Optimal. Leaf size=116 \[ -\frac{\left (a^2+a b x+1\right ) \sqrt{a^2+2 a b x+b^2 x^2+1}}{2 \left (a^2+1\right ) x^2}-\frac{b^2 \tanh ^{-1}\left (\frac{a^2+a b x+1}{\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}}\right )}{2 \left (a^2+1\right )^{3/2}}-\frac{a}{2 x^2}-\frac{b}{x} \]
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Rubi [A] time = 0.0858755, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5907, 14, 720, 724, 206} \[ -\frac{\left (a^2+a b x+1\right ) \sqrt{a^2+2 a b x+b^2 x^2+1}}{2 \left (a^2+1\right ) x^2}-\frac{b^2 \tanh ^{-1}\left (\frac{a^2+a b x+1}{\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}}\right )}{2 \left (a^2+1\right )^{3/2}}-\frac{a}{2 x^2}-\frac{b}{x} \]
Antiderivative was successfully verified.
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Rule 5907
Rule 14
Rule 720
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\sinh ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac{a+b x+\sqrt{1+(a+b x)^2}}{x^3} \, dx\\ &=\int \left (\frac{a}{x^3}+\frac{b}{x^2}+\frac{\sqrt{1+a^2+2 a b x+b^2 x^2}}{x^3}\right ) \, dx\\ &=-\frac{a}{2 x^2}-\frac{b}{x}+\int \frac{\sqrt{1+a^2+2 a b x+b^2 x^2}}{x^3} \, dx\\ &=-\frac{a}{2 x^2}-\frac{b}{x}-\frac{\left (1+a^2+a b x\right ) \sqrt{1+a^2+2 a b x+b^2 x^2}}{2 \left (1+a^2\right ) x^2}+\frac{b^2 \int \frac{1}{x \sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx}{2 \left (1+a^2\right )}\\ &=-\frac{a}{2 x^2}-\frac{b}{x}-\frac{\left (1+a^2+a b x\right ) \sqrt{1+a^2+2 a b x+b^2 x^2}}{2 \left (1+a^2\right ) x^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{4 \left (1+a^2\right )-x^2} \, dx,x,\frac{2 \left (1+a^2\right )+2 a b x}{\sqrt{1+a^2+2 a b x+b^2 x^2}}\right )}{1+a^2}\\ &=-\frac{a}{2 x^2}-\frac{b}{x}-\frac{\left (1+a^2+a b x\right ) \sqrt{1+a^2+2 a b x+b^2 x^2}}{2 \left (1+a^2\right ) x^2}-\frac{b^2 \tanh ^{-1}\left (\frac{1+a^2+a b x}{\sqrt{1+a^2} \sqrt{1+a^2+2 a b x+b^2 x^2}}\right )}{2 \left (1+a^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.19399, size = 129, normalized size = 1.11 \[ \frac{1}{2} \left (-\frac{\left (a^2+a b x+1\right ) \sqrt{a^2+2 a b x+b^2 x^2+1}}{\left (a^2+1\right ) x^2}-\frac{b^2 \log \left (\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}+a^2+a b x+1\right )}{\left (a^2+1\right )^{3/2}}+\frac{b^2 \log (x)}{\left (a^2+1\right )^{3/2}}-\frac{a}{x^2}-\frac{2 b}{x}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.009, size = 457, normalized size = 3.9 \begin{align*} -{\frac{1}{ \left ( 2\,{a}^{2}+2 \right ){x}^{2}} \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{ab}{2\, \left ({a}^{2}+1 \right ) ^{2}x} \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}{b}^{2}}{ \left ({a}^{2}+1 \right ) ^{2}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}-{\frac{{a}^{3}{b}^{3}}{2\, \left ({a}^{2}+1 \right ) ^{2}}\ln \left ({({b}^{2}x+ab){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+{\frac{{a}^{2}{b}^{2}}{2}\ln \left ({\frac{1}{x} \left ( 2\,{a}^{2}+2+2\,xab+2\,\sqrt{{a}^{2}+1}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ) } \right ) \left ({a}^{2}+1 \right ) ^{-{\frac{3}{2}}}}-{\frac{a{b}^{3}x}{2\, \left ({a}^{2}+1 \right ) ^{2}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}-{\frac{a{b}^{3}}{2\, \left ({a}^{2}+1 \right ) ^{2}}\ln \left ({({b}^{2}x+ab){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+{\frac{{b}^{2}}{2\,{a}^{2}+2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}+{\frac{a{b}^{3}}{2\,{a}^{2}+2}\ln \left ({({b}^{2}x+ab){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{{b}^{2}}{2}\ln \left ({\frac{1}{x} \left ( 2\,{a}^{2}+2+2\,xab+2\,\sqrt{{a}^{2}+1}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}+1}}}}-{\frac{b}{x}}-{\frac{a}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.63687, size = 423, normalized size = 3.65 \begin{align*} \frac{\sqrt{a^{2} + 1} b^{2} x^{2} \log \left (-\frac{a^{2} b x + a^{3} + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{\left (a^{2} - \sqrt{a^{2} + 1} a + 1\right )} -{\left (a b x + a^{2} + 1\right )} \sqrt{a^{2} + 1} + a}{x}\right ) - a^{5} -{\left (a^{3} + a\right )} b^{2} x^{2} - 2 \, a^{3} - 2 \,{\left (a^{4} + 2 \, a^{2} + 1\right )} b x -{\left (a^{4} +{\left (a^{3} + a\right )} b x + 2 \, a^{2} + 1\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - a}{2 \,{\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2}} \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 \frac{a + b x + \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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