Optimal. Leaf size=52 \[ \frac {\left (2 a d^2+c\right ) \cosh ^{-1}(d x)}{2 d^3}+\frac {\sqrt {d x-1} \sqrt {d x+1} (2 b+c x)}{2 d^2} \]
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Rubi [B] time = 0.07, antiderivative size = 135, normalized size of antiderivative = 2.60, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {901, 1815, 641, 217, 206} \[ \frac {\sqrt {d^2 x^2-1} \left (2 a d^2+c\right ) \tanh ^{-1}\left (\frac {d x}{\sqrt {d^2 x^2-1}}\right )}{2 d^3 \sqrt {d x-1} \sqrt {d x+1}}-\frac {b \left (1-d^2 x^2\right )}{d^2 \sqrt {d x-1} \sqrt {d x+1}}-\frac {c x \left (1-d^2 x^2\right )}{2 d^2 \sqrt {d x-1} \sqrt {d x+1}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 641
Rule 901
Rule 1815
Rubi steps
\begin {align*} \int \frac {a+b x+c 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}{\sqrt {-1+d^2 x^2}} \, dx}{\sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {c x \left (1-d^2 x^2\right )}{2 d^2 \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\sqrt {-1+d^2 x^2} \int \frac {c+2 a d^2+2 b d^2 x}{\sqrt {-1+d^2 x^2}} \, dx}{2 d^2 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {b \left (1-d^2 x^2\right )}{d^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {c x \left (1-d^2 x^2\right )}{2 d^2 \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (\left (c+2 a d^2\right ) \sqrt {-1+d^2 x^2}\right ) \int \frac {1}{\sqrt {-1+d^2 x^2}} \, dx}{2 d^2 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {b \left (1-d^2 x^2\right )}{d^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {c x \left (1-d^2 x^2\right )}{2 d^2 \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (\left (c+2 a d^2\right ) \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 )}{2 d^2 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {b \left (1-d^2 x^2\right )}{d^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {c x \left (1-d^2 x^2\right )}{2 d^2 \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (c+2 a d^2\right ) \sqrt {-1+d^2 x^2} \tanh ^{-1}\left (\frac {d x}{\sqrt {-1+d^2 x^2}}\right )}{2 d^3 \sqrt {-1+d x} \sqrt {1+d x}}\\ \end {align*}
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Mathematica [B] time = 0.22, size = 126, normalized size = 2.42 \[ \frac {4 \sqrt {1-d x} \tanh ^{-1}\left (\sqrt {\frac {d x-1}{d x+1}}\right ) (d (a d-b)+c)+d \sqrt {-(d x-1)^2} \sqrt {d x+1} (2 b+c x)+2 \sqrt {d x-1} (2 b d-c) \sin ^{-1}\left (\frac {\sqrt {1-d x}}{\sqrt {2}}\right )}{2 d^3 \sqrt {1-d x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.76, size = 61, normalized size = 1.17 \[ \frac {{\left (c d x + 2 \, b d\right )} \sqrt {d x + 1} \sqrt {d x - 1} - {\left (2 \, a d^{2} + c\right )} \log \left (-d x + \sqrt {d x + 1} \sqrt {d x - 1}\right )}{2 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 80, normalized size = 1.54 \[ \frac {\sqrt {d x + 1} \sqrt {d x - 1} {\left (\frac {{\left (d x + 1\right )} c}{d^{2}} + \frac {2 \, b d^{5} - c d^{4}}{d^{6}}\right )} - \frac {2 \, {\left (2 \, a d^{2} + c\right )} \log \left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}{d^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 120, normalized size = 2.31 \[ \frac {\sqrt {d x -1}\, \sqrt {d x +1}\, \left (2 a \,d^{2} \ln \left (\left (d x +\sqrt {d^{2} x^{2}-1}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )+\sqrt {d^{2} x^{2}-1}\, c d x \,\mathrm {csgn}\relax (d )+2 \sqrt {d^{2} x^{2}-1}\, b d \,\mathrm {csgn}\relax (d )+c \ln \left (\left (d x +\sqrt {d^{2} x^{2}-1}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )\right ) \mathrm {csgn}\relax (d )}{2 \sqrt {d^{2} x^{2}-1}\, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 90, normalized size = 1.73 \[ \frac {a \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - 1} d\right )}{d} + \frac {\sqrt {d^{2} x^{2} - 1} c x}{2 \, d^{2}} + \frac {\sqrt {d^{2} x^{2} - 1} b}{d^{2}} + \frac {c \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - 1} d\right )}{2 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.40, size = 312, normalized size = 6.00 \[ \frac {b\,\sqrt {d\,x-1}\,\sqrt {d\,x+1}}{d^2}+\frac {2\,c\,\mathrm {atanh}\left (\frac {\sqrt {d\,x-1}-\mathrm {i}}{\sqrt {d\,x+1}-1}\right )}{d^3}-\frac {4\,a\,\mathrm {atan}\left (\frac {d\,\left (\sqrt {d\,x-1}-\mathrm {i}\right )}{\left (\sqrt {d\,x+1}-1\right )\,\sqrt {-d^2}}\right )}{\sqrt {-d^2}}-\frac {\frac {14\,c\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {d\,x+1}-1\right )}^3}+\frac {14\,c\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {d\,x+1}-1\right )}^5}+\frac {2\,c\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {d\,x+1}-1\right )}^7}+\frac {2\,c\,\left (\sqrt {d\,x-1}-\mathrm {i}\right )}{\sqrt {d\,x+1}-1}}{d^3-\frac {4\,d^3\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+\frac {6\,d^3\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {d\,x+1}-1\right )}^4}-\frac {4\,d^3\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {d\,x+1}-1\right )}^6}+\frac {d^3\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {d\,x+1}-1\right )}^8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 48.29, size = 277, normalized size = 5.33 \[ \frac {a {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} - \frac {i a {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} + \frac {b {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} + \frac {i b {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} + \frac {c {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4} & - \frac {1}{2}, - \frac {1}{2}, 0, 1 \\-1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{3}} - \frac {i c {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 1 & \\- \frac {5}{4}, - \frac {3}{4} & - \frac {3}{2}, -1, -1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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