Optimal. Leaf size=153 \[ \frac {a^3 \text {sech}^{-1}(a+b x)}{3 b^3}-\frac {\left (6 a^2+1\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{6 b^3}+\frac {5 a \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{6 b^3}-\frac {x \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{6 b^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x) \]
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Rubi [A] time = 0.10, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6321, 5468, 3782, 3770, 3767, 8} \[ -\frac {\left (6 a^2+1\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{6 b^3}+\frac {a^3 \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {5 a \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{6 b^3}-\frac {x \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{6 b^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3782
Rule 5468
Rule 6321
Rubi steps
\begin {align*} \int x^2 \text {sech}^{-1}(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x \text {sech}(x) (-a+\text {sech}(x))^2 \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^3}\\ &=\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int (-a+\text {sech}(x))^3 \, dx,x,\text {sech}^{-1}(a+b x)\right )}{3 b^3}\\ &=-\frac {x \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int \left (-2 a^3+\left (1+6 a^2\right ) \text {sech}(x)-5 a \text {sech}^2(x)\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{6 b^3}\\ &=-\frac {x \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^2}+\frac {a^3 \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)+\frac {(5 a) \operatorname {Subst}\left (\int \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{6 b^3}-\frac {\left (1+6 a^2\right ) \operatorname {Subst}\left (\int \text {sech}(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{6 b^3}\\ &=-\frac {x \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^2}+\frac {a^3 \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)-\frac {\left (1+6 a^2\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{6 b^3}+\frac {(5 i a) \operatorname {Subst}\left (\int 1 \, dx,x,-i \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)\right )}{6 b^3}\\ &=\frac {5 a \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^3}-\frac {x \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{6 b^2}+\frac {a^3 \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)-\frac {\left (1+6 a^2\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{6 b^3}\\ \end {align*}
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Mathematica [C] time = 0.28, size = 200, normalized size = 1.31 \[ \frac {-2 a^3 \log (a+b x)+2 a^3 \log \left (a \sqrt {-\frac {a+b x-1}{a+b x+1}}+b x \sqrt {-\frac {a+b x-1}{a+b x+1}}+\sqrt {-\frac {a+b x-1}{a+b x+1}}+1\right )+\sqrt {-\frac {a+b x-1}{a+b x+1}} \left (5 a^2+a (4 b x+5)-b x (b x+1)\right )+i \left (6 a^2+1\right ) \log \left (2 \sqrt {-\frac {a+b x-1}{a+b x+1}} (a+b x+1)-2 i (a+b x)\right )+2 b^3 x^3 \text {sech}^{-1}(a+b x)}{6 b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.92, size = 327, normalized size = 2.14 \[ \frac {2 \, b^{3} x^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + a^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) - a^{3} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) - {\left (6 \, a^{2} + 1\right )} \arctan \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (b^{2} x^{2} - 4 \, a b x - 5 \, a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{6 \, b^{3}} \]
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 \[ \int x^{2} \operatorname {arsech}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 189, normalized size = 1.24 \[ \frac {\frac {\left (b x +a \right )^{3} \mathrm {arcsech}\left (b x +a \right )}{3}-\mathrm {arcsech}\left (b x +a \right ) \left (b x +a \right )^{2} a +\mathrm {arcsech}\left (b x +a \right ) \left (b x +a \right ) a^{2}-\frac {\mathrm {arcsech}\left (b x +a \right ) a^{3}}{3}+\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (2 a^{3} \arctanh \left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right )+6 a^{2} \arcsin \left (b x +a \right )-\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+6 a \sqrt {1-\left (b x +a \right )^{2}}+\arcsin \left (b x +a \right )\right )}{6 \sqrt {1-\left (b x +a \right )^{2}}}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, b^{3} x^{3} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right ) - 2 \, b^{3} x^{3} \log \left (b x + a\right ) - 2 \, b x + {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right ) - 2 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (b x + a\right ) + {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (-b x - a + 1\right )}{6 \, b^{3}} + \int \frac {b^{2} x^{4} + a b x^{3}}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} e^{\left (\frac {1}{2} \, \log \left (b x + a + 1\right ) + \frac {1}{2} \, \log \left (-b x - a + 1\right )\right )} - 1\right )}}\,{d x} \]
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 \[ \int x^2\,\mathrm {acosh}\left (\frac {1}{a+b\,x}\right ) \,d x \]
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 \[ \int x^{2} \operatorname {asech}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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