Optimal. Leaf size=203 \[ -\frac{\left (17 a^2+2\right ) \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{12 b^4}+\frac{\left (2 a^2+1\right ) a \tan ^{-1}\left (\frac{\sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{2 b^4}-\frac{a^4 \text{sech}^{-1}(a+b x)}{4 b^4}-\frac{x^2 \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{12 b^2}+\frac{a (a+b x) \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{3 b^4}+\frac{1}{4} x^4 \text{sech}^{-1}(a+b x) \]
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Rubi [A] time = 0.155323, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {6321, 5468, 3782, 4048, 3770, 3767, 8} \[ -\frac{\left (17 a^2+2\right ) \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{12 b^4}+\frac{\left (2 a^2+1\right ) a \tan ^{-1}\left (\frac{\sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{2 b^4}-\frac{a^4 \text{sech}^{-1}(a+b x)}{4 b^4}-\frac{x^2 \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{12 b^2}+\frac{a (a+b x) \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{3 b^4}+\frac{1}{4} x^4 \text{sech}^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 6321
Rule 5468
Rule 3782
Rule 4048
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int x^3 \text{sech}^{-1}(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x \text{sech}(x) (-a+\text{sech}(x))^3 \tanh (x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{b^4}\\ &=\frac{1}{4} x^4 \text{sech}^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int (-a+\text{sech}(x))^4 \, dx,x,\text{sech}^{-1}(a+b x)\right )}{4 b^4}\\ &=-\frac{x^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac{1}{4} x^4 \text{sech}^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int (-a+\text{sech}(x)) \left (-3 a^3+\left (2+9 a^2\right ) \text{sech}(x)-8 a \text{sech}^2(x)\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{12 b^4}\\ &=-\frac{x^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac{a (a+b x) \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{3 b^4}+\frac{1}{4} x^4 \text{sech}^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int \left (6 a^4-12 a \left (1+2 a^2\right ) \text{sech}(x)+2 \left (2+17 a^2\right ) \text{sech}^2(x)\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{24 b^4}\\ &=-\frac{x^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac{a (a+b x) \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{3 b^4}-\frac{a^4 \text{sech}^{-1}(a+b x)}{4 b^4}+\frac{1}{4} x^4 \text{sech}^{-1}(a+b x)+\frac{\left (a \left (1+2 a^2\right )\right ) \operatorname{Subst}\left (\int \text{sech}(x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{2 b^4}-\frac{\left (2+17 a^2\right ) \operatorname{Subst}\left (\int \text{sech}^2(x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{12 b^4}\\ &=-\frac{x^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac{a (a+b x) \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{3 b^4}-\frac{a^4 \text{sech}^{-1}(a+b x)}{4 b^4}+\frac{1}{4} x^4 \text{sech}^{-1}(a+b x)+\frac{a \left (1+2 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 )}{2 b^4}-\frac{\left (i \left (2+17 a^2\right )\right ) \operatorname{Subst}\left (\int 1 \, dx,x,-i \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)\right )}{12 b^4}\\ &=-\frac{\left (2+17 a^2\right ) \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^4}-\frac{x^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac{a (a+b x) \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{3 b^4}-\frac{a^4 \text{sech}^{-1}(a+b x)}{4 b^4}+\frac{1}{4} x^4 \text{sech}^{-1}(a+b x)+\frac{a \left (1+2 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 )}{2 b^4}\\ \end{align*}
Mathematica [C] time = 0.493848, size = 225, normalized size = 1.11 \[ -\frac{\sqrt{-\frac{a+b x-1}{a+b x+1}} \left (\left (9 a^2-4 a+2\right ) b x+13 a^3+13 a^2+(1-3 a) b^2 x^2+2 a+b^3 x^3+2\right )-3 a^4 \log (a+b x)+3 a^4 \log \left (a \sqrt{-\frac{a+b x-1}{a+b x+1}}+b x \sqrt{-\frac{a+b x-1}{a+b x+1}}+\sqrt{-\frac{a+b x-1}{a+b x+1}}+1\right )+6 i \left (2 a^2+1\right ) a \log \left (2 \sqrt{-\frac{a+b x-1}{a+b x+1}} (a+b x+1)-2 i (a+b x)\right )-3 b^4 x^4 \text{sech}^{-1}(a+b x)}{12 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.292, size = 250, normalized size = 1.2 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{ \left ( bx+a \right ) ^{4}{\rm arcsech} \left (bx+a\right )}{4}}-{\rm arcsech} \left (bx+a\right ) \left ( bx+a \right ) ^{3}a+{\frac{3\,{\rm arcsech} \left (bx+a\right ) \left ( bx+a \right ) ^{2}{a}^{2}}{2}}-{\rm arcsech} \left (bx+a\right ) \left ( bx+a \right ){a}^{3}+{\frac{{\rm arcsech} \left (bx+a\right ){a}^{4}}{4}}-{\frac{bx+a}{12}\sqrt{-{\frac{bx+a-1}{bx+a}}}\sqrt{{\frac{bx+a+1}{bx+a}}} \left ( 3\,{a}^{4}{\it Artanh} \left ({\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}} \right ) +12\,{a}^{3}\arcsin \left ( bx+a \right ) +\sqrt{1- \left ( bx+a \right ) ^{2}} \left ( bx+a \right ) ^{2}-6\,a \left ( bx+a \right ) \sqrt{1- \left ( bx+a \right ) ^{2}}+18\,{a}^{2}\sqrt{1- \left ( bx+a \right ) ^{2}}+6\,a\arcsin \left ( bx+a \right ) +2\,\sqrt{1- \left ( bx+a \right ) ^{2}} \right ){\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, b^{4} x^{4} \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^{4} x^{4} \log \left (b x + a\right ) - b^{2} x^{2} + 6 \, a b x -{\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right ) - 2 \,{\left (b^{4} x^{4} - a^{4}\right )} \log \left (b x + a\right ) -{\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (-b x - a + 1\right )}{8 \, b^{4}} + \int \frac{b^{2} x^{5} + a b x^{4}}{4 \,{\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52586, size = 772, normalized size = 3.8 \begin{align*} \frac{6 \, b^{4} x^{4} \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 ) - 3 \, a^{4} \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 ) + 3 \, a^{4} \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 ) + 12 \,{\left (2 \, a^{3} + a\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 ) - 2 \,{\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} + 13 \, a^{3} +{\left (9 \, a^{2} + 2\right )} b x + 2 \, 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}}}}{24 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{asech}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arsech}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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