Optimal. Leaf size=55 \[ \frac{a \sqrt{d x-1} \sqrt{d x+1}}{x}+b \tan ^{-1}\left (\sqrt{d x-1} \sqrt{d x+1}\right )+\frac{c \cosh ^{-1}(d x)}{d} \]
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Rubi [B] time = 0.179683, 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, 1807, 844, 217, 206, 266, 63, 205} \[ -\frac{a \left (1-d^2 x^2\right )}{x \sqrt{d x-1} \sqrt{d x+1}}+\frac{b \sqrt{d^2 x^2-1} \tan ^{-1}\left (\sqrt{d^2 x^2-1}\right )}{\sqrt{d x-1} \sqrt{d x+1}}+\frac{c \sqrt{d^2 x^2-1} \tanh ^{-1}\left (\frac{d x}{\sqrt{d^2 x^2-1}}\right )}{d \sqrt{d x-1} \sqrt{d x+1}} \]
Antiderivative was successfully verified.
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Rule 1610
Rule 1807
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^2 \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^2 \sqrt{-1+d^2 x^2}} \, dx}{\sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{a \left (1-d^2 x^2\right )}{x \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\sqrt{-1+d^2 x^2} \int \frac{b+c x}{x \sqrt{-1+d^2 x^2}} \, dx}{\sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{a \left (1-d^2 x^2\right )}{x \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (b \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 (c \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{a \left (1-d^2 x^2\right )}{x \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (b \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 (c \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{a \left (1-d^2 x^2\right )}{x \sqrt{-1+d x} \sqrt{1+d x}}+\frac{c \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 (b \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{a \left (1-d^2 x^2\right )}{x \sqrt{-1+d x} \sqrt{1+d x}}+\frac{b \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{c \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 [A] time = 0.16739, size = 89, normalized size = 1.62 \[ \frac{a \left (d^2 x^2-1\right )+b x \sqrt{d^2 x^2-1} \tan ^{-1}\left (\sqrt{d^2 x^2-1}\right )}{x \sqrt{d x-1} \sqrt{d x+1}}+\frac{2 c \tanh ^{-1}\left (\sqrt{\frac{d x-1}{d x+1}}\right )}{d} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0., size = 96, normalized size = 1.8 \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{dx} \left ( -\arctan \left ({\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}} \right ){\it csgn} \left ( d \right ) dxb+{\it csgn} \left ( d \right ) d\sqrt{{d}^{2}{x}^{2}-1}a+\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}+dx \right ){\it csgn} \left ( d \right ) \right ) xc \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 = 3.56079, size = 86, normalized size = 1.56 \begin{align*} -b \arcsin \left (\frac{1}{\sqrt{d^{2}}{\left | x \right |}}\right ) + \frac{c \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} a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.1004, size = 203, normalized size = 3.69 \begin{align*} \frac{a d^{2} x + 2 \, b d x \arctan \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right ) + \sqrt{d x + 1} \sqrt{d x - 1} a d - c x \log \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right )}{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 28.1729, size = 216, normalized size = 3.93 \begin{align*} - \frac{a d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i a d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{b{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 b{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{c{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 c{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.62059, size = 112, normalized size = 2.04 \begin{align*} -\frac{2 \, b d \arctan \left (\frac{1}{2} \,{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2}\right ) - \frac{8 \, a d^{2}}{{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{4} + 4} + c \log \left ({\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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