Optimal. Leaf size=55 \[ a \tan ^{-1}\left (\sqrt{d x-1} \sqrt{d x+1}\right )+\frac{b \cosh ^{-1}(d x)}{d}+\frac{c \sqrt{d x-1} \sqrt{d x+1}}{d^2} \]
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Rubi [B] time = 0.184061, antiderivative size = 135, normalized size of antiderivative = 2.45, number of steps used = 8, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1610, 1809, 844, 217, 206, 266, 63, 205} \[ \frac{a \sqrt{d^2 x^2-1} \tan ^{-1}\left (\sqrt{d^2 x^2-1}\right )}{\sqrt{d x-1} \sqrt{d x+1}}+\frac{b \sqrt{d^2 x^2-1} \tanh ^{-1}\left (\frac{d x}{\sqrt{d^2 x^2-1}}\right )}{d \sqrt{d x-1} \sqrt{d x+1}}-\frac{c \left (1-d^2 x^2\right )}{d^2 \sqrt{d x-1} \sqrt{d x+1}} \]
Antiderivative was successfully verified.
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Rule 1610
Rule 1809
Rule 844
Rule 217
Rule 206
Rule 266
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{a+b x+c x^2}{x \sqrt{-1+d x} \sqrt{1+d x}} \, dx &=\frac{\sqrt{-1+d^2 x^2} \int \frac{a+b x+c x^2}{x \sqrt{-1+d^2 x^2}} \, dx}{\sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{c \left (1-d^2 x^2\right )}{d^2 \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\sqrt{-1+d^2 x^2} \int \frac{a d^2+b d^2 x}{x \sqrt{-1+d^2 x^2}} \, dx}{d^2 \sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{c \left (1-d^2 x^2\right )}{d^2 \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (a \sqrt{-1+d^2 x^2}\right ) \int \frac{1}{x \sqrt{-1+d^2 x^2}} \, dx}{\sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (b \sqrt{-1+d^2 x^2}\right ) \int \frac{1}{\sqrt{-1+d^2 x^2}} \, dx}{\sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{c \left (1-d^2 x^2\right )}{d^2 \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (a \sqrt{-1+d^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1+d^2 x}} \, dx,x,x^2\right )}{2 \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (b \sqrt{-1+d^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-d^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+d^2 x^2}}\right )}{\sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{c \left (1-d^2 x^2\right )}{d^2 \sqrt{-1+d x} \sqrt{1+d x}}+\frac{b \sqrt{-1+d^2 x^2} \tanh ^{-1}\left (\frac{d x}{\sqrt{-1+d^2 x^2}}\right )}{d \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (a \sqrt{-1+d^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{d^2}+\frac{x^2}{d^2}} \, dx,x,\sqrt{-1+d^2 x^2}\right )}{d^2 \sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{c \left (1-d^2 x^2\right )}{d^2 \sqrt{-1+d x} \sqrt{1+d x}}+\frac{a \sqrt{-1+d^2 x^2} \tan ^{-1}\left (\sqrt{-1+d^2 x^2}\right )}{\sqrt{-1+d x} \sqrt{1+d x}}+\frac{b \sqrt{-1+d^2 x^2} \tanh ^{-1}\left (\frac{d x}{\sqrt{-1+d^2 x^2}}\right )}{d \sqrt{-1+d x} \sqrt{1+d x}}\\ \end{align*}
Mathematica [B] time = 0.399322, size = 128, normalized size = 2.33 \[ \frac{\frac{a d^2 \sqrt{d^2 x^2-1} \tan ^{-1}\left (\sqrt{d^2 x^2-1}\right )+c d^2 x^2-2 c \sqrt{1-d^2 x^2} \sin ^{-1}\left (\frac{\sqrt{1-d x}}{\sqrt{2}}\right )-c}{\sqrt{d x-1} \sqrt{d x+1}}-2 (c-b d) \tanh ^{-1}\left (\sqrt{\frac{d x-1}{d x+1}}\right )}{d^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.019, size = 95, normalized size = 1.7 \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{{d}^{2}} \left ( -{\it csgn} \left ( d \right ) \arctan \left ({\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}} \right ) a{d}^{2}+{\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}c+\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{ \left ( dx+1 \right ) \left ( dx-1 \right ) }+dx \right ){\it csgn} \left ( d \right ) \right ) bd \right ) \sqrt{dx-1}\sqrt{dx+1}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.91204, size = 86, normalized size = 1.56 \begin{align*} -a \arcsin \left (\frac{1}{\sqrt{d^{2}}{\left | x \right |}}\right ) + \frac{b \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - 1} \sqrt{d^{2}}\right )}{\sqrt{d^{2}}} + \frac{\sqrt{d^{2} x^{2} - 1} c}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08568, size = 184, normalized size = 3.35 \begin{align*} \frac{2 \, a d^{2} \arctan \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right ) - b d \log \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right ) + \sqrt{d x + 1} \sqrt{d x - 1} c}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 26.2475, size = 240, normalized size = 4.36 \begin{align*} - \frac{a{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} + \frac{c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{i c{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.98647, size = 96, normalized size = 1.75 \begin{align*} -2 \, a \arctan \left (\frac{1}{2} \,{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2}\right ) - \frac{b \log \left ({\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2}\right )}{d} + \frac{\sqrt{d x + 1} \sqrt{d x - 1} c}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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