Optimal. Leaf size=142 \[ \frac {(d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e}-\frac {b d^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 e}-\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{2 c^2}+\frac {b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sin ^{-1}(c x)}{c} \]
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Rubi [A] time = 0.12, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6288, 1809, 844, 216, 266, 63, 208} \[ \frac {(d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e}-\frac {b d^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 e}-\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{2 c^2}+\frac {b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sin ^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1809
Rule 6288
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(d+e x)^2}{x \sqrt {1-c^2 x^2}} \, dx}{2 e}\\ &=-\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}+\frac {(d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-c^2 d^2-2 c^2 d e x}{x \sqrt {1-c^2 x^2}} \, dx}{2 c^2 e}\\ &=-\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}+\frac {(d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e}+\left (b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx+\frac {\left (b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx}{2 e}\\ &=-\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}+\frac {(d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e}+\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{c}+\frac {\left (b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{4 e}\\ &=-\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}+\frac {(d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e}+\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{c}-\frac {\left (b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{2 c^2 e}\\ &=-\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 c^2}+\frac {(d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e}+\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{c}-\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 122, normalized size = 0.86 \[ a d x+\frac {1}{2} a e x^2-\frac {b d \sqrt {\frac {1-c x}{c x+1}} \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c (c x-1)}+b e \left (-\frac {1}{2 c^2}-\frac {x}{2 c}\right ) \sqrt {\frac {1-c x}{c x+1}}+b d x \text {sech}^{-1}(c x)+\frac {1}{2} b e x^2 \text {sech}^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 177, normalized size = 1.25 \[ \frac {a c e x^{2} + 2 \, a c d x - b e x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 4 \, b d \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - {\left (2 \, b c d + b c e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + {\left (b c e x^{2} + 2 \, b c d x - 2 \, b c d - b c e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, c} \]
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 \[ \int {\left (e x + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 125, normalized size = 0.88 \[ \frac {\frac {a \left (\frac {1}{2} c^{2} x^{2} e +c^{2} d x \right )}{c}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{2} x^{2} e}{2}+\mathrm {arcsech}\left (c x \right ) c^{2} x d +\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (2 c d \arcsin \left (c x \right )-e \sqrt {-c^{2} x^{2}+1}\right )}{2 \sqrt {-c^{2} x^{2}+1}}\right )}{c}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 70, normalized size = 0.49 \[ \frac {1}{2} \, a e x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arsech}\left (c x\right ) - \frac {x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c}\right )} b e + a d x + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b d}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.53, size = 99, normalized size = 0.70 \[ \frac {a\,x\,\left (2\,d+e\,x\right )}{2}+\frac {b\,d\,\mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}}\right )}{c}+\frac {b\,e\,x^2\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{2}+b\,d\,x\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )-\frac {b\,e\,x\,\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}}{2\,c} \]
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 \[ \int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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