} \sinh \left (1\right )^{2} + 3 \, {\left (b c^{3} d x^{2} - b c^{3} d\right )} \cosh \left (1\right ) + {\left (3 \, b c^{3} d x^{2} - 3 \, b c^{3} d + 2 \, {\left (b c^{3} x^{3} - b c^{3}\right )} \cosh \left (1\right )\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 ) + 2 \, {\left (2 \, a c^{3} x^{3} \cosh \left (1\right ) + 3 \, a c^{3} d x^{2}\right )} \sinh \left (1\right ) - {\left (b c^{2} x^{2} \cosh \left (1\right )^{2} + b c^{2} x^{2} \sinh \left (1\right )^{2} + 6 \, b c^{2} d x \cosh \left (1\right ) + 2 \, {\left (b c^{2} x^{2} \cosh \left (1\right ) + 3 \, b c^{2} d x\right )} \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\right )^{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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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